TWO-COMMODITY PERISHABLE INVENTORY SYSTEM WITH BULK DEMAND FOR ONE COMMODITY

Transcription

1 h://aeournalacza TWO-COMMOTY PRHL NVNTORY YTM WTH ULK MN FOR ON COMMOTY V Yadavall O deun vauar and G rvargnan 4 earen of ndural & ye ngneerng Unvery of Preora ouh frca arayadavall@uacza olufeadeun@uacza 4 earen of led Maheac and ac Madura Kaara Unvery nda vabuar@yahooco arvargnan@yahooco TRCT Th arcle conder a wo-coody connuou revew nvenory ye n whch he arrvng cuoer belong o any one of hree ye uch ha ye cuoer deand a ngle e of he fr coody ye cuoer deand bul e of he econd coody and ye cuoer deand one e of he fr coody and bul e of he econd coody The arrval of all hree ye of cuoer are aued o be a Marovan arrval roce MP alo aued ha he nuber of e deanded for he econd coody a rando varable The orderng olcy o lace order for boh coode when he nvenory level are below refxed level for boh coode The lead e aued o have a hae ye drbuon and he deand ha occur durng oc ou erod are aued o be lo The on robably drbuon for boh coode obaned n he eady ae cae Varou eaure of ye erforance and he oal execed co rae n he eady ae are derved The reul are lluraed wh nuercal exale OPOMMNG n Tweeroduelel e voorraad vr lane word onnu heren e vraag na de rodue word geener deur lanvooreure e veroeeo van lane vr voorraad word aanvaar a n Marovroe annae word geaa oor vraaghoeveelhede en aanlooyd e reulae van de onderoe word voorgehou va yfervoorbeelde ouh frcan Journal of ndural ngneerng May Vol : 7-55

2 h://aeournalacza NTROUCTON One of he facor ha conrbue o he colexy of reen day nvenory ye he ulude of e oced neceang ul-coody nvenory ye n early dealng wh uch ye any odel were rooed wh ndeendenly eablhed re-order on u n uaon where everal roduc coee for led orage ace or hare he ae ranor facly or are roduced on rocured fro he ae equen uler he above raegy overloo he oenal avng aocaed wh on orderng and hence wll no be oal Thu coordnaed alo nown a on relenhen reduce he orderng and eu co and allow he uer o ae advanage of quany dcoun f any Varou odel and reference ay be found n Mller [] garwal [5] lver [] Thoone and lver [6] Kalaa and rvargnan[5] and rnvaan and Ravchandran [5] and furher reference ha hey conan n connuou revew nvenory ye alnfy [6] and lver [] have condered a coordnaed re-orderng olcy rereened by he rle c where hree araeer c and are ecfed for each e wh c for a un zed Poon deand and conan lead e n h olcy f he level of he -h coody a any e below an order laced for e; and a he ae e f any oher e wh avalable nvenory a or below can-order level c an order laced o a o brng level bac o axu caacy Many ubequen arcle have aeared wh odel nvolvng he above olcy furher arcle of nere ha of Federgruen e al [9] whch deal wh he general cae of coound Poon deand and non-zero lead e Throughou he year he wor on ehod o olve he on relenhen roble ha been exenve Reader are referred o he ublcaon of Fung and Ma [] Goyal [] Goyal and ar [4] Ka and Roenbla [6] Nlon e al [7] Nlon and lver [8] Olen [9] lver [] Van [7] Vwanahan [894] and Wldean e al [4] and reference ha hey conan Kalaa and rvargnan [5] have nroduced an olcy wh a ngle re-order level defned n er of he oal nuber of e n he oc The olcy avod earae orderng of each coody Hence a ngle roceng of order for boh coode ha oe advanage n a uaon where rocureen ade fro he ae ule e are roduced on he ae achne or e have o be uled by he ae ranor facly n he cae of wo-coody nvenory ye nbazhagan and rvargnan [4] have rooed varou orderng olce Yadavall e al [4] have analyed a odel wh a on orderng olcy and varable order quane vauar e al [] have condered a wo-coody ubuable nvenory ye n whch he e deanded are delvered afer a rando e vauar e al [4] have condered a wo-coody erhable nvenory ye wh a on orderng olcy There are oe uaon n whch a ngle e deanded for one coody and ulle e are deanded for anoher coody For nance a cuoer ay buy a ngle razor or a e of blade or boh noher exale he ale of a V wrer and a e of V ay be noed ha he eller would be lacng a on order for boh coode a hee are avalable fro he ae ource Moreover a eller ay no be wllng o lace order frequenly and ay refer o have one order o relenh oc n a gven cycle Thee uaon are odelled n h wor by aung deand rocee ha requre a ngle e for one coody ulle e for he oher coode or boh coode and by aung a on re-order for boh coode Th aer organed a follow n econ he aheacal odel and noaon 8

3 h://aeournalacza followed n he re of he aer are decrbed The eady ae oluon of he on robably drbuon for boh coode he hae of he deand roce and he hae of he lead e roce are gven n econ n econ 4 varou eaure are derved of ye erforance n he eady ae; and he oal execed co rae calculaed n econ 5 econ 6 reen he co analy of he odel ung nuercal exale Noaon : zero arx : an deny arx x f x > H x f x { } { } e a colun vecor of one MOL CRPTON Fgure : ace of nvenory level Conder a wo-coody erhable nvenory ye wh he axu caacy un for he -h coody ue ha he deand for he fr coody for a ngle e and he deand for he econd coody for bul e n arrvng cuoer ay deand only he fr coody or only he econd coody or boh The nuber of e deanded for he econd coody a any deand on a rando varable Y wh a robably funcon Pr{ Y } The hree ye of deand for hee wo coode occur accordng o a Marovan arrval roce MP The lfe e of each coody exonenal wh araeer The re-order level for he -h coody fxed a and he orderng quany for he -h coody > e when boh he nvenory level are le han or equal o her reecve re-order level ue ha deand durng oc-ou erod are lo a well a unafed deand The requreen > enure ha afer a relenhen he nvenory level of boh coode wll alway be above he reecve re-order level Oherwe ay no be oble o lace a re-order accordng o he olcy leadng o a ereual horage Tha f L rereen nvenory level of -h coody a e hen a re-order ade when L and L ee Fgure The e o delver 9

4 h://aeournalacza he e aued o be of hae ye PH wh rereenaon T of order Noe ha he hae ye drbuon defned a he e unl aboron n a fne ae rreducble Marov chan wh one aborbng ae The ean of he hae ye drbuon T gven by T e Le denoe he recrocal of h ean Tha T e gve he rae of relenhen once an order laced Le T be uch ha Te T For he deand roce he decron of MP a gven n Lucanon [] wa ued Conder a connuou-e Marov chan on he ae ace The deand roce conrucvely defned a follow When he chan ener a ae rean for an exonenal e wh araeer he end of he oourn e n ae here are four oble ranon: wh robable a he chan ener he ae when a deand for he fr coody occur; wh robable b he chan ener he ae when a deand for he econd coody occur; wh robable c he chan ener he ae when a deand for boh coode occur; wh robable he ranon correond o no deand and he d ae of he chan The Marov chan can go fro ae o ae only hrough a deand efnng he quare arce of ze by [ ] and [ ] d [ ] a[ ] b and [ ] c een ha an nfneal generaor of a connuou-e Marov e chan ue ha rreducble and Le be he aonary robably vecor of he connuou-e Marov chan wh generaor Tha he unque robably vecor afyng e Le be he nal robably vecor of he underlyng Marov chan governng he MP y choong aroraely he e orgn can be odelled o be: an arbrary arrval on he end of an nerval durng whch here are a lea arrval; or he on a whch he ye n a ecfc ae uch a when he buy erod end or begn The oran cae ha of he aonary veron of he MP for The conan e referred o a he fundaenal rae gve he execed nuber of deand er un e n he aonary veron of he MP The quane e e and e gve he arrval rae of deand for fr coody for econd coody and for boh reecvely Noe ha For furher deal on MP and hae-ye drbuon and her uefulne n ochac odellng he reader ay refer o Chaer n Neu [4] Chaer 5 n Neu [5] Raawa [] Lucanon [] Lucanon e al [] Laouche and Raawa [7] L and L [9] Lee and Jeon [8] Charavarhy and udn [8] and reference heren for a dealed nroducon of he MP and hae-ye drbuon oe recen revew can be found n Neu [6] and Charavarhy [7] Le J and J reecvely denoe he hae of he deand roce and he hae of 4

5 4 he lead e roce Then he ochac roce } { J J L L ha he ae ace \ \ \ \ 4 4 Fro he auon ade on he deand and relenhen rocee can be hown ha { } J J L L a Marov roce on he ae ace y lacng he e of ae ace n lexcograhc order he nfneal generaor of he Marov chan governng he ye n bloc aroned for gven by ] [ oherwe C P where ] [ oherwe T C For ] [ oherwe ' n ' n For ] [ oherwe ' or h://aeournalacza

6 4 For ] [ oherwe ' ' For 4 ] [ oherwe ' For ] [ oherwe T T or ' ' h://aeournalacza

7 4 For ] [ oherwe T T or ' ' ay be noed ha he arx C of order he arce are of order he arx of order he arce are of order he arce are of order and he arce are of order TY TT NLY can be een fro he rucure of P ha he hoogeneou Marov roce } { J J L L on he fne ae ace rreducble Hence he lng drbuon l J J L L J J L L Pr ex Le F F where F \ \ F \ \ h://aeournalacza

8 44 and Then he vecor of lng robable afe e and P The fr equaon of he above yeld he followng e of equaon: C C C 4 The equaon exce can be recurvely olved o oban 5 where C ubung he value of n equaon and n he noralng condon follow ha C C 6 and C e 7 Fro he equaon 6 he value of ay be obaned u o a conan ullcaon Th conan can be deerned by ubung he value of n he equaon 7 ubung he value of n he equaon 5 we ge he value of h://aeournalacza

9 h://aeournalacza 4 YTM PRFORMNC MUR n h econ ceran aonary erforance eaure of he ye are derved Ung hee eaure he oal execed co er un e ay be deerned 4 Mean nvenory level Le denoe he ean nvenory level of h coody n he eady ae nce he eady ae robably vecor for he nvenory level of fr coody and he econd coody follow ha and e e 4 Mean re-order rae re-order for boh coode ade when he on nvenory level dro o or < or < Le R denoe he ean re-order rae for boh coode n he eady ae gven by u e e R u u e 4 Mean horage rae u e e Le h denoe he ean horage rae of h ye deand n he eady ae Conequenly h e h e and h e e 44 Mean falure rae Le he ean falure rae of coody- n he eady ae be denoed by F falure occur when any one of he oced e ceae o funcon or erhe nce he rae of falure of a ngle e for coody he rae a whch any one of e for he h coody fal gven by When he roce n ae 45

10 h://aeournalacza he rae of falure of any one e of he fr coody gven by rovded > and he falure rae of any one e of he econd coody rovded > Then we have F and e F e 5 COT NLY The oal execed co er un e oal execed co rae n he eady-ae for h odel defned o be TC ch c h c R ch h c h h c h h c f F c f F where c h : The nvenory carryng co of -h coody er un e er un e c : Jon orderng co er order c : The falure co of -h coody er un e er un e f c : horage co due o ye deand er un e h nce he oal execed co rae only lcly nown he analycal roere uch a convexy of he oal execed co rae canno be carred ou n he reen for However he followng nuercal exale o deonrae he couably of he reul derved n our wor and o llurae he exence of local oa when he oal co funcon reaed a a funcon of only wo varable are reened 6 LLUTRTV NUMRCL XMPL nce he oal execed co rae obaned n a colex for he convexy of he oal execed co rae canno be uded by analycal ehod Hence le nuercal earch rocedure are ued o fnd he local oal value for any wo of he decon varable { } by conderng a all e of neger value for hee varable Wh a large nuber of nuercal exale ha been found ha he oal co rae er un e n he long run eher a convex funcon of boh varable or an ncreang funcon of any one varable The followng fve MP for arrval of deand are condered and ay be noed ha hee rocee can be noralzed o have a ecfc gven deand rae when condered for arrval of deand xonenal x H H rlang rl H H 46

11 h://aeournalacza Hyer-exonenal Hx 9 H H 9 4 MP wh negave correlaon MNC H 8 H MP wh ove correlaon MPC H 8 H ll he above MP are qualavely dfferen n ha hey have dfferen varance and correlaon rucure The fr hree rocee are ecal cae of renewal rocee and he correlaon beween arrval e The deand roce labelled a MNC ha correlaed arrval wh a correlaon coeffcen of -54 and he deand correondng o he roce labelled MPC have a ove correlaon coeffcen of nce rlang ha he lea varance aong he fve arrval rocee condered here he rao of he varance of he oher four arrval rocee labelled x Hx MNC and MPC above wh reec o he rlang roce are reecvely The rao are gven raher han he acual value nce he varance deend on he arrval rae whch vared n he dcuon PH drbuon are condered For he lead e drbuon he followng hree gan hee rocee can be noraled o have a ecfc gven rae when condered for relenhen xonenal x T rlang rl T Hyer-exonenal Hx 9 T xale : n h exale he effec of he deand rae he lead e rae he fve ye of deand rocee and he hree ye of lead e rocee on he oal * * * * value and he oal co rae TC 4 lluraed The followng fxed value for he araeer and co are aued: H H 4H H * 4 c h 5 c c c 8 c 5 c c c h h h h f f 47

12 h://aeournalacza Table gve he ou value * * and ha ne he oal execed co rae for each of he fve MP for arrval of deand condered agan each of he hree PH for lead e The aocaed oal execed co rae value are alo gven n he able The lower enry n each cell gve he oal execed co rae and he uer enre correond o * * and The followng obervaon fro Table ay be noced: ncreae he oal oal co rae decreae for all fve deand rocee and for all hree lead e rocee larly a ncreae he oal oal co rae decreae The oal oal execed co rae ha a hgher value for a deand roce havng an hyer-exonenal drbuon and a lower value for he rlang deand roce The rlang drbued lead e ha a low oal oal co rae exce for an Hx drbued deand roce; and Hx drbued lead e ha hgh oal oal co rae exce for Hx drbued deand roce For Hx drbued deand roce h obervaon revere e Hx drbued lead e ha a low oal oal co rae and rl drbued lead e ha a hgh oal oal co rae xale : n h exale he effec of he arrval rae he lead e rae and he * * ye of arrval and lead e rocee on he oal value and oal co rae * * TC 5 lluraed ue he followng fxed value for he araeer and co: H ch ch H c 4H c h H 6 8 c h 5 c h 5 c f 55 * c f 45 * The ou value and * ha ne he execed oal co for each of he fve MP for arrval of deand condered agan each of he hree PH for lead e gven n Table The aocaed oal execed co rae value are alo gven The lower enry n each cell gve he oal execed co rae and he uer enre correond o * * and The ey obervaon are uared below MP deand drbuon 48 Lead e drbuon 5 x rl Hx x rl Hx x rl Hx MNC MPC x rl Hx MNC MPC Table : Toal execed co rae a a funcon of

13 h://aeournalacza ncreae he oal oal co rae ncreae exce for an Hx drbued deand roce For an Hx drbued deand roce he oal oal co rae decreae a he deand rae ncreae When ncreae he oal oal co rae ncreae for all cobnaon of fve arrval rocee and hree deand rocee The oal co rae hgh n cae where he deand roce an Hx and low when he deand roce rlang 4 The oal oal co rae low when he lead e rl exce for he Hx drbued deand roce For Hx drbued lead e he oal oal co rae hgh exce for Hx drbued deand roce For Hx drbued deand roce h obervaon revere e he Hx drbued lead e aocaed wh a low oal oal co rae and rl aocaed wh a hgh oal oal co rae MP eand rbuon Lead e drbuon 5 x rl Hx x rl Hx x rl Hx MNC MPC x rl Hx MNC MPC Table : Toal execed co rae a a funcon of xale : Nex conder he ac of c f and c f on he oal execed co rae For h conder he followng value for he araeer and co: H H 4H H ch c 8 5 h c ch c h c h The grah of he oal execed co rae a a funcon of c and f 55 * 45 c f for he hree lead e rocee and he fve deand rocee are hown n Fgure o 6 n all he fgure he lead e drbuon x rl and Hx are coloured blue blac and red reecvely Noe he followng: n all fve arrval rocee a c f and co rae ncreae u he ncreang rae for c f ncreae ulaneouly he oal execed c hgh coared wh c The rlang lead e roce aocaed wh a low oal execed co rae and for he hyer exonenal lead e roce cae he oal execed co rae hgh f f 49

14 h://aeournalacza 5 5 TC c f c f 5 5 Fgure : x deand roce TC c f c f 5 5 Fgure : rl deand roce TC c f c f 5 Fgure 4: Hx deand roce 5

15 h://aeournalacza 5 5 TC c f c f 5 5 Fgure 5: MNC deand roce TC c f c f 5 Fgure 6: MPC deand roce xale 4: n he fnal exale he ac of c h and c h on he oal execed co rae hown Conder he followng value for he araeer and co: H H 4H H c c 8 c 5 c c c h h h The grah of he oal execed co rae a a funcon of f f c f and 6 * 4 c f for he hree lead e rocee and he fve deand rocee are hown n Fgure 7 o n all he fgure he lo for he lead e drbuon x rl and Hx are coloured blue blac and red reecvely The followng ay be oberved: n all fve arrval rocee a c h and c h ncreae he oal execed co rae ncreae u he ncreang rae for c h hgh coared wh ha of c h For all he deand rocee he rlang lead e roce ha a low oal execed co rae and a hyer exonenal lead e roce ha a hgh oal execed co rae The dfference beween he oal execed co rae for any wo lead e rocee hgh exce for he Hx deand roce For he Hx deand roce he dfference beween he oal execed co rae for any wo lead e rocee low 5

16 h://aeournalacza TC c h c h 5 Fgure 7: x deand roce TC c h c h 5 Fgure 8: rl deand roce TC ch c h 5 Fgure 9: Hx deand roce 5

17 h://aeournalacza TC c h c h 5 Fgure : MNC deand roce TC c h c h CONCLUON Fgure : MPC deand roce Th reearch ha exended he exng wor on wo-coody connuou revew nvenory ye by nroducng erhably for boh coode Marov arrval rocee for deand e on and a hae ye drbuon for lead e alo aued ha one of he coode ay acce bul deand eady ae oluon are alo rovded for he on drbuon of nvenory level For a uable co rucure he oal execed co rae n he eady ae ha been deerned To deonrae he couably of reul derved here ale nuercal lluraon are gven Nuercal analy of he effec of he araeer and co on he oal execed co rae gven 8 CKNOWLGMNT V Yadavall would le o han he Naonal Reearch Foundaon NRF for her fnancal uor G rvargnan would le o han he CR - nda for her fnancal uor No 5 56/7/MR- 9 RFRNC [] nbazhagan N & rvargnan G Two-coody connuou revew nvenory ye wh coordnaed reorder olcy nernaonal Journal of nforaon and Manageen cence 9-5

18 h://aeournalacza [] nbazhagan N & rvargnan G wo-coody coordnaed nvenory ye wh renewal deand lecronnoe odelrovane 5 6 n Ruan [] nbazhagan N & rvargnan G naly of wo-coody Marovan nvenory ye wh lead e The Korean Journal of Couaonal and led Maheac [4] nbazhagan N & rvargnan G Two-coody nvenory ye wh ndvdual and on orderng olce nernaonal Journal of Manageen and ye [5] garwal V 984 Coordnaed order cycle under on relenhen ul-e nvenore Naval Logc Reearch uarerly -6 [6] alnfy JL 964 On a bac cla of nvenory roble Manageen cence [7] Charavarhy The bach Marovan arrval roce: revew and fuure wor dvance n Probably and ochac Procee Krhnaoorhy e al ed Noable Publcaon nc New Jerey U -49 [8] Charavarhy & udn naly of a reral queueng odel wh MP arrval and wo ye of cuoer Maheacal and Couer Modellng [9] Federgruen Groenvel H & T HC 984 Coordnaed relenhen n a ul-e nvenory ye wh coound Poon deand Manageen cence [] Fung RYK & Ma X new ehod for on relenhen roble Journal of he Oeraonal Reearch ocey [] Goyal K 97 eernaon of econoc acagng frequency of e only relenhed Manageen cence -5 [] Goyal K 974 eernaon of oal acagng frequency of only relenhed e Manageen cence [] Goyal K 988 conoc orderng olcy for only relenhed e nernaonal Journal of Producon Reearch [4] Goyal K & ar T 989 Jon relenhen nvenory conrol: eernc and ochac odel uroean Journal of Oeraon Reearch 8 - [5] Kalaa & rvargnan G 99 coordnaed ul-coody nvenory ye Maheacal and Couer Modellng [6] Ka M & Roenbla MJ 99 On he econoc orderng quany for only relenhed e nernaonal Journal of Producon Reearch [7] Laouche G & Raawa V 999 nroducon o arx analyc ehod n ochac odellng M Phladelha [8] Lee G & Jeon J new aroach o an N /G/ queue ueueng ye 5 7- [9] L L & L JJ 994 n alcaon of Marov-odulaed Poon roce o wo-un ere rearable ye Chnee Journal of ngneerng Maheac [] Lucanon M 99 New reul on he ngle erver queue wh a bach Marovan arrval roce ochac Model 7-46 [] Lucanon M 99 The MP /G/ queue : uoral n Model and echnque for erforance evaluaon of couer and councaon ye L onaello and R Nelon ed rnger-verlag New Yor -58 [] Lucanon M Meer-Hellern K & Neu MF 99 ngle erver queue wh erver vacaon and a cla of non-renewal arrval rocee dvance n led Probably [] Mller L 97 ul-e nvenory odel wh on robably bac-order creron Oeraon Reearch [4] Neu MF 994 Marx-geoerc oluon n ochac odel: n algorhc aroach over Publcaon nc New Yor [5] Neu MF 989 rucured ochac arce of M /G/ ye and her alcaon Marcel eer [6] Neu MF 995 Marx-analyc ehod on he heory of queue n dvance n 54

19 h://aeournalacza queueng: Theory ehod and oen roble JH halalow ed CRC 65-9 [7] Nlon egered & van der lu 7 new erave heurc o olve he on relenhen roble ung a readhee echnque nernaonal Journal of Producon conoc [8] Nlon & lver 8 le roveen on lver` heurc for he on relenhen roble Journal of he Oeraonal Reearch ocey [9] Olen L 5 n evoluonary algorh o olve he on relenhen roble ung drec groung Couer and ndural ngneerng 48-5 [] Raawa V 98 The N /G/ queue and dealed analy dvance n led Probably -6 [] lver 974 conrol ye for coordnaed nvenory relenhen nernaonal Journal of Producon Reearch [] lver 976 le ehod of deernng order quane n on relenhen under deernc deand Manageen cence 5-6 [] vauar nbazhagan N & rvargnan G 5 wo-coody erhable nvenory ye ORON 57-7 [4] vauar nbazhagan N & rvargnan G 6 Two-coody connuou revew erhable nvenory ye nernaonal Journal of nforaon and Manageen cence [5] rnvaan K & Ravchandran N 994 Mul-e nvenory odel wh Poon eand general lead e and aduable re-order ze n GV Krhna Reddy e al Revewed eded Proceedng of he Conference of ochac Model Ozaon and Couer lcaon Wley aern Led 6-6 [6] Thoone TM & lver 975 coordnaed nvenory conrol ye nernaonal ournal of Producon Reearch [7] Van MJG 99 noe on he on relenhen roble under conan deand Journal of he Oeraonal Reearch ocey [8] Vwanahan 996 new oal algorh for he on relenhen roble Journal of he Oeraonal Reearch ocey [9] Vwanahan On oal algorh for he on relenhen roble Journal of he Oeraonal Reearch ocey [4] Vwanahan 7 n algorh for deernng he be lower bound for he ochac on relenhen roble Oeraon Reearch [4] Wldean R Fren JG & eer R 997 n effcen oal oluon ehod for he on relenhed roble uroean Journal of Oeraonal Reearch [4] Yadavall V nbazhagan N & rvargnan G 4 wo-coody connuou revew nvenory ye wh lo ale ochac naly and lcaon

20 56 h://aeournalacza