OPTIMAL BATCH QUANTITY MODELS FOR A LEAN PRODUCTION SYSTEM WITH REWORK AND SCRAP. A Thesis

Transcription

1 OTIMAL BATH UANTITY MOELS FOR A LEAN ROUTION SYSTEM WITH REWORK AN SRA A Thei Submied o he Graduae Faculy of he Louiiana Sae Univeriy and Agriculural and Mechanical ollege in parial fulfillmen of he requiremen for he degree of Maer of Science in Indurial Engineering in The eparmen of Indurial & Manufacuring Syem Engineering By ablo Biwa B.Sc.Eng., Bangladeh Univeriy of Engineering and Technology, Bangladeh, 998 May, 003

2 AKNOWLEGEMENTS I would like o hank he Almighy for wha I have become oday. My incere graiude and appreciaion o rofeor Bhaba R. Sarker for hi preciou ime and guidance hroughou he ime of hi reearch and my graduae udy a LSU. The ime he pen for me o complee hi reearch i exremely appreciable. My pecial hank o r. Lawrence Mann, Jr. and r. enni B. Weber for reviewing my work and for erving on my commiee. Their commen and uggeion have cerainly improved he qualiy of my work. I alo hank my paren for heir acrifice and uppor hroughou my life. I alo hank my couin Lopa and my wife Sumi for heir coninuou encouragemen and conan moral uppor. Finally I would like o hank all my friend and oher family member. ii

3 TABLE OF ONTENTS AKNOWLEGEMENTS ii LIST OF TABLES vi LIST OF FIGURES vii ABSTRAT ix HATER : INTROUTION. The roblem.. Rework roblem 3.. Scrap eecion roblem Applicaion 5.3 Reearch Goal Reearch Objecive 8 HATER : LITERATURE REVIEW 9. Opimal Lo Sizing roblem 9. Opimal Bach uaniy wih Rework and Scrap 0.3 Imperfec roducion roce.4 Oher Reearch wih Rework and Inpecion 3.5. rawback of reviou Reearch 4 HAER 3: MOEL EVELOMENT 8 3. Aumpion 8 3. Noaion The Model Fir olicy: Wihin ycle Rework roce 3.4. ae I: Scrap eeced before Rework Average Invenory alculaion Makeup Buffer Invenory alculaion ae II: Scrap eeced during Rework Average Invenory alculaion Makeup Buffer Invenory alculaion ae III: Scrap eeced afer Rework Special ae of ae II: δ Toal o for Fir olicy Seup o Invenory arrying o Invenory arrying o for All ae Makeup Buffer Invenory arrying o for All ae roceing o roceing o for Rework Scrap Handling o 34 iii

4 3.5.5 Toal Syem o for All ae Opimaliy Special ae Numerical Example 37 HATER 4: REWORK AFTER N YLES Second olicy: Rework afer N ycle Invenory of Finihed roduc Invenory for Reworked Iem ae I: Scrap eeced before Rework ae II: Scrap eeced during Rework ae III: Scrap eeced afer Rework, δ Special ae of ae II: δ In-roce Invenory of Rejeced Maerial Toal o for Second olicy Seup o Invenory arrying o enaly o Scrap Handling o Toal Syem o Opimaliy Special ae Numerical Example 53 HATER 5: OERATIONAL SHEULE Operaional Schedule for Fir olicy: Wihin ycle Rework Operaional Schedule for ae I of Fir olicy Operaional Schedule for ae II of Fir olicy Operaional Schedule for Second olicy: Rework afer N ycle Operaional Schedule for ae I of Second olicy Operaional Schedule for ae II of Second olicy 60 HATER 6: SENSITIVITY ANALYSIS Effec of on * and T(* Effec of α on * and T(* Effec of Boh α and on * and T(* Effec of S on * and T(* Effec of on * and T(* Effec of H on * and T(* Effec of δ on * and T(* for ae II 77 HATER 7: RESEARH SUMMARY AN ONLUSIONS Summary oncluion Significance of Reearch Fuure Reearch 8 iv

5 REFERENES 83 AENIX 86 A. EXAMLE : YLE TIME VERIFIATION 86 B. EXAMLE : YLE TIME VERIFIATION 87. ROOF OF ONVEXITY OF T( FOR WITHIN YLE REWORK 88. ROOF OF ONVEXITY OF T( FOR REWORK AFTER N YLES 89 VITA 90 v

6 LIST OF TABLES Table. omparion beween ome reearch wih rework opion 6 Table. omparion of differen feaure beween curren reearch and oher reearch 7 Table 3. A comparion beween ae I and ae II 8 Table 3. Reul of he numerical example for fir policy 37 Table 4. Reul of he numerical example for econd policy Table 5.: Table 5.: alculaion of operaional chedule for wihin cycle rework policy 58 alculaion of operaional chedule for rework afer N cycle policy 6 Table 6. Effec of over T( * and * for ae I and ae II 66 Table 6. Effec of over T( * and * for ae III and Special ae 66 Table 6.3 Effec of α over T( * and * for ae I and ae II 68 Table 6.4 Effec of α over T( * and * for ae III and Special ae 68 Table 6.5 Effec of boh α and on T( * and * for ae I and ae II 70 Table 6.6 Effec of boh α and on T( * and * for ae III and Special ae 7 Table 6.7 Effec of S on T( * and * for ae I and ae II... 7 Table 6.8 Effec of S on T( * and * for ae III and Special ae 73 Table 6.9 Effec of on T( * and * for ae I and ae II 74 Table 6.0 Effec of on T( * and * for ae III and Special ae 74 Table 6. Effec of H on T( * and * for Fir olicy 76 Table 6. Effec of H n on T( * and * for Second policy 76 vi

7 LIST OF FIGURES Figure. A muli-age producion yem wih defecive iem Figure. roblem rucure and heir relaionhip in hierarchical order 4 Figure 3. Block diagram of he proce wih rework and crap for wihin cycle rework Figure 3. Invenory buil-up when crap i found before rework proce Figure 3.3 Invenory when crap i deeced during rework proce 6 Figure 4. Block diagram of he producion proce wih rework and crap for rework afer N cycle policy 38 Figure 4. Invenory of finihed good in one cycle 39 Figure 4.3 Invenory of finihed good during rework for ae I 4 Figure 4.4 Invenory of finihed good during rework for ae II 43 Figure 4.5 In-proce invenory of he rejeced maerial over N cycle for Second olicy 45 Figure 5.: Operaional chedule for ae I of Fir olicy 55 Figure 5.: Operaional chedule for ae II of Fir olicy 57 Figure 5.3: Operaional chedule for ae I of Second olicy 60 Figure 5.4: Operaional chedule for ae II of Second olicy 6 Figure 6. Effec of proporion of defecive on bach ize 64 Figure 6. Effec of proporion of defecive on oal co Figure 6.3 Effec of proporion of crap on bach ize and oal co 67 Figure 6.4 Figure 6.5 Effec of boh proporion of crap and proporion of defecive on bach ize 69 Effec of boh proporion of crap and proporion of defecive on oal co 70 Figure 6.6 Effec of eup co on bach ize 7 vii

8 Figure 6.7 Effec of crap handling co on bach ize 73 Figure 6.8 Effec of holding co on T( * and * for Fir olicy.. 75 Figure 6.9 Effec of crap producion facor on bach ize 77 viii

9 ABSTRAT In an imperfec manufacuring proce, he defecive iem are produced wih finihed good. Rework proce i neceary o conver hoe defecive ino finihed good. A he yem i no perfec, ome crap i produced during hi proce of rework. In hi reearch, invenory model for a ingle-age producion proce are developed where defecive iem are produced and reworked, where crap i produced, deeced and dicarded during he rework. Two policie of rework procee are conidered (a Fir policy: rework i done wihin he cycle, and (b Second policy: rework i done afer N cycle of normal producion. Alo, hree ype of crap producion and deecion mehod are conidered for each policy, uch a (i crap i deeced before rework, (ii crap i deeced during rework and (iii crap i deeced afer rework. Baed on hee invenory iuaion, he oal co funcion for a ingle-age imperfec manufacuring yem are developed o find he opimum operaional policy. Some numerical example are provided o validae he model and a eniiviy analyi i carried ou wih repec o differen parameer ued o develop he model. ix

10 HATER INTROUTION Manufacuring procee are omeime imperfec; for ha reaon heir oupu may conain defecive iem. Thee defecive iem can be reworked, crapped, ubjeced o oher correcive procee or old a reduced price, bu he reul i increaed exenive co in every cae. In recen decade, reearcher ried o deermine he opimal bach quaniy of imperfec producion yem conidering differen operaing condiion. Reearch i focued on he pracical iuaion involved in a ingle or muli-age producion yem. To evaluae an invenory yem policy properly require deermining he opimal bach-izing model along wih eup co, invenory holding co, proceing co and horage co. The demand and producion paern of a manufacuring faciliy affec he opimal bach quaniy or he economic lo ize. When raw maerial are proceed hrough a producion proce ino finihed good, hree ype of finihed produc can be delivered due o variou producion qualiie and maerial defec. Thee are (a qualiy of finihed produc, (b reworkable defecive produc, and (c crap. Addiional reource wih ubanial co are needed in cae where rework i involved. In a muli-age producion proce where produc move from one age o he nex age, he number of defecive may vary. epending on he proporion of defecive, he opimal bach quaniy alo varie hee affec he co of proceing, eup, and holding of invenory. Whenever a producion yem ha rework or repair faciliie, ome crap could ill be produced and, if no uch faciliie exi, all defecive produc go o crap, incurring addiional co a well a he lo of goodwill if he

11 company fail o mee cuomer demand. The flow diagram of a muli-age, imperfec producion yem ha produce boh good produc and defecive iem imulaneouly, i hown in Figure. Inpu Working Sage Working Sage Working Sage n Finihed roduc o uomer efecive efecive efecive Figure.: A muli-age producion yem wih defecive iem.. The roblem eermining he opimal bach quaniy of familiar invenory model ha been a primary miion among he reearcher for year. Mo of he reearch i devoed o developing well-known invenory model wih ideal condiion. In mo manufacuring procee, wih ingle or muliple age, ome defecive iem are produced. Even if he producion proce may have he rework capabiliy, ome percenage of crap may no be avoided even afer he rework proce. For hee reaon, ome manufacuring yem experience a horage of produc. Thee reul in cuomer diaifacion (ariing from unfilled demand and lo of goodwill. Thi reearch focue on reworking of defecive iem wih le han 00% recovery, reuling in a cerain percenage of crap during or afer he rework proce. In an imperfec manufacuring proce wih reworking faciliy, he invenory of finihed

12 good can build up if he producion rae i higher han he demand rae. The buil-up of invenory conain boh good and defecive iem. The defecive iem are reproceed hrough he yem reworking faciliy. The invenory build up again during he reworking proce reuling in crap being produced. epending on he crap producion, here migh be a reducion in he invenory of finihed good. Hence, he oal co of he yem, which i ignificanly dependen on he invenory of he finihed good, will be affeced. To produce he required quaniy, he yem need machinery eup, incurring a eup co. Holding he invenory hrough ou he producion period require an invenory carrying co. roceing he raw maerial for finihed good producion and reworking he defecive involve proceing co. In hi reearch he oal co of he producion yem will vary due o buil-up of differen invenorie and proceing co of regular and rework procee. In order o dicourage defecive and crap producion, a penaly co will be incorporaed in he oal co funcion... Rework roblem The rework proce i nohing bu he correcion proce of he defecive iem produced during normal producion. Thi reearch deal wih wo ype of rework procee: (a wihin-cycle rework, where he defecive are reworked wihin he ame cycle, and (b rework afer N cycle, in which he defecive iem from each cycle are accumulaed unil compleion of N cycle of normal producion, afer which he defecive par are reproceed. For boh policie he buil-up invenory iuaion are differen from he ideal one. A a reul, he modeling perpecive are alo differen. 3

13 .. Scrap eecion roblem Some of he defecive migh no be ranformed ino good iem hrough he rework proce decribed above. Hence, hey are dicarded a crap and he invenory decreae. Thi crap can be deeced in hree way before, during and afer he rework proce. In hi reearch, he crap producion and deecion echnique are involved for boh operaional policie: (a wihin-cycle rework, and (b rework afer N cycle a decribed above. Thu he invenory of he yem will form a differen paern from a radiional one due o rework proce and crap producion. Figure. how he problem rucure and heir relaionhip in a heirarchial order. The hree cenario of crap deecion are decribed below: Manufacuring proce wih rework Single-age Wihin ycle rework Rework afer N ycle Scrap deeced before rework Scrap deeced during rework Scrap deeced afer rework Figure.: roblem rucure and heir relaionhip in hierarchical order. (a In an imperfec producion yem, he good and defecive iem produced ogeher and he invenory build up a he proce coninue. Thee defecive iem are reworked and he correced iem are added o he invenory. Some crap produced 4

14 during he enire producion proce dicarded. efecive iem excluding he crap idenified a he beginning of rework are reproceed. (b Someime crap i produced and deeced during he rework proce. uring rework crap may ake le ime o produce han a good iem. A reducion in he finihed good invenory occur during he rework. For ha reaon, a he end of he producion a horage arie ha reul in unaified cuomer demand. (c Scrap may be deeced a he end of he rework proce of he defecive iem. Afer he end of ha rework crap i dicarded which reul in a furher horfall. In hi reearch, i i hypoheized ha a large porion of he defecive iem can be ranformed ino good iem hrough rework proce. The opimal bach quaniy migh be obained by opimizing he oal co of he producion yem wih repec o he finihed good invenory of he yem.. Applicaion Gla and ilicon wafer probe are illuraion of modern produc. Every office or home need gla for heir door, window, ec., aembly indurie need faner and cellular phone indurie require wafer probe. In a nu manufacuring indury he eel bar i he raw maerial for manufacuring nu. A he beginning of he producion hee bar are heared o lengh. Nex one end of he bar i heaed in he inducion furnace. Afer ha head of he nu i forged by an upeer and hread are eiher cu or rolled. A he end of he proce he produced nu are hea reaed or galvanized. uring he galvanizaion, ome defec may occur and he nu urn ino a defecive produc. Then, ha iem i fed again ino he galvanizing age for correcion. Before he correcion proce crap reul due o improper hread cuing 5

15 and are dicarded immediaely. Thu, crap i found before rework and good iem invenory build up. The ilicon wafer i an imporan par for he cellular phone manufacuring. The fir ep in he wafer manufacuring proce i he formaion of a large, ilicon ingle cryal or ingo. Thi proce begin wih he meling of polyilicon, wih minue amoun of elecrically acive elemen uch a arenic, boron, phophorou or animony in a quarz crucible. Once he mel ha reached he deired emperaure, a ilicon eed cryal i lower ino he mel. The mel i lowly cooled o he required emperaure, and cryal growh begin around he eed. A he growh coninue, he eed i lowly exraced from he mel. The emperaure of he mel and he peed of exracion govern he diameer of he ingo, and he concenraion of an elecrically acive elemen in he mel govern he elecrical properie of he ilicon wafer o be made from he ingo. Thi i a complex and proprieary proce requiring many conrol feaure on he cryal-growing equipmen. Afer producion of he ingo, hey are exraced from he cryal pulling furnace and allow hem o cool. Then he ingo are grinded o he pecified diameer. Nex, he ingo i liced ino hin wafer uing a 0-on wire aw. The baic principle of wire awing i o feed he ingo ino a web of ulra-hin, fa moving wire. Afer he awing proce, he individual lice have harp, fragile edge. Thee edge mu be rounded in order o provide rengh o he wafer. rofiling will ulimaely preven chipping or breakage in ubequen inernal proceing and during device fabricaion. Lapping remove conrolled amoun of ilicon from a wafer uing lurry. Thi proce remove aw damage and final polihing and cleaning procee give he wafer he clean and uper-fla mirror polihed urface required for he fabricaion of emiconducor 6

16 device. Some of he produc are furher proceed ino epiaxial wafer. Thu, he wafer producion i compleed and during he proce ome of he wafer are urned ino defecive due o exra ilicon paricle or breakage or chipping. They are proceed again hrough he profiling or fabricaion for correcion. ue o variou reaon, ome of he defecive wafer are urned ino crap during he rework proce and are dicarded. The correced wafer are added o he invenory of good iem and upplied o he cuomer. Thu, crap can be produced and deeced during he rework proce. Someime he wafer crap i deeced afer all rework procee are compleed and a he end of he producion he wafer crap i dicarded. Thi crap reul from variou caue, uch a imperfec polihing, awing damage, ec. Thu, he oal invenory of good wafer i reduced due o wafer crap producion. Gla manufacuring, bol manufacuring alo have he ame ype of rework wih crap problem. Some example of uch indurie are ardinal Gla Indury, Eden rairie, MN, orland Bol & Mfg. o., Inc, and GGB Indurie, Inc., Naple, FL which produce gla, nu and faner, wafer probe, repecively. They have a large invenory wih ingle and muli-age producion yem. Thee companie alo produce defecive iem and reproce hem in heir reworking faciliy. Thi reearch will ignificanly affec he invenory yem of hee companie, which evenually migh increae heir profiabiliy ha hey loe due o crap producion..3 Reearch Goal The inenion of he reearch i o udy and model he invenory yem in an imperfec manufacuring faciliy where he rework opion i available. In uch a faciliy, defecive iem are produced wih he finihed good, and hee defecive are 7

17 reproceed. Scrap i alo produced and deeced in differen age of producion: before, during, and afer he reproceing of he defecive. Thi reearch alo propoe a echnique o aify he cuomer demand, which may no be obained due o crap producion. The principal moivaion of hi reearch i o minimize he oal yem co of he invenory of an imperfec manufacuring proce..4 Reearch Objecive The behavior of invenory paern in producion proce wih rework capabiliie i differen from he radiional invenory paern for an ideal producion proce. efecive iem and crap i produced during he normal producion due o imperfec yem. To repair hee defecive, a manufacuring proce may incorporae variou reproceing echnique in rework faciliie. Some crap ha i produced before, during and afer he reproce i dicarded cauing a reducion in he invenory. ue o he above reaon, he naure of he invenorie of hee yem are differen from radiional one. Hence, he primary objecive of hi reearch are: (i To udy he behavior of he invenorie in differen reworking policie and crap producion. (ii (iii To find he opimal order policy for raw maerial. To deermine an opimal afey ock o mee he horage of he invenory due o crap (rejecion. (iv (v To e up he opimal bach ize for producion of he iem. To find he operaional chedule (implemenaion of he producion proce. 8

18 HATER LITERATURE REVIEW rior reearch having o do or repair opion and crap in he producion yem are rare. Several reearcher have developed economic lo ize or opimal bach quaniy model conidering he perfec invenory model wihou any rework or crap opion. They have conidered he perfec producion proce wherea mo producion procee are ofen imperfec. Kumar and Vra (979 ried o focu in hi area and developed opimal bach ize of finihed good invenory for a muli-age producion yem. Goyal (978 alo menioned he effec of amplified in-proce invenory economic bach ize model for a muli-age producion yem. Neiher of hem ha conidered he defecive producion. handra e al. (997 have conidered a model of bach quaniy in a muliage producion yem wih differen proporion of defecive in every age, bu hey ignored he rework opion. Sarker e al. (00b have developed an opimum bach quaniy conidering rework bu hey alo have ignored he crap opion during rework proce.. Opimal Lo Sizing roblem Goyal (976 poined ou ha, in a ypical indurial purchaing iuaion, he buyer order quaniy i o mall from a producer perpecive ha he producer eup co per bach i uually larger han buyer ordering co. He uggeed an inegraed lo izing approach ha would minimize he join oal co o boh parie. Banerjee (986 and Monahan (989 developed lo-izing model for he vendor, where he approach i o induce he cuomer o order in larger lo hrough offer of quaniy price dicoun. Thee model did no conider he work-in-proce, which occupy he ignifican amoun 9

19 of oal invenorie. Banerjee (99 coniderd periodic, dicree cuomer demand and aumed ha he vendor producion rae wa finie. onidering he effec of uch a finie producion rae on work-in-proce invenorie and conequenly on he baching deciion ielf, hey developed wo model o deermine he producer opimal and independen coure of acion in erm of lo izing, in repone o cuomer periodic ordering policy. In 990, Banerjee and Buron made effor o accoun for work-inproce invenorie in heir ingle and muli-age bach izing model under uniform demand, imulaneouly. lark and Armenano (995 propoed a heuriic for he reource-limied muli-age lo-izing problem wih general produc rucure, eup co and reource uage, work-in-proce invenory co and lead ime. oreu (986 derived a ignifican relaionhip beween qualiy and lo ize. He howed clearly ha he improved oupu qualiy i achievable by reducing lo ize. oreu (985 alo propoed a model ha can obain he opimum eup co and invemen required in achieving hi eup co in dicouned and undicouned economic order quaniy model. Goyal and Gunahekharan (990 developed a mahemaical model, which howed how he oal co yem could be affeced by he invemen in qualiy. They alo conidered he invemen in qualiy and producion bach ize.. Opimal Bach uaniy wih Rework and Scrap In a manufacuring faciliy, producion of defecive iem i a common. Thee defecive can reduce profiabiliy. For a long ime, reearcher have avoided he defecive iem producion problem. Gupa and hakrabary (984 conidered hi problem and have brough i o reearcher aenion. They deal wih he rework proce in a muli-age producion yem. They formed a model of a yem where all defecive 0

20 iem are colleced afer producing he finihed good, and hoe are fed ino he fir age of producion proce for reworking. In heir reearch, hey developed a model for opimal producion bach quaniy and opimal recycling lo ize o minimize he oal operaional co of uch circumance. hakrabary and Rao (988 inroduced he rework proce in a muli-age producion yem incurring wo operaional policie for proceing reworked lo. In one cae he rework i done in he ame age where i occur. For he fir cae hey inroduced a buffer o deliver he horage occurred due o defecive iem producion. The oher cae he rework i done immediaely a he ame age from where hey are produced before he whole lo i en o he ubequen age and ubequen bache are aken only afer proceing of reworked lo hrough all he age i compleed. They alo developed he opimal bach quaniy for boh cae o minimize he oal yem co. Wein (99 conidered a yield problem wihin emiconducor fabricaion and developed a mahemaical model for rework and crap deciion in a muli-age producion yem o deermine he effec of rework opion uing Markov deciion model. handra e al. (997 udied a problem ha coni of opimum producion bach ize in a muli-age producion faciliy wih crap ignoring he rework opion of he defecive. They alo conidered he opimal amoun of invemen in ha manufacuring faciliy. Recenly, Sarker e al. (00a conidered a ingle producion yem wih rework opion incorporaing wo cae of rework proce. In fir cae hey conidered ha he rework i done wihin he ame cycle and he ame age where i produced. In he econd cae, he defecive iem are accumulaed up o N cycle and he accumulaed iem are

21 reworked in nex cycle. They developed he economic bach quaniy o minimize he oal yem co and increae he profiabiliy of he yem. Sarker e al. (00b alo developed he opimal bach quaniy for a muli-age producion yem o minimize he yem co under he ame echnique..3 Imperfec roducion roce Lee and Roenbla (986 conidered a circumance when manufacuring faciliy goe from in-conrol o ou-of-conrol during producion. They developed a model o deermine he opimal order quaniie wih imperfec producion proce and eablihed he relaionhip of qualiy, lo ize, eup and holding co and he deerioraion of he producion proce. Teuner and Flapper (000 conidered a producion line ha produce a ingle iem in muli-age. The produced lo are nondefecive, reworkable defecive or non-reworkable defecive. The rework proce i done in he ame producion line. The auhor developed he model for perihable iem auming ha he rework ime and he rework co increae linearly wih he ime ha a lo i held in ock and hey derived an explici expreion for he average profi (ale revenue minu co. Lee (99 chooe he lo-izing problem conaining he key characeriic of imperfecion in a producion proce and developed a model which include proce hifing o ou-of-conrol ae, deecion of he ou-of-conrol producion. orrecive acion follow he deecion, and fixed eup and variable ime of rework. He found he problem from he wafer probe operaion in emi conducor manufacuring.

22 .4 Oher Reearch wih Rework and Inpecion Agnihohri and Kene (995 deal wih a producion proce which ha an inpecion proce where he defecive iem are ored ou and en o he reworking age. Afer reworking, he finihed good are delivered o he cuomer. They have developed a model, conidering he number of defec i a random variable having geomeric diribuion, and inveigaed he impac of he defec diribuion on yem performance meaure. Tay and Ballou (988 conidered ha he produc i produced in bache which are ranpored inac from age o age. The proceing a one age begin only afer he compleion of proceing a previou age. Each producion age ha been conidered a few defecive iem producion, which are ored ou and en for rework a one or more age. They developed he model for any pecified inpecion configuraion in equenial producion proce and have obained a cloed-form oluion o deermine he opimal lo ize and rework bach ize o minimize he oal yem co. So and Tang (995a preened a model of a boleneck yem ha perform wo diinc ype of operaion uch a regular producion and rework proce. In heir reearch, each job i paed hrough an inpecion, and he job ha pae he inpecion i fed o he downream of he producion proce, oherwie i i fed o ha age for rework. They formulaed he problem a a emi Markov deciion proce. They alo developed a imple procedure o compue he criical value ha idenifie he opimal hrehold policy and evaluaed he impac of bach ize, yield, and wichover ime on an opimal hrehold policy. So and Tang (995b preened anoher model of a boleneck faciliy which perform wo eparae ype of operaion uch a regular and repair. They conidered wo policie, repair none and repair all, and found he opimaliy 3

23 condiion for boh policie. Hong e al. (998 developed an economic deign of inpecion procedure when he crap iem are reworked. Their co model coni of he co incurred by imperfec qualiy, reproceing co and inpecion co and hey developed a probabiliic mehod for olving ha problem..5. rawback of reviou Reearch The above lieraure udy indicae ha deerminaion of opimal bach quaniie of producion procee wa he principal earch for mo of he reearcher. Reearch ha focued on developing opimal lo ize of radiional invenory model, economic bach ize for imperfec producion proce, comparion for differen producion policie and inpecion policie for variou invenory model wih rework or reproceing opion. Thi reearch may no be ufficien o olve he problem adequaely. From he above urvey, ome drawback are found ha exi in previou oluion mehod. The drawback are in he following area: (a A decribed in he previou ecion, reearcher [(Goyal, 976, 978, (Monahan, 989, (Kumar and Vra, 978, (Banerjee, 986, 99, (lark and Armenano, 995] ried o develop opimal bach quaniy for perfec producion proce in ideal condiion. racically, he producion faciliy involve a lo of imperfecion, which reul defecive iem. (b Many reearcher [(Gupa and hakrabary, 984, (hakrabary and Rao, 988, (handra e al., 997] deal wih defecive producion procee, and ried o develop opimal bach ize o improve he qualiy and minimizing he oal co of he yem. Some of hem did no conider rework which reul in maerial waage. Some of hem conider rework, 4

24 bu during he rework proce, he producion equipmen i conidered o be a perfec reworking faciliy. (c A defecive iem are produced and rework in an imperfec producion yem, ome of he defecive iem canno be reworked and hoe go o crap. For ha reaon ome horage may occur in cuomer demand aifacion. Few reearcher [(Sarker e al., 00, (Tay and Ballou, 988, (Lee, 99] focued on developing opimal bach quaniie in hi ype of producion yem, bu hey did no conider abou he horage ha may occur due o crap produced during rework proce. Table. how ome comparion beween ome previou reearch which conidered he rework faciliy in producion yem. In hi reearch, opimal bach quaniy model are o be developed conidering rework proce and crap producion during or before he rework, which will overcome he drawback of he previou reearch. I i anicipaed ha hi reearch will provide beer reul o he real problem. Table. how he comparion of he model feaure beween hi reearch and oher reearch. 5

25 Table.: omparion beween ome reearch wih rework opion. # Tile Summery Aribue In a muli-age producion yem, a model for opimum producion bach Gupa and hakrabary quaniy a well a opimal recycling (IJR, (, 984, 99- lo ize ha been developed wih 3: repec o minimize he oal co Looping in a muli-age funcion of he iuaion where he producion yem defecive iem are produced during normal producion of finihed good. 8. Balanced work load hakrabary and Rao (Opearch, 988, 5(, 75-88: EB for a muli-age producion yem conidering rework Tay and Ballou (IJR, 988, 6(8, 99-35: An inegraed producioninvenory model wih reproceing and inpecion Agnihohri and Kene (EJOR, 995, 80(, : The impac of defec on a proce wih rework handra e al. (, 997, 8(6, : Opimal bach ize and invemen in muli-age producion yem wih crap. Sarker e al. (Working paper, 00a: Manufacuring bach izing for rework proce in a ingle-age producion yem. Sarker e al. (Working paper, 00b: Manufacuring bach izing for rework proce in a muli-age producion yem. Model are developed o deermine opimum number of cycle for rework and opimum bach quaniy in a muli-age producion yem conidering wo differen policie of proceing reworked lo. The co and qualiy level of producion yem have examined hrough a model, and obained a cloed form oluion for opimal lo ize and reproceing bach ize uing Markovian proce. To quanify he impac of defec on variou yem performance meaure for a producion yem wih 00% inpecion followed by rework. The model ha developed auming he number of defec o be random variable wih geomeric diribuion and inveigaion ha done for he impac of he defec diribuion on yem performance meaure. The problem are elecing he opimum producion bach ize in a muli-age manufacuring yem wih crap and deermining he opimal amoun of invemen. The effec of invemen for qualiy improvemen on he yem parameer have alo analyzed. To deermine he opimal bach quaniy of a ingle-age producion yem wih rework faciliy conidering wo policie of rework proce o minimize he oal co of he yem. The opimal bach quaniy of a muli-age producion yem ha deermined wih rework faciliy conidering wo policie of rework proce o minimize he oal co of he yem.. Muli-age yem. Uniform demand 3. eerminiic proceing ime 4. eerminiic eupime 5. onan defecive 6. Negligible waiing ime 7. No crap. Muli-age yem. Uniform demand 3. Variable defecive 4. No crap 5. No defecive during rework 6. Balanced workload.. Muli-age yem. robabiliic demand 3. Bach producion 4. efecive are produced every age 5. Rework i conidered 6. No crap. Three age manufacuring yem. robabiliic demand 3. efec are deeced one a a ime 4. Number of defec have a dicree probabiliy diribuion 5. FFS dicipline applied 6. Unlimied buffer 7. erfec rework condiion 8. Seady ae yem 9. Muliple parallel ever. Muli-age yem. onan and uniform demand 3. onan price per uni of produc 4. onan eup co 5. All defecive are craped 6. roducion i greaer han demand 7. Each age defecive producion 8. Normally diribued produc qualiy. Single-age yem. onan demand 3. onan producion rae 4. No crap 5. No defecive during rework Inpecion co i ignored. Muli-age yem. onan demand 3. onan producion rae 4. No crap 5. No defecive during rework 6. Inpecion co i ignored 6

26 Table.: omparion of differen feaure beween curren reearch and oher reearch. roperie conidered Gupa and hakrabary hakrabary and Rao (984 (988 Tay and Ballou (988 Agnihohri and Kene (995 handra e al. (997 Sarker e al. (00a Sarker e al. (00b urren Reearch roduc Single Single Single Single Single Single Single Single Sage N N N N N N emand rae Uniform Uniform rob. rob. onan on. on. onan roducion rae onan onan on. onan onan on. on. onan Rework opion Ye Ye Ye Ye No Ye Ye Ye Scrap conidered No No No No Ye No No Ye Rework policy R T & U V W - Y & Z Y & Z Y & Z Way of Scrap deecion A - - B, & Inpecion co No No Ye Ye Ye No No No Buffer conidered No Ye No No No No No Ye Shorage co No No No No No Ye Ye Ye rob. probabiliic, on. conan. A Scrap deecion during he producion, B Scrap deecion before he rework, Scrap deecion during he rework proce, and Scrap deecion afer he rework proce. N Muli-age R Rework opion: afer defecive producion, hey fed in o he fir age, T Rework i done in he ame age where defecive produced U Rework i done immediaely afer a defecive produced a ame age, V Rework i done afer he end of producion, W Rework i done afer producion in a differen age, Y Rework i done immediaely wihin he cycle where he defecive produced, and Z Rework i done afer N cycle of producion in a differen age. 7

27 HATER 3 MOEL EVELOMENT Thi ecion of he reearch conain he formulaion of he invenory model a decribed previouly for differen policie wih rework and idenificaion of differen cae for crap in a ingle-age producion yem. The formulaion of he model depend on ome aumpion and noaion. They are decribed a he beginning afer which he average invenorie for differen cae and he oal co funcion of he invenorie are derived. 3. Aumpion The following aumpion are made o develop he model: (a A ingle ype of produc in a ingle-age producion yem i conidered, (b roducion rae i conan and greaer han demand rae, (c roporion of defecive i conan in each cycle, (d Only one ype of defecive i produced in each cycle, (e Scrap i produced and deeced in differen way, (f The defecive are reproceed once, afer which hey are dicarded a crap, if no correced properly, (g roporion of crap i le han he proporion of defecive, (f Inpecion co i ignored ince i i negligible wih repec o oher co. 3. Noaion The following noaion will be neceary o explain and formulae he problem. α roporion of crap during rework wih repec o oal produced defecive iem. 8

28 roporion of defecive in each cycle wih repec o oal produced iem. δ Scrap producion facor, where 0 δ. H H n roceing co in fir operaional policy, dollar/uni. Invenory carrying co for wihin cycle rework, dollar/uni/year. Uni invenory carrying co for uni finihed produc per uni ime for rework afer N cycle, dollar/uni/year. p d K S K S K T w N Uni penaly co, dollar/uni/year. Seup co, dollar/minue. Seup co for defecive iem, dollar/minue. Seup co, dollar/year, Seup co for defecive, dollar/year, Scrap handling co, dollar/year, Uni in-proce invenory carrying co, dollar/uni/year, Uni crap handling co, dollar/uni, emand rae, uni/year. Number of producion cycle afer which he defecive iem are reworked. 3 roducion rae, uni per planning period, uni/year. Bach quaniy per cycle, uni/bach. Time of normal producion, year. Time of rework proce, year. Time of conumpion afer producion op for he ae I, and crap producion ime for ae II, year. 9

29 4 onumpion ime afer producion op for ae II, year. The quaniy of good iem remaining afer conumpion a he end of ime. The quaniy of good iem ha hould remain afer conumpion a he end of ime. 3 The quaniy remaining afer conumpion a he end of ime, when rework i compleed wihou he crap. 4 5 The quaniy ha i produced a he end of rework. The quaniy remaining afer he conumpion during he producion, deecion and eparaion of crap. S d T T Seup co in wihin cycle rework policy, dollar/bach. Seup ime, minue/bach. Seup ime for defecive, minue/bach. Toal cycle ime for ae I, year. Toal cycle ime for ae II, year. 3.3 The Model In hi reearch, he invenory model are developed for wo operaional policie. The fir policy cover rework i done whenever defecive are produced. Therefore, he finihed produc are delivered o he cuomer a he end of he cycle and crap i dicarded. The econd policy encompae defecive iem produced in each cycle and accumulaed unil N cycle of producion are compleed, and hen he rework i performed. A a reul, he reworking cycle may be differen from he normal cycle. The crap i alo accumulaed during reproceing. When good iem are produced hey are 0

30 delivered o he cuomer direcly, for crap producion horage occur and penaly co i impoed on boh crap iem and buffer invenory. 3.4 Fir olicy: Wihin ycle Rework roce A decribed above, he rework in he fir policy i done wihin he ame cycle in which he defecive are produced, and ome crap i produced and deeced before, during and afer he rework proce. Figure 3. how he block diagram of he producion proce for he fir policy. efecive Iem Inpu roducion ycle Finihed Good o The uomer Scrap Makeup Buffer Figure 3.: Block diagram of he proce wih rework and crap for wihin cycle rework. Scrap i divided ino hree differen cae: (a crap deeced before rework, (b crap deeced during rework and (c crap deeced afer rework. All cae are conidered below.

31 3.4. ae I: Scrap eeced before Rework In hi cae, crap can be deeced before he rework proce ar o produce good iem from he defecive. Figure 3. how invenory when defecive iem are reworked wihin he cycle and crap i deeced before rework ar. Accordingly, Figure 3.,,, and 3, are he ime egmen, which repreen he proceing ime (upime, rework ime wihou crap and downime or conumpion ime, repecively. Toal defecive, Scrap, α α 4 3 B X - (- - L J A I G O F E h h - H 3 ae I Scrap i deeced before rework. No crap during he rework. Rework ime normal proceing ime. (-α/ Z Y T Figure 3.: Invenory buil-up when crap i found before rework proce. The invenory level repreened by he riangle (BAG wih dahed line indicae he ideal cae of invenory when no defecive or crap iem are produced in he producion cycle. From he beginning o he end of he producion proce boh good and

32 defecive iem are produced ogeher. When a defecive iem i produced, he iem i immediaely reworked. In Figure 3., i i hown ha he defecive iem are produced a a rae of during ime. The riangle BLG repreen invenory when he defecive iem are produced during he upime,. The line BL indicae he lope of (--, i.e., he ne replenihmen rae when he defecive iem are produced during ime a a defecive proporion of. The ne amoun of defecive produced during ime i. I i aumed ha α % of defecive i crap. Hence, a he end of ime, he crap uni α are idenified and eparaed from he main invenory, and ay, line AI indicae ha amoun. The remaining defecive ( α uni repreened by LI are reworked a he rae of uni/year, a he rework rae i aumed a he ame a producion rae. Therefore, invenory build up again a he rework proce coninue from poin L o poin F during ime. The riangle EH indicae he pure conumpion occurred afer he producion op a he end of ime, and only pure conumpion coninue during ime 3. A ome crap i deeced during he ime of, herefore, o mainain he goodwill of he company, an equivalen quaniy of buffer, α repreened by he riangle XYZ a he boom of he invenory diagram i mainained during he ime period Average Invenory alculaion According o he definiion, T. Again, which lead o /. In Figure 3., repreen he quaniy of good iem remaining afer conumpion a he end of ime ; repreen he quaniy of iem ha hould remain afer conumpion, if no defecive iem i produced a he end of ime ; and 3 indicae he quaniy 3

33 remaining afer conumpion a he end of ime, when rework i compleed wihou he crap. Hence, i can be hown ha ( / and ( /, and number of defecive iem produced i AL. Here, line AJ repreen he conumpion during upime, o AJ, and during ime ( α/, he invenory ued i EF ( α/. Therefore, 3 EH AI EF ( / α ( α / [ α ( α/]. Now 3 can be found a follow: [ α ( α / ] EH 3 3. (3. Hence, he oal cycle ime T 3 can be calculaed a T 3 ( α [ α ( α / ] ( α. (3.a If α 0, hen T /, which i he andard invenory model. follow: Therefore, according o Figure 3., he average invenory, I, can be calculaed a I I h T h T ( h h T h T ( T h T ( h T ( T [ ( ] ( α [ ( ] ( α ( α ( ( α ( α, which, upon implificaion, yield I I [ ( α ( α α]. (3. ( α 4

34 When α 0, equaion (3. reduce o I, which indicae he andard finie producion invenory model. If α 0, equaion (3. reduce o [ ( ] I. (3.3 which indicae he finie producion model wih rework opion and no crap Makeup Buffer Invenory alculaion A makeup buffer XYZ i mainained during ime due o crap producion o aify cuomer demand. Hence, he average invenory of area XYZ can be calculaed a I α α IB ( α. ( ae II: Scrap eeced during Rework In hi cae, crap i deeced during he rework proce. I i aumed ha he ime o qualify a reworking iem a crap i le han he ime o produce a good iem. Figure 3.3 how invenory during he enire cycle when crap i deeced during rework proce. In Figure 3.3, he proceing ime (upime, rework ime, crap producion ime and pure conumpion ime are repreened by he ime egmen,, 3 and 4, repecively. The procee of producion of good and defecive iem and crap declaraion have already been decribed in previou ecion. Here, i i aumed ha he reworked good iem are produced a he rae of uni/year, and crap i produced a he rae of /δ uni/year, where δ i he crap producion facor, (0 δ. 5

35 Good reworked, (- α Toal defecive, L J A E /δ F - M N S R U h ae II: δ Scrap i deeced during rework. Scrap produced a rae of /δ. Scrap, α /δ F E 3 α α αδ M N S R U - Scrap, α 3 - O V - h (- - B G X H W 3 4 α Z T Y Figure 3.3: Invenory when crap i deeced during rework proce. According o he aumpion, he crap iem are α % of he oal defecive iem. The oal good iem produced during rework ime are ( α. Thu, he 6

36 invenory coninue o build a he rework proce proceed from poin L o poin E during ime, and afer he producion, α uni of crap i deeced. Though crap i produced more or le uniformly during he rework proce, in order o iolae he crap producion, i i hown eparaely from poin F o M in he Figure 3.3, where in FM indicae he lope /δ. A crap i eparaed during he rework proce, he ime o produce hem i added o he acual invenory producion ime. A he end of ime 3, producion op and conumpion repreened by he line U occur hrough he ime period 4. The riangle UW how he invenory conumpion during ime 4. A crap i found during ha ime period and 3 a compenaing buffer i mainained during he ime of rework o mee he demand. In Figure 3.3, he riangle XYZ repreen he compenaing buffer invenory mainained from he beginning o he end of rework proce Average Invenory alculaion The repreenaion and value of,, 3, AJ, AL and EF have been decribed in previou ecion. In hi cae, 4 i he quaniy ha i produced a he end of rework, 6 i he quaniy remaining afer he conumpion during he producion deecion and eparaion of crap (a he end of ime 3. Hence, ( α/, and he invenory conumed during ime i EF ( α/. Therefore, 4 FH EO EF ( / ( ( α/ ( α/ [ ( /], o he acual invenory wihou he crap i 3, and 3 EH EM ( / ( ( α/ [ α ( α/]. Afer he producion, α % crap i found, which i α. Thee iem are produced during he rework, bu in Figure 3.3 i i hown eparaely from poin F o M. A hee are eparaed during he rework proce, he ime 7

37 o produce crap i added o he acual invenory producion ime. Hence, 3 can be calculaed a 3 αδ/. uring he ime 3, he invenory i conumpion repreened by line RU can be calculaed a RU 3 αδ/. Afer he end of he producion, he acual invenory of good iem remaining i 6 uni: 6 UW 3 RU ( α αδ α [ α ( α αδ / ]. (3.5 Now 4 can be found a follow: 4 [ α ( α αδ / ]. (3.6 UW 4 Hence, he oal cycle ime T 3 4 can be calculaed a T ( α αδ ( α αδ α ( α αδ α. (3.7 If α 0, hen T /, which i he andard invenory model. A comparion beween differen invenory level of ae I and ae II i hown in Table 3.. Table 3.: A comparion beween ae I and ae II. arameer Scrap Before Rework uring Rework (0 δ ( α/ ( α/ 3 [ α ( α/]/ αδ/ α ( α αδ / / T ( / ( / ( / ( / 3 [ α ( α/] [ α ( α/] 4 [ α /] [ ( /] α ( α αδ / 4 ( 6 ( 8

38 fahion below, According o Figure 3.3, he average invenory, I, can be evaluaed in hi I II h T h ( T 3 ( h h ( T 3 h ( T T 3 T [ h ( h 3 ( T ] ( α αδ ( ( ( α ( α αδ ( α α. Simplifying he above equaion, he average invenory can be evaluaed a I II [ ( ( α α ( δ αδ α α]. (3.8 When α 0 and δ, equaion (3.8 reduce o I, which i he ame a he andard finie producion invenory model and when α 0, ha mean when no crap i producing during rework, equaion (3.8 reduce o equaion ( Makeup Buffer Invenory alculaion In hi cae, a crap i produced during he rework a makeup buffer XYZ i mainained from he beginning of ime o he end of ime 3. Therefore, he average invenory of he buffer can be found from he area XYZ a follow: α ( α αδ I IIB α( 3. (3.9 9

39 ae III: Scrap eeced afer Rework Thi ecion deal wih he deecion of crap afer he rework i compleed. Thi i a pecial cae of ae II, decribed in he previou ecion. In hi cae, δ, he crap producion facor, i aumed o be, which mean crap i producing a he ame rae. Hence, he equaion (3.8 become I III [ α ( α], (3.0 ( α and he average invenory for he makeup buffer reduce o α I IIIB α ( 3. ( Special ae of ae II, δ 0 Thi i anoher pecial cae, where he crap producion facor, δ 0, which mean crap i deeced a he beginning of producion. Hence, equaion (3.8 reduce o I S [ α ( α α], (3.a ( α and he buffer invenory become α ( α I SB α( 3. (3.b 3.5 Toal o for Fir olicy In he curren and previou ecion, he configuraion of invenory are decribed under he aumpion of he fir policy. Generally, he oal co of a producion yem coni of hree major co: uch a (a eup co, (b invenory carrying co, and (c proceing co. A a makeup buffer i mainained o overcome he horage of he cuomer demand due o crap producion, a buffer mainenance co i alo included in he oal co. In hi cae, he oal co funcion coniing eup co, invenory carrying 30

40 co for all cae, and buffer mainenance and proceing co can be calculaed a follow: 3.5. Seup o Each and every producion proce need a eup for proceing he raw maerial o manufacure finihed produc. Hence, eup co (K can be calculaed a K S, (3. where i demand rae (uni/year, i he bach quaniy (uni/year and S i eup co per bach Invenory arrying o Invenory build up during he upime becaue of higher producion rae han he demand rae. Thu, he producion faciliy incur invenory-carrying co due o he accumulaed invenory in each cycle. According o hi policy, he invenory carrying co are calculaed a follow: Invenory arrying o for All ae Uually invenory-carrying co of finihed produc i proporional o he average invenory of he produc in a cycle. Hence, invenory-carrying co can be calculaed a average invenory of he produced iem in he cycle muliplied by uni invenory carrying co of he produc. Uing equaion (3., (3.8, (3.0 and (3.a invenory carrying co i calculaed for ae I, ae II, ae III, and Special ae are repecively a follow: I I H [ ( α ( α α], (3.3 ( α 3

41 I II H [ ( ( α α ( δ αδ α α], (3.4 I III H [ α ( α], and (3.5 ( α I S H [ ( α ( α α]. (3.5a ( α Makeup Buffer Invenory arrying o for All ae ue o crap a horage occur. To compenae for hee horage, in every cae a makeup buffer i mainained o ha he carrying co for he makeup buffer play a role in he oal co funcion. The buffer carrying co can be evaluaed by muliplying invenory carrying co H and average invenorie of he makeup buffer. Uing he equaion (3.4, (3.9, (3. and (3.b he makeup buffer carrying co can be evaluaed a αh B I, (3.6 ( α α ( α αδ H B II, (3.7 ( α α H B III, and (3.8 ( α α ( α H B S. (3.8a ( α roceing o Here, in each cycle, he bach quaniy i proceed and he defecive iem are reworked, incurring proceing co. Hence, he oal proceing co of he yem i he 3

42 accumulaion of he proceing co of bach quaniy, and defecive iem found during producion of oal bach quaniy. If he proceing co per bach i, hen he proceing co of bach quaniy i. Hence, he oal proceing co for he whole planning period, K can be calculaed a K /. ( roceing o for Rework In hi model he defecive iem are produced in every cycle a proporion of oal bach quaniy, o he oal quaniy ha will be defecive i. According o he fir policy, he defecive iem are proceed wihin he ame cycle for correcion, and here i no eup co involved for hem. Only he proceing co play he role in he oal co funcion. Again, he invenory co for he defecive iem i alo aken ino conideraion and i i already added o he invenory carrying co, o he proceing co of he rework of defecive iem over he whole planning period can be calculaed a K R /. (3.0 Here, a proceing co for he buffer quaniy α i alo aeed during he producion period o makeup he horage due o crap. Therefore, he proceing co of he buffer quaniy α i K B α / α. (3.0a Hence, he oal proceing co for rework and buffer mainenance can be evaluaed by adding equaion (3.0a and (3.0b a follow: K RB α ( α. (3.0b 33

43 3.5.4 Scrap Handling o To handle crap i i neceary o add a co for he amoun of α uni crap. I can be evaluaed by muliplying he uni crap handling co, and he amoun of crap produced in enire period of ime a K T α α. ( Toal Syem o for All ae The oal co of he yem T(, i he accumulaion of he eup co, invenory-carrying co, makeup buffer carrying co, proceing co and proceing co due o reworking and buffer mainenance. Hence, he oal co of he yem for ae I can be calculaed by adding equaion (3., (3.3, (3.6, (3.9, (3.0b and (3. a follow: S H T I ( ( α α [ ( ( α αh α ( α α]. (3. ( α The oal co of he yem for ae II can be calculaed by adding equaion (3., (3.4, (3.7, (3.9, (3.0b and (3., a T II S H ( ( α α [ ( ( α α ( α αδ H α ( δ αδ α α] ( α. (3.3 The oal co of he yem ae III can be calculaed by adding equaion (3., (3.5, (3.8, (3.9, (3.0b and (3. a follow: T III S H ( ( α α [ ( ( α 34

44 α H α ( 3 α]. (3.4 ( α The oal co of he yem for Special ae of ae II can be calculaed by adding equaion (3., (3.5a, (3.8a, (3.9, (3.0b and (3. a follow: T S S H ( ( α α [ ( ( α α ( α H α ( α α]. (3.5 ( α A hi age, i i neceary o find he naure of he above funcion for opimizaion purpoe. If he funcion are convex, hen he parial differeniaion wih T( repec o bach quaniy can be e o zero; ha mean, 0 and he opimal bach quaniy * can be evaluaed. If he funcion are non-convex, a ingle-variable direc earch mehod uch a (a random earch mehod, (b univariae mehod, (c paern earch mehod, and (d Roenbrock mehod of roaing coordinae mehod can be applied o find he opimum order quaniy. 3.6 Opimaliy I can be eaily hown ha T( i a convex funcion in (ee Appendix. Hence, an opimum bach quaniy *, can be calculaed from T( / 0, which yield T I S H α [ ( ( α ( αh α ( α α] 0, (3.6 ( α 35

45 * I S. (3.7 H γ α ( α where γ [ ( α ( 3 α α], Similarly, from equaion (3.3, (3.4, and (3.5 he opimal bach quaniy *, can be evaluaed, repecively, a * II S, (3.8 H π α ( α where π [ ( α ( 3 δ αδ α α], * III S, and (3.9 H λ α ( α where λ [ ( α ( 4 α], * S S, (3.30 H ω α ( α where ω [ ( α ( 3 α α] Special ae If he producion yem i conidered o be ideal, i.e., no defecive iem and crap are produced, mean he value of and α i e o zero. In ha cae, equaion (3.7, (3.8, (3.9 and (3.30 reduce o he claical economic bach quaniy model a follow: * S. (3.3 H( / 36

46 When he defecive iem are produced and crap i no, he equaion (3.7, (3.8, (3.9 and (3.30 reduce o he economic bach quaniy model wih defecive a follow: S *. (3.3 H [ ( ] The oluion above are validaed hrough numerical example in he following ecion. 3.7 Numerical Example The opimum value of * for all cae can be obained by ubiuing he parameer value in equaion (3.7, (3.8, (3.9 and (3.30. Aume, 300 uni/year, 550 uni/year, $7/uni, S $50/bach, H $50/uni/year, $5/uni, 0.05 and α 0.0 and δ 0.07 (meaning i aumed a 0.05 i.e., 5% defec, and α i aumed a 0.0 i.e., 0% crap from he defecive. The opimum bach quaniie * and he oal co wih repec o * are obained for all cae and are calculaed by uing above parameric value and equaion (3., (3.7, (3.3, (3.8, (3.4, (3.9, (3.5, (3.30, and (3.3, repecively. All reul are repreened in abular form below: Table 3.: Reul of he numerical example for fir policy. ae arameer ae I ae II ae III Special ae Ideal ae *, uni T(*, $ According o he above able, i can be oberved ha he opimal oal co and opimum bach ize of hi reearch are greaer han he ideal invenory model. 37

47 HATER 4 REWORK AFTER N YLES In hi ecion afer-n-cycle policy of reworking proce i decribed. Here, he reworking cycle occur afer he compleion of N cycle of regular producion. The invenory for hi policy i decribed, average invenory i calculaed, and he oal co funcion baed on hee invenorie i formulaed. 4. Second olicy: Rework afer N ycle Under hi policy, he defecive iem are accumulaed up o N cycle of normal producion, and afer which hey are reworked. A he producion in a cycle coninue, he finihed good are upplied o he cuomer. Inpu ycle ycle ycle 3 ycle N Finihed good o he cuomer Makeup Buffer Reworking age efecive for rework Scrap Figure 4.: Block diagram of he producion proce wih rework and crap for rework afer N cycle policy. 38

48 The yem i aeed wih a penaly co for horage a each cycle conain ome of he defecive iem, ill he rework i accomplihed. Figure 4. how he block diagram of he enire producion proce wih rework. Under hi policy, he defecive iem from each cycle are accumulaed unil compleion of N cycle of producion, afer which he defecive par are reproceed. The crap i deeced during he ime of rework and he makeup buffer i ued. 4. Invenory of Finihed roduc Figure 4. how he invenory buil-up for one cycle during he producion, a he defecive iem are eparaed. [(--] Invenory level h p d Time Figure 4.. Invenory of finihed good in one cycle According o Figure 4., he invenory of one cycle can be calculaed a I h( p d. (4. Since, h [ ( ] /, he producion upime p /, downime d h /, and he oal ime of he cycle i T ( /. 39

49 Hence, he average invenory I, in equaion (4. can be wrien a I ( ( [ ( ], o he average invenory for he enire period i given by I ( (. ( Invenory for Reworked Iem Under hi policy he rework i done afer accumulaion of he defecive hrough N cycle of producion and ome porion of defecive crapped. A decribed in haper 3, crap deecion and producion ake place in hree way; hey are decribed below and he invenorie of he reworked iem are calculaed. Alo a buffer i mainained in each cae o makeup he horage due o he crap producion a he ime of rework, and boh normal invenory during rework and buffer invenory are calculaed ogeher ae I: Scrap eeced before Rework In hi cae he crap i deeced a he beginning of he rework proce. The invenory for reworked good iem i hown in he Figure 4.3. The oal crap for one cycle found i α, a he proporion of crap i aumed 00α % of oal defecive produced. Tha i why, he oal reworked iem i evaluaed a ( α. From he Figure 4.3 he average invenory can be calculaed a follow: Here, p producion upime for a reworked lo ( α/, down ime d h/, where h ( ( α/. Therefore, he average invenory can be calculaed a I IR h ( pr dr ( ( α ( α ( ( α 40

50 ( α. (4.3 Invenory level α - h - pr dr Time α Figure 4.3: Invenory of finihed good during rework for ae I. A he boom of he Figure 4.3, i i hown ha he invenory for he makeup buffer of α quaniy i mainained unil producion of good iem i compleed from he defecive. The horage migh occur a ha ime due o he deecion of crap before rework, o he makeup buffer invenory i I IB α( α α pr. (4.4 Therefore, he oal aggregaed invenory during he rework proce for ae I can be evaluaed by adding equaion (4.3 and (4.4 a I IF I IR I IB ( α α( α 4

51 ( α ( α α. (4.5 Hence, he average invenory for he enire period i I ITF ( α ( α α ( α ( α α. ( ae II: Scrap eeced during Rework In hi cae crap i deeced when rework proce coninue and crap producion rae i aumed a /δ, where δ i he crap producion facor (0 δ. The oal crap produced for one cycle i α, a he proporion of crap i aumed a 00α%, of he oal defecive produced. Tha i why, he oal reworked iem are calculaed a ( α. The invenory buil-up for reworked good iem i hown in Figure 4.4. The average invenory can be calculaed a follow: here, he producion upime for he reworked lo i p pr p ( α/ αδ/ ( ααδ/, down ime d h/, where ( α ( α αδ h p α α αδ. (4.7 Therefore, he average invenory can be calculaed a I IIR h ( p dr ( α αδ ( α αδ α ( α αδ ( α α ( α αδ α. (4.8 4

52 A hown in Figure 4.4, he makeup buffer invenory of α uni i mainained unil he compleion of good iem and crap producion from he defecive, o he makeup buffer invenory i given by I IIB α ( α αδ α p. (4.9 /δ Invenory level α - p - h α α αδ pr dr Time p α Figure 4.4: Invenory of finihed good during rework for ae II. Therefore, he oal aggregaed invenory during he rework proce for ae II i found from equaion (4.8 and (4.9 a I IIF I IIR I IIB ( α ( α αδ α ( α αδ α 43

53 ( αδ ( α αδ α α. (4.0 Hence, he average invenory for he enire period i I IITF ( αδ ( α αδ α α ( αδ ( α αδ α α. ( ae III: Scrap eeced afer Rework, δ ae III i a pecial cae of ae II, where δ. A decribed in Secion , equaion (4. reduce o ( α α I IIITF α α. ( Special ae of ae II: δ 0 Thi i anoher pecial cae of ae II, where δ 0. Equaion (4. reduce o ( α I STF α α. (4.a 4.4 In-roce Invenory of Rejeced Maerial In he rework afer N cycle policy, ome defecive iem are produced in each cycle and are accumulaed unil he end of N cycle of producion. Figure 4.5 how he in-proce invenory of he rejeced maerial for N cycle. The in-proce invenory of rejeced maerial depend on he waiing ime for he enire bach of quaniy during he eup and proceing ime for one componen a N h cycle. The eup waiing ime can be given a T w N( N ( N ( N ( N (4.3 44

54 uaniy, N- N Time Figure 4.5. In-proce invenory of he rejeced maerial over N cycle for Second olicy. The waiing ime for proceing he bache can be calculaed a ( N ( N ( N 3 T wp... N [( N ( N ( N 3... ] N N( N N. (4.4 Hence, he oal waiing ime i found by adding equaion (4.3 and (4.4: T w N( N N Tw Twp. (4.5 The oal in-proce invenory of he rejeced maerial for he enire period can be evaluaed by accumulaing he in-proce invenorie for all lo: I WI N( N N ( N. ( Toal o for Second olicy In he rework afer N cycle policy, he rework proce occur in he (N h cycle afer compleion of N cycle of producion. To formulae he oal co funcion for hi 45

55 model, i i neceary o calculae he eup co for hi proce, in-proce invenory carrying co, reworked and buffer invenory carrying co, penaly co due o horage creaed by he defecive iem aken ou during normal producion, and proceing co. Thee co are decribed and calculaed in hi ecion Seup o Each producion faciliy need a eup for producing finihed good, which incur a co for eup. Hence, he eup co i found for he whole bach quaniy a K. (4.7 S In rework afer N cycle policy, anoher eup co i needed for rework a he rework i done afer compleion of normal producion proce. Hence, he eup co for rework proce for enire bach of defecive quaniy i K. (4.7a S d d 4.5. Invenory arrying o Four ype of invenorie are found in hi policy hey are finihed good invenory for N cycle, reworked finihed good invenory, invenory of rejeced iem during regular proce, and makeup buffer invenory. Thee invenorie are calculaed in previou ecion, o he invenory carrying co can be evaluaed by muliplying he uni carrying co and he average invenory of he cycle. By accumulaing differen invenorie and muliplying hem by uni invenory carrying co, he oal average invenory carrying co can be calculaed. The invenory carrying co can be found for ae I by uing he equaion (4., (4.6 and (4.6, for ae II by uing he equaion (4., (4. and (4.6, for ae III 46

56 47 and Special cae of ae II by uing he equaion (4., (4., (4.6 and (4.a repecively, a follow: ( H H I n n I ( ( α α α N w (, (4.8 ( H H I n n II ( ( αδ α αδ α α N w (, (4.9 ( H H I n n III ( α α α α N w (, and (4.0 ( H H I n n S ( α α α N w (. (4.0a enaly o The producion of defecive iem in every cycle reul in horage by iem, and hee horage are fulfilled afer he rework i compleed a he end of (N h cycle. For ha reaon, a penaly co i aeed in he oal co funcion. To calculae he penaly co, i i neceary o calculae he oal ime elaped in horage. The horage ime coni of producion runime and downime, which i he ame for N cycle. A he end of N h cycle he rework i performed i.e., in (N h cycle. Hence, he oal horage

57 ime i differen in hi cycle a he accumulaed defecive iem up o N cycle are reworked here. Shorage ime for hee wo cae are calculaed below. The horage ime up o N cycle i calculaed from Figure 4. by adding he producion upime and down ime for each cycle, ( p d p h p / ( /. Therefore, he oal horage ime up o N cycle, T n, i given by T n [( N ( N... ] ( ( ( N, (4. where N /. The horage ime for (N h cycle can be calculaed in hree way, a rework ime varie due o crap producion. Alo hey coni of rework producion ime and conumpion ime, o hey are calculaed for differen cae, repecively, a follow: T ( n h IS ( α N ( ( α N ( α N ( α, (4. ( n h ( α αδ N ( α ( α αδ N TIIS α αδ, ( α N ( α, and (4.3 ( n h ( n h ( α N TIIIS TSS ( α. (4.4 Hence, he oal horage ime can be evaluaed for differen cae by uing equaion (4., (4., (4.3, and (4.4 a T IS T IIS T IIIS T SS T n S T ( n h IS ( N ( / ( α [ N N 3 α ]/, (4.5 The oal penaly co over a planning period for differen cae can be obained a 48

58 I II III S p [ N( 3 α ] ( p 3 α N. (4.6 ( Scrap Handling o Scrap need a co for handling he amoun of α uni crap during producion. I can be evaluaed by muliplying he uni crap handling co, and he amoun of crap produced in enire period a K T α. ( Toal Syem o The producion of oal bach quaniy, i divided ino N cycle o minimize he invenory carrying co, o i i neceary o find he oal co for he enire producion period. Hence, i i required o form he oal co funcion wih repec o number of cycle, N ( /. Thu, he oal yem co of he producion and rework proce over N cycle for he econd policy wih differen cae of crap producion can be evaluaed from equaion (4.7, (4.8, (4.9, (4.0, (4.0a, (4.6, and (4.7. Since he oal co i a funcion of N ( /, i can be wrien a α H n T I ( N N dd ( ( α N N ( α w ( N α p 3 α N, (4.8 ( T ( N N II d d α H n ( N N 49

59 50 ( ( ( αδ α αδ α N N p w α 3 ( (, (4.9 N H N N N T n d d III α ( ( N w ( ( α α α N p α 3 (, and (4.30 N H N N N T n d d S α ( ( N w ( ( α α α N p α 3 (. (4.3 which need o be minimized. For he minimizaion purpoe, i i neceary o find he naure of hee funcion and i i found ha he above funcion are convex funcion [ee Appendix ]. 4.6 Opimaliy I can be eaily hown ha T(N i a convex funcion in N (ee Appendix. Hence, an opimum number of cycle N*, can be calculaed from N N T / ( 0, which yield ( ( ( α α N H N N N T n I

60 5 0 ( 3 ( ( ( N p w α α α, which yield H N n ( ( ( α α α α ( ( 3 ( α w p, from which w p n I H N α θ α ( 3 ( ( ( * (4.3 where ( ( ( α α α θ. Similarly, from equaion (4.9, (3.30, and (3.3 he opimum number of cycle N*, can be evaluaed, repecively, a w p n II H N α ϕ α ( 3 ( ( ( *, (4.33 where ( ( ( ( αδ α αδ α ϕ, w p n III H N α ψ α ( 3 ( ( ( *, (4.34 where ( ( ( α α ψ, and w p n S H N α φ α ( 3 ( ( ( *, (4.35

61 5 where ( ( α α α φ. From he above equaion, he opimum bach quaniy for all cae can be evaluaed a i * /N i * (where i I, II, III and S. ( Special ae If no defecive and crap are produced during he producion, i.e., α 0, hen equaion (4.3, (4.33, (4.34, and (4.35 reduce o n H N / ( *. (4.37 Since * /N*, he opimal bach quaniy reduce o he claical model for he opimum bach quaniy which i / ( * H n. (4.38 When no crap i produced during he rework, i.e., α 0, equaion (4.3, (4.33, (4.34, and (4.35 reduce o w p n H N ( 3 ( ( ( *. (4.39 Hence, p n w H 3 ( ( ( ] [( *. (4.40 The derived oluion above are validaed hrough numerical example in he following ecion.

62 4.7 Numerical Example Uing he ame 0.05, α 0., and δ 0.07 N i obained uing equaion (4.36, (4.37, (4.38, and (4.39 and hown in a abular form. For $.00/min (equivalenly $,68,000/year, 50 min/eup (equivalenly year/eup, d $.00/min, d 50 min, 300 uni/year, 550 uni/year, H n $8/ uni/year, p $77/ uni/year, and w $88.5/ uni/year, he following value can be obained by uing equaion (4.6, (4.7, (4.8, (4.9, (4.3, (4.33, (4.34, (4.35, (4.36, (4.37, and (4.38, repecively. Table 4.: Reul of he numerical example for econd policy. ae arameer ae I ae II ae III Special ae Ideal ae N* *, uni T(N*, $ According o Table 4. i can be concluded ha he value of *, and T(N* for ideal invenory model i le han for he evaluaed invenory model. 53

63 HATER 5 OERATIONAL SHEULE I i neceary o implemen evaluaed model wih numerical daa for pracical iuaion of a producion proce. Thee operaional chedule deermine he opimum order quaniy, ime of producion and rework, quaniy of produced iem, quaniy of crap and buffer, ec., in a producion proce wih rework and crap. Thi chaper deal wih he operaional chedule of he model of hi reearch for enire producion proce. 5. Operaional Schedule for Fir olicy: Wihin ycle Rework Under hi policy, he rework i done wihin he ame cycle in which he defecive are produced, ogeher wih defecive iem produced and deeced before, during and afer he rework proce. epending on he everiy of defec, hey are reworked or crapped. To evaluae he operaional chedule, i i neceary o calculae he ime of producion and rework, quaniy of produced good iem, defecive iem and crap, opimum order quaniy, ec. The following value of parameer are ued o calculae he operaional chedule: 300 uni/year, 550 uni/year, $7/uni, S $50/bach, H $50/uni/year, $5/uni, 0.05 and α 0.0 and δ 0.07 (meaning i aumed a 0.05, i.e., 5% defec, and α i aumed a 0.0, i.e., 0% crap from he defecive. The operaional chedule for differen cae i calculaed a follow: 5.. Operaional Schedule for ae I of Fir olicy ae I of he fir policy ae ha, he crap can be deeced before he rework proce ar o produce good iem from he defecive. Uing he above parameric 54

64 value, he opimum bach quaniy for ae I i found by uing equaion (3.7 a * I 37 uni/year. From hi poin he following value are evaluaed a * I / 37/ year, he number of defecive produced i * I uni and he number of crap i α * I , which i he buffer quaniy a well. Hence, ( - - / * I ( / uni/year and ( - / * I ( 300/ uni/year. roducion wih defecive and crap Rework wihou crap 0.4 Invenory Level L A O F E onumpion ae I Scrap i deeced before rework. No crap during he rework. Rework ime normal proceing ime. roducion wih defecive and crap G Time 3 Y 0. Figure 5.: Operaional chedule for ae I of Fir olicy. Again, he rework ime i (-α I * / ( / year and, from equaion (3.a, 55

65 [ α ( α / ] [ ( / 550] year. Hence, he oal cycle ime i T year. Uing he above value, he operaional chedule for ae I of fir policy i graphically repreened in Figure 5.. The relaed co involved in operaional chedule of ae I are calculaed uing he above value and equaion (3., (3.3, (3.6, (3.9, (3.0b, (3. and (3. a follow: Seup co $40.8/eup, invenory carrying co $394./year, makeup buffer invenory carrying co $5.6/year, proceing co $00, proceing co for rework $6, crap handling co $.87 and he opimum oal co $ The opimum oal co found here i ame a he opimum oal co evaluaed in numerical example for ae I of wihin cycle rework policy. 5.. Operaional Schedule for ae II of Fir olicy In ae II of fir policy, crap i deeced during he rework proce, and i i aumed ha he ime o produce a reworking iem a crap i le han he ime o produce a good iem. Hence, anoher parameer i conidered in hi cae which i known a crap producion facor, δ Uing equaion (3.8 i i found ha II * 38 uni/year. Hence, II */ 38/ year, 5 uni/year and 7 uni/year, number of defecive produced i II * uni and number of crap i α II * 0.4 are calculaed a for ae I. The rework ime i ( α II */ ( / year, 3 αδ II */ and uing equaion (3.6 4 i found a 56

66 ( Hence, T year year. The relaed co of operaional chedule for ae II are calculaed uing he above value and equaion (3., (3.4, (3.7, (3.9, (3.0b, (3. and (3.3 a follow: Seup co $398.7/eup, invenory carrying co $396.54/year, makeup buffer invenory carrying co $0.0007/year, proceing co $00, proceing co for rework $6, crap handling co $.87 and he opimum oal co $ The opimum oal co i mached wih he opimum oal co calculaed in numerical example for ae II of wihin cycle rework policy. Invenory Level roducion wih defecive Rework wih crap onumpion 4 ae II: δ Scrap i deeced during rework. Scrap produced a rae of /δ Time 0.3 Figure 5.: Operaional chedule for ae II of Fir olicy. 57

67 Figure 5. repreen he operaion chedule for ae II of fir policy and he calculaed parameer of operaional chedule for fir policy are hown in Table 5.. Table 5.: alculaion of operaional chedule for wihin cycle rework policy. arameer ae I ae II Uni * uni/year * uni/year * uni/year * uni α * uni year year year year T year K $/eup I $ B $ K $ K RB $ K T $ T( * $ 5. Operaional Schedule for Second olicy: Rework afer N ycle In hi policy, he defecive iem are accumulaed up o N cycle of normal producion, afer which hey are reworked. A he producion in a cycle coninue, he finihed good are upplied o he cuomer. For he econd policy, he operaional chedule i compued uing he following value: 300 uni/year, 550 uni/year, 0.05, α 0., δ 0.07, $.00/min (equivalenly $,68,000/year, 50 min/eup (equivalenly year/eup, d $.00/min, d 50 min, H n $8/ uni/year, p $77/ uni/year, and w $88.5/ uni/year. The operaional chedule for differen cae i calculaed below. 58

68 5.. Operaional Schedule for ae I of Second olicy To calculae he operaional chedule of ae I (crap deeced before rework of he econd policy, i i fir required o calculae he opimum number of cycle, N * I. * Applying above value in equaion (4.3, he opimum number of cycle i found a N I 0.59, and from equaion (4.36, he opimum quaniy of iem produced in a cycle i * I uni. Hence, he upime p * I/N * I 8/( , quaniy produced in a cycle * I/N I * 3 uni and downime i d [( ] * * I/N I [550( ]8/( year. The quaniy i remained afer he end of he producion and conumpion of he fir cycle i [550 ( ] 3/550 uni. The number of defecive produced in hi cycle i 0.5 uni. The oal number of defecive iem produced in cycle i According o econd policy, he rework i compleed in (N h cycle, here i i ( h h cycle. Scrap produced i α uni and number of defecive iem for rework remain uni. According o Figure 4.3 he rework ime can be calculaed a pr.6/[ ] year and conumpion ime for rework i dr.6/ year. Hence, he oal ime of he enire cycle i T N * I ( p d ( pr pr 0.08 year. Figure 5.3 repreen he operaional chedule for ae I of econd policy. The co relaed o hi operaional chedule are calculaed uing he equaion (4.7, (4.7a, (4.8, (4.6, (4.7, and (4.8, repecively, a follow: Seup co for enire bach, K S $59.50/eup, eup co for rework, K S $50/eup, invenory carrying co, I I $05. 66/year, penaly co, I $.7, crap handling co, K T $.4 and he opimum oal co, T(N * I $89.8. The 59

69 opimum oal co found in hi ecion i equal o he opimum oal co evaluaed in numerical example for ae I of rework afer N cycle policy, becaue he ame value of he parameer are ued o calculae. 3 roducion wih defecive and crap and conumpion for N cycle Reworking age wihou crap Invenory Level N N N Time Figure 5.3: Operaional chedule for ae I of Second olicy 5.. Operaional Schedule for ae II of Second olicy In hi ecion he operaional chedule for ae II (crap deeced during rework of econd policy i calculaed by uing he daa decribed before. Uing equaion (4.33, he opimum number of cycle calculaed i N * II 0.58, and from equaion (4.36 he opimum bach quaniy i calculaed a * II 8 uni/year. Some value uch a he quaniy of iem produced in a cycle 3 uni, p 0.005, d year, number of 60

70 defecive produced in hi cycle 0.5 uni and defecive iem produced in cycle * II, number of crap produced α * II 0.4 uni and number of defecive iem remained for rework.6 uni a calculaed previouly. According o Figure 4.4 he rework ime can be calculaed a pr p ( - α * II /N * II αδ * II /N * II year and conumpion ime for rework i dr ( * II α * II p / year. Hence, he oal ime of he enire cycle i T N * II ( p d ( pr r dr.58( ( year. The graphical repreenaion of he operaional chedule for ae II of econd policy i hown in Figure 5.4. roducion wih defecive and conumpion for N cycle Reworking age wih crap Invenory Level N N N Time Figure 5.4: Operaional chedule for ae II of Second olicy. The co involved in hi operaional chedule for ae II are calculaed uing he equaion (4.7, (4.7a, (4.9, (4.6, (4.7, and (4.9, repecively, below: 6

71 Seup co for enire bach, K S $59.00/eup, eup co for rework, K S $50/eup, invenory carrying co, I II $05.48/year, penaly co, II $.59, crap handling co, K T $.4 and he opimum oal co, T(N II * $ The opimum oal co found in hi ecion i equal o he opimum oal co evaluaed in numerical example for ae II of rework afer N cycle policy where he ame parameric value are ued for numerical compuaion of he equaion. The calculaed value of he operaional chedule for econd policy are hown in Table 5.. Table 5.: alculaion of operaional chedule for rework afer N cycle policy. arameer ae I ae II Uni * uni/year N * * uni α * uni p year d year pr year dr year T year K S $/eup K S $ I $.7.59 $ K T.4.4 $ T(N * $ In he operaional chedule, ae III and Special ae are no hown a hey are he pecial cae of ae II. 6

72 HATER 6 SENSITIVITY ANALYSIS The oal co funcion are he real oluion in which he model parameer (bach quaniy, proporion of defecive, proporion of crap are aumed o be aic value. I i reaonable o udy he eniiviy, i.e., he effec of making change in he model parameer over a given opimum oluion. I i imporan o find he effec on differen yem performance meaure, uch a co funcion, invenory yem, ec. For hi purpoe, eniiviy analye of variou yem parameer for he model of hi reearch are required o oberve wheher, (a The curren oluion remain unchanged, (b The curren oluion become ub-opimal, (c The curren oluion become infeaible, ec. In hi reearch, wo alernaive model wih hree differen cae are developed for he opimal producion lo ize wih allowance for rework of defecive iem and crap. A eniiviy analyi i carried ou for boh policie o deermine how he oal co of he yem and he opimum bach quaniy are affeced due o he change of defecive rae, percenage of crap α, boh α and, crap producion facor δ for ae II, eup co S, and crap handling co. 6. Effec of on * and T(* The proporion of he defecive i a major parameer in developing he model. Boh he bach quaniy and he oal co are affeced due o variaion of he proporion of he defecive. Mahemaically, for ae I of he fir policy, 63

73 d * S ( H ξ α ( α, (6. 3 / d ( α ν 3 where ξ α( α ( α 4α α α α α, and ν [ H(( ( α ( 3 α α α ( α ]. The rae and he direcion of change of wih repec o depend upon he parameric value ued in numerical example. Hence, d * /d > 0 hold on he real value of. According o equaion (6., d * /d Ω if [0.0, 0.593] and d * /d Ω if [0. 593, ] and he effec of over he defecive proporion i hown in Figure 6.. From he Figure 6., i can be oberved ha he change over * i occurring lowly up o poin [0.55, 700] due o change of, and afer ha * increae wih an increae of. d * /d roporion of defecive, % Figure 6.: Effec of proporion of defecive on bach ize. Again, he effec of proporion of he defecive iem over he T( * can be hown mahemaically for ae I of fir policy by equaion (6. a follow: 64

74 * T( ( α α * * H ( α [( α 3 ( α 4α α α α α ] (6. Uing equaion (6., i can be found ha dt( * /d Φ, if [0.0, 0.8] and dt( * /d Φ if [0. 8, ] he effec of over he defecive proporion i hown in Figure 6.. dt( * /d roporion of defecive, % Figure 6.: Effec of proporion of defecive on oal co. The effec of proporion of defecive i udied by changing he value of from 0.05 o 0.3. I i oberved ha he oal co, T( * and opimal bach quaniy, * are direcly relaed wih defecive rae and heir value increae a increae. Thi udy i hown in Table 6. and 6. for all cae for boh policie. I i oberved ha, in econd policy, he opimum bach ize increae up o a cerain level wih he increae of he proporion of defecive. Afer ha i ar o decreae, which indicae ha econd policy i more eniive han fir policy. 65

75 Table 6.: Effec of over T( * and * for ae I and ae II. ASE I ASE II Fir olicy Second olicy Fir olicy Second olicy * T( * * T( * * T( * * T( * Table 6.: Effec of over T( * and * for ae III and Special ae. ASE III SEIALASE Fir olicy Second olicy Fir olicy Second olicy * T( * * T( * * T( * * T( * Effec of α on * and T(* Anoher imporan parameer in developing he model i he proporion of he crap, α wih repec o defecive iem produced. The effec of α on opimal bach quaniy, * i repreened mahemaically in equaion (6.3. S( [ [( ( * α α α α α H ν 3 / ] ( α (6.3 where ν [ H(( ( α ( 3 α α α ( α ]. ] 66

76 The rae and direcion of change of *, wih repec o α, depend on he value of he parameer ued in numerical example for ae I of wihin cycle rework policy. The effec of α on he oal co, T( * can be repreened mahemaically by equaion (6.4 a * T( α H [( α * * ( α α ].(6.4 ( α Hence, uing equaion (6.3 and (6.4 and α [0.0, 0. 7], he effec of α on * and T( * i hown in Figure 6.3. According o he Figure 6.3 i can be oberved ha he rae of change in T( * wih repec o α value i very mall. d * hange on * hange on T( * dt( * dα dα roporion of crap, α % Figure 6.3: Effec of proporion of c rap on bach ize and oal co. The effec of crap rae i udied by changing α value over he range from 0.05 o 0.7 and i i oberved ha change of opimum bach quaniy * i inverely proporional o he change of α for he ae I of fir policy. In econd policy, * increae, bu he oal co, T( * decreae omewha. The value of α are varied from 67

77 0. o 0.8 and he change in * and T( * for all cae of boh policie are hown in Table 6.3 and Table 6.4 repecively. Table 6.3: Effec of α over T( * and * for ae I and ae II. ASE I ASE II α Fir olicy Second olicy Fir olicy Second olicy * T( * * T( * * T( * * T( * Table 6.4: Effec of α over T( * and * for ae III and Special ae. ASE III SEIALASE α Fir olicy Second olicy Fir olicy Second olicy * T( * * T( * * T( * * T( * Effec of Boh α and on * and T(* When he wo imporan parameer α and are boh changed, he effec on opimum bach quaniy can be repreened by equaion (6.5 a * 3 α S [ H [( α ( α α ] ( α ] ( α 3 / 4ν 5 / 3 [ H {( α α ( α 4α α α α α } 68

78 S ( α 3 / 3 α( α ] [ H 3 / { ( α ν ( α 3 α 3α α } ( α ], (6.5 where ν [ H(( ( α ( 3 α α α ( α ]. Alo he effec over oal co due o variaion of α and can be hown mahemaically by equaion (6.6 a * T( α * * H ( α [( α ( 3 α 3α α 3α α ]. (6.6 Uing he above equaion and α [0.0, 0. 5], and [0.0, 0. 5], he effec i hown in Figure 6.4 and Figure 6.5, repecively. * α α Figure 6.4: Effec of boh proporion of crap and proporion of defecive on bach ize. 69

79 * T( α α Figure 6.5: Effec of boh proporion of crap and proporion of defecive on oal co. A udy on he effec of changing boh α and over T( * and * i repreened in Table 6.5 and 6.6 for all cae, which how ha if boh α and increae imulaneouly, he value of T( * and * alo increae. In hi udy, he ame parameric value are ued wih he variaion of α from 0. o 0.8 and from 0.05 o 0.7 and all he value are a hown in he Table 6.5 and 6.6. Table 6.5: Effec of boh α and on T( * and * for ae I and ae II. ASE I ASE II α Fir olicy Second olicy Fir olicy Second olicy * T( * * T( * * T( * * T( *