A multi-item production lot size inventory model with cycle dependent parameters

Transcription

1 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE A mul-em producon lo ze nvenory model wh cycle dependen parameer Zad. Balkh, Abdelazz Foul Abrac-In h paper, a mul-em producon nvenory model condered whn a gven me horzon ha con of dfferen me perod. For each produc, producon, demand, and deeroraon rae n each perod are known. horage for each produc allowed bu compleely backlogged. he objecve o fnd he opmal producon and rearng me for each produc n each perod o ha he overall oal nvenory co for all produc mnmzed. In h paper, a formulaon of he problem developed and opmzaon echnque are performed o how unquene and global opmaly of he oluon. Keyword - Mul-em producon, Invenory, Varyng demand, Deeroraon, Opmaly. I. INRODUCION Invenory known a maeral, commode, produc,..ec, whch are uually carred ou n ock n order o be conumed or benefed from when needed. In fac, mo of economc, radng, manufacurng, admnrave, ec, yem regardle of ze, need o deal wh own Invenory Conrol yem. Keepng nvenory n ore ha own varou co whch may, omeme, be more han he value of he commody beng carred ou n ore. A example, nuclear and bologcal weapon, blood n blood bank, and ome knd of enve medcaon. However, any nvenory yem mu anwer he followng wo man queon. () How much o order or o produce for each nvenory cycle?. () When o order or o produce a new quany?. Anwerng hee wo queon for ceran nvenory yem lead o he o called Opmal Invenory Polce whch Zad. Balkh, wh he Kng aud Unvery, Ryadh 45, aud Araba ( phone : , e-mal: [email protected]) Abdelazz Foul wh he Kng aud Unvery, Ryadh 45, aud Araba ( e-mal: [email protected]) Mnmze he oal Invenory Co of h yem. I expeced ha all yem, n whch conrollng and managng nvenory an mporan facor ha ha grea effec on performance, can grealy benef from h reearch reul o a o mnmze her relevan nvenory co operaon. In fac, many clacal nvenory model concern wh ngle em. Among hee are Reh, Fredman, and Barboa [9] who condered a clacal lo ze nvenory model wh lnearly ncreang demand. Hong, andropory, and Hayya [5] condered an nvenory model n whch he producon rae unform and fne where he nroduced hree producon polce for lnearly ncreang demand. A new nvenory model n whch produc deerorae a a conan rae and n whch demand, producon rae are allowed o vary wh me ha been nroduced by Balkh and Benkherouf []. In h model, an opmal producon polcy ha mnmze he oal relevan co eablhed.ubequenly, Balkh [], [3], [4], [5], [6], [8], [9], and Balkh, Goyal, and Gr [7] have nroduced everal nvenory model n each of whch, he demand, producon, and deeroraon rae are arbrary funcon of me, and n ome of whch, horage are allowed bu are compleely backlogged. In each of he la menoned even paper, cloed form of he oal nvenory co wa eablhed, a oluon procedure wa nroduced and he condon ha guaranee he opmaly of he oluon for he condered nvenory yem were nroduced. hough o many paper have deal wh ngle em opmal nvenory polcy and hough he leraure concerned wh mulem are pare, he analy of mul-em opmal nvenory polce,, almo, parallel o ha of ngle em.he mul-em nvenory clacal nvenory model under reource conran are avalable n he well known book of Hadley and Whn [4] and n Nador [8]. Ben-Daya and Raouf [] have developed an Iue, Volume 3, 9 94

2 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE approach for more realc and general ngle perod for mul-em wh budgeary and floor,or helf pace conran, where he demand of em follow a unform probably drbuon ubjec o he rercon on avalable pace and budge. Bhaacharya [] ha uded wo-em nvenory model for deerorang em. enard and Roy [7] have ued dfferen approache for he deermnaon of opmal nvenory polce baed on he noon of effcen polcy and exended h noon o mul-em nvenory conrol by defnng he concep of famly and aggregae em. Kar, Bhuna, and Ma [6] obaned ome nereng reul abou mul deerorang em wh conran pace and nvemen. Roenbla [] ha dcued mulem nvenory yem wh budgeary conran comparon beween he agrangan and he fxed cycle approach, wherea, Roenbla and Rohblum [] have uded a ngle reource capacy where h capacy wa reaed a a decon varable. Recenly, Balkh and Foul [] have appled a mul-em producon nvenory model o he aud Bac Indure Corporaon (ABIC), whch one of he world leadng manufacurer of ferlzer, plac, chemcal,and meal,n aud Araba.For more deal abou mul-em nvenory yem, he reader are adved o conul he urvey of Yaemn and Erenguc [3] and he reference heren. Our man concern n h udy o fnd he opmal producon and rearng me for each produc n each perod o ha he overall oal nvenory co for all produc mnmzed. In h paper, a formulaon of he problem developed and opmzaon echnque are performed o how unquene and global opmaly of he oluon. Opmal number of un o be produced from each of he produc are deermned by a mple lnear program. Havng found hee opmal number of un, we eablh opmal nvenory polce for he dfferen produc, whch mean ha we deermne he opmal oppng and rearng producon me for each produced em o ha he oal relevan nvenory co of all em mnmum. he paper organzed a follow. Fr, we nroduce our aumpon and noaon, hen we buld he mahemacal model of he underlyng problem. he oluon procedure of he developed model eablhed n econ 4, and he opmaly of he obaned oluon proved n econ 5. Fnally we nroduce a concluon n whch we ummarze he man reul of he paper a well a our propoal for furher reearch. II. AUMPION AND NOAION Our aumpon and noaon for our model are a follow:. m dfferen em are produced and held n ock over a known and fne plannng horzon of H un long whch dvde no n dfferen cycle.. he em are ubjec o deeroraon when hey are effecvely n ock and here no repar or replacemen of deeroraed em. 3. he demand, producon and deeroraon rae of em n cycle j are em and cycle dependen, and are denoed by D, P and θ repecvely. 4. horage are allowed for all em, bu are compleely backlogged. 5. he followng noaon are ued n he equel, where =,, m and j =,, n I () Invenory level of em n cycle j a me. me a whch he nvenory level of em n cycle j reache maxmum. me a whch he nvenory level of em n cycle j ar o fall below zero and horage ar o accumulae. Begnnng of cycle j for em, wh o = and n =H. me a whch he horage for em reache maxmum n cycle j. Invenory holdng co n cycle j per un of h em per un of me. b horage co n cycle j per un of em per un of me. k e up co for em n cycle j. c Un producon co of em n cycle j. III. MODE FORMUAION For each em ( =,, m) and each cycle me j (j =,, n), he yem operae a follow. he producon ar a me - o buld up he nvenory level a a rae P - D - θ I () unl me where he producon op. hen he ock level deplee a a rae D - θ I () unl reache zero a me where horage ar o accumulae wh rae -D up o me, afer whch he producon reared wh rae P - D unl me o fulfll boh he horage and he demand. A ypcal behavor of he yem hown n Fg.. Iue, Volume 3, 9 95

3 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE Invenory level me = - H= n Fg. he varaon of he nvenory yem n he gven perod Before wrng he mahemacal formulaon of he problem, we noe ha he jh cycle can be dvded no four nerval. ha, - <, <, <, <. For em and cycle j, he nvenory level I () governed by he followng dfferenal equaon : d I ()/d = P - D - θ I () - <, () wh nal condon I ( - ) =, d I ()/d = - D - θ I () < () wh endng condon I ( ) =, d I ()/d = - D < (3) wh nal condon I ( ) =, and d I ()/d = P - D - θ I () < (4) wh endng condon I ( ) =, he oluon o (), (), (3), and (4) gven by: I () = repecvely.inegrang he rgh hand de of he la four equaon, we oban : I () = ; I () = ; (5) (6) I () = ; (7) I () = (8) Now, he amoun of em beng held n ock n perod [ gven by I ( I () = I () = - I () = - Hence, he holdng co. mlarly he amoun of em beng held n ock n perod [ gven by Iue, Volume 3, 9 96

4 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE I ( he oal ne nvenory co for all em n he gven me horzon of n cycle gven by W = And hence he holdng co. he amoun of horage of em n perod [ gven by I ( hu, our problem o mnmze W a a funcon of -,,,, and ubjec o he followng conran : - < < < < () = () Hence he horage co n perod [. mlarly, he horage co n perod [ where I ( Fnally, he number of un produced from em n cycle j equal o =,, m. and j =,, n Conran () a naural conran whch mu be afed, oherwe he whole problem wll be meanngful. Conran () ay ha he nvenory level gven by (5), and (6) a = are equal. mlarly, conran (3) ay ha he nvenory level gven by (6), and (7) a = are equal. hu, our problem, call (Q), : Mnmze W = (Q) Hence, he producon co of em n cycle j For convenence, le u rewre () and (3) a equaly conran : gven by hu he oal ne nvenory co of em n cycle j, ay gven by = (4) = (5) IV. OUION PROCEDURE Conder problem (Q) where conran () gnored and uppoe ha he number of cycle n fxed. e call he new problem a (P). Clearly problem (P) a nonlnear program wh equaly Iue, Volume 3, 9 97

5 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE conran. herefore he oluon procedure ued he agrangean echnque. he agrangean funcon for problem (P) gven by = W + Where = for =,, m, and j =,, n-.. For =,, m, and j =,, n. he fr order neceary condon for havng a mnmum are : for =,, m and j =,,.,n-(ee model aumpon). Ung () and (), we can expre equaon (8) and () a : Condon (7) can be wren explcly a : From whch, we ge (4) (4) mple ha (9a) and (3a) can be expreed a : e u denoe by (), he nonlnear yem conng of equaon (9b),,(),(), (3b), and (4). Nex we how ha any oluon ha afe yem () afe conran (). heorem. Any oluon ha afe yem () afe conran (). Proof : From (4), we have: Iue, Volume 3, 9 98

6 whch mple ha (a). Now, for j =, we have herefore (a) rue for j=. By nducon on j, we can how ha (a) hold for any j.from (5), we have: whch mple ha (b). ubung (4) no (9a) and recallng (a), we ge : (c) Bu, he relaon need no o be condered nce he correpondng mulpler equal zero a an mplcaon of Kuhn-ucker opmaly condon. Combnng (a), (b), and (c), we ge: - < < < <. h complee he proof of he heorem. A a conequence of heorem, any oluon o yem () a feable oluon o problem (P). V. OPIMAIY OF HE OUION In h econ, we derve condon ha guaranee he exence, unquene, and global opmaly of oluon o problem (Q). For ha purpoe, we fr eablh uffcen condon under whch he Hean marx of he agrangean funcon pove defne a any feable oluon of (P). o compue he Hean marx of we conder he followng block marce : Afer ome calculaon, we can ealy how ha he Hean marx ha he followng form : n n n j INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE Iue, Volume 3, 9 99

7 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE By Balkh and Benkherouf [, heorem ], and ewar [], he above marx pove defne f (5a), and +, j=,3.n (5b) and j=,,.n (6) j=,,.n- (7a) (7b) (3) (Here we noe ha (3) (3), o no need for (3)). From (9a) : j=,,.n- (8) Recallng (4) and (5), we have : hu (6) For =,, m, and j =,3, n, from (6) we have: Combnng (3) wh (3), we oban : (3) (33) = (P D )(h + c θ )e θ (9) (33) a gnfcan relaon beween and. However, we mu ake whch afe (3) o ha all econd order condon of opmaly are fulflled. From (), we have Hence (5a) (3) hu (7a) and (5b) whch alway afed. mlarly, we can ealy verfy ha (7b) alway afed. Fnally, from (3) we have Iue, Volume 3, 9

8 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE Recallng (4), and replacng j by j+ n (9), we oban + = c + (P + o (8) D + )θ + e θ +( + ) + h + (P + D + )θ + e θ +( + ) = hen we have he followng mporan reul : emma :, for j=,,.n- and,for j=,,.n-. (34) Proof : nce for j= for j=. From (4) and wh j=, we have whch alway hold nce he holdng co uually le han em producon co. hu we have he followng reul: heorem. Any oluon for whch (3) hold a mnmzng oluon for problem (P). emma. All urnng pon are funcon of Proof : Gven ha hen from (4), a funcon of, whch n urn mple ha (recall (9a)) a funcon of and ha (recall(5)) a funcon of. Now for j= and by he relaon (4), a funcon of and herefore a funcon of. From (4), a funcon of and from (9a), a funcon of and hence a funcon of. ubung n (5), we ge ha a funcon of. Repeang he ame proce, we oban ha all varable are funcon of. h complee he proof of he lemma. Now le Hence. Alo from (9a) and (4) wh j=, we have h D (. )e θ ( ) + c D ( )e θ ( ) = b D ( ) < Fnally, from() wh j=, we have:. hu (34) hold for ome k and by nducon we can hen how ha (34) hold for k+. h complee he proof of he lemma. Corollary : All urnng pon are ncreang funcon of and of each oher. Proof : We have hown n lemma ha he urnng pon -,,,, and are funcon of each oher and ha all of hem are funcon of. Hence, by he chan rule of dfferenaon and he reul of lemma, we can conclude ha all hee urnng pon are ncreang funcon of and ha an ncreang funcon of an ncreang funcon of, and an ncreang. Iue, Volume 3, 9

9 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE funcon of have :. h o, nce for example, we Recallng relaon (34), we have. ha an ncreang funcon of and ha ake a negave value for =. h mple ha equaon (35) ha a unque oluon whch a mnmzng oluon by heorem. hu problem (P) ha a unque and global opmal oluon. h complee he proof of he heorem. Hence, all urnng pon are ncreang funcon of and of each oher. h complee he proof of he corollary. heorem 3 : Under condon (3), problem (P) ha a unque and global opmal oluon. Proof: nce all urnng pon are funcon of, we conclude ha f ha been choen adequaely, hen all oher urnng pon are alo choen adequaely and we hen mu have. Now, le u conder arbrary arng pon. If near he correc value, hen wll be near H. However, for any choce of and for any em, we alway mu have Nex, we how ha problem (P) ha a unque and global opmal oluon for any value of n. For more deal, ee Eme [3]. Fr, recall ha he whole yem depend on whch o be deermned correcly for any value of n. he followng reul gve u an ngh abou uch a deermnaon. Before ha, le u conder wo dfferen chedule wh he ame arng and fnhng pon, ay : chedule : ( and chedule : ( wh and. wh emma 3: he urnng pon of chedule le beween he urnng pon of chedule. ha wh (Recall ()). Recallng ha all urnng pon are funcon of, and ha f = mple (from(4)) ha whch n urn mple (from(9a)) ha and from (5). By nducon, we can ealy how ha f =, hen. Now, le hen F() = -H and Proof : By corollary, f we reduce o, hen all oher urnng pon are o be reduced. Now, uppoe our concluon fal for ome value k of j and for one nequaly, whle all oher nequale of (37) hold for j=k. ha uppoe we, for nance, have. hen, f we pa o he end pon, we oban, whch a conradcon. If he concluon fal for wo nequale, ay and, hen whch alo a conradcon wh (37). Repeang he ame argumen, we reach he dered reul. h complee he proof of he emma. A an mporan corollary of he prevou lemma he followng : Corollary : If condon (3) hold hen he quany Iue, Volume 3, 9

10 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE a decreang funcon of n. Proof : e j= n (9). he we have (Recall (4)). Hence E an ncreang funcon of.now, conder he wo chedule and a defned above. If we ncreae n by, hen by emma 3, we have whch mple ha E(n+, ) E(n, ). nce E an ncreang funcon of he la nequaly mean ha E a decreang funcon of n. h complee he proof of he corollary. Our la reul n h paper he followng : heorem 4 : Under condon (3), he underlyng nvenory yem(q) ha a unque and global opmal oluon. Proof : A a drec conequence of all above reul, we ar wh a uable value of for n=. If we ncreae n by, hen E would decreae. uch decreae of E hall op afer choong new value of le han he prevou one. Connung he procedure n h manner, we hall evenually reach o a value of n, ay n *, a whch he funcon E ar o ncreae. hen he opmal value of n n * -. h opmal value of n wh he correpondng opmal value of -,,,, and ay and are our unque and global opmal oluon for problem (Q). h complee he proof of he heorem. VI. CONCUION In h paper, we have condered a general mulem producon lo ze nvenory problem for a gven fne me horzon of H un long. he me horzon dvded no n dfferen cycle n each of whch a number of m em are produced. We have bul an nvenory model wh he objecve of mnmzng he overall oal relaed nvenory co. hen we have nroduced a oluon procedure by whch we could deermne he opmal oppng and rearng producon me for each em n each cycle n he gven me horzon when horage are allowed bu are compleely backordered. hen, que mple and feable uffcen condon ha guaranee he unquene and global opmaly of he obaned oluon are eablhed.uch opmal oluon can lead o opmal nvenory polce for he dfferen produc. Furher reearch may nclude he pobly of havng ome parameer of uch yem ncludng he co parameer a known funcon of me or a known probably drbuon. ACKNOWEDGMEN he auhor would lke o expre her hank o he ABIC Company for fnancal uppor of h reearch, and o he Reearch Cener n he College of cence n Kng aud Unvery for role n h uppor. REFERENCE [] Z. Balkh, On he global opmal oluon o an negrang nvenory yem wh general me varyng demand, producon, and deeroraon rae. European Journal of Operaonal Reearch, 4, 996, pp [] Z. Balkh, Z. and. Benkherouf, A producon lo ze nvenory model for deerorang em and arbrary producon and demand rae, European Journal of Operaonal Reearch 9, 996, pp [3] Z. Balkh, On he Global Opmaly of a General Deermnc Producon o ze Invenory Model for Deerorang Iem, Belgan Journal of Operaonal Reearch, ac and Compuer cence, 38, 4, 998, pp [4] Balkh, Z. (3). he effec of learnng on he opmal producon lo ze for deerorang and parally backordered em wh me varyng demand and deeroraon rae. Journal of Appled Mahemacal Modelng 7, [5] Z. Balkh, Vewpon on he opmal producon oppng and rearng me for an EOQ wh deerorang em. Journal of operaonal Reearch ocey, 5,, pp [6] Z. Balkh, On a fne horzon producon lo ze nvenory model for deerorang em: An opmal Iue, Volume 3, 9 3

11 INERNAIONA JOURNA OF MAHEMAICA MODE AND MEHOD IN APPIED CIENCE oluon, European Journal of Operaonal Reearch 3,, pp. -3. [7] Z. Balkh, C. Goyal and. Gr, Vewpon, ome noe on he opmal producon oppng and rearng me for an EOQ model wh deerorang em, Journal of Operaonal Reearch ocey, 5,, pp [8] E. Nador, Invenory yem, John Wley & on Inc., New York, 996 [9] M. Reh, M. Fredman, and.c. Barboa, On a general oluon of a deermnc lo ze problem wh me proporonal demand, Operaon Reearch, 4, 976, pp [8] Z. Balkh, An opmal oluon of a general lo ze nvenory model wh deeroraed and mperfec produc, akng no accoun nflaon and me value of money, Inernaonal Journal of yem cence, 35, 4, pp [9] Z. Balkh, On he opmaly of a varable parameer nvenory model for deerorang em under rade cred polcy, Proceedng of he 3 h WEA Inernaonal Conference on Appled Mahemac, Mah 8, 8, pp [] Z. Balkh, and A. Foul, Improvng nvenory conrol for ABIC produc, Recen Advance n Aplled Mahemac and Compuonal and Informaon cence,, 9, pp [] M. Ben-Daya, and A. Raouf, On he conraned mul-em ngle-perod nvenory problem, Inernaonal journal of Producon Managemen, 3, 993, pp.4-. [] D.K. Bhaacharya, (5), Producon, manufacurng and logc on mul-em nvenory, European Journal of Operaonal Reearch 6, 993, pp [3]. Eme, A mxed neger approach for opmzng producon plannng, Proceedng of he 3 h WEA Inernaonal Conference on Appled Mahemac, Mah 8,, pp [4] G. Hadley, J.M. Whn, Analy of nvenory yem, Prence Hall Company, 963. [5] J.D. Hong, R.R. andropory, and J.C. Hayya, On producon polce for lnearly ncreang demand and fne producon rae, Compuaonal and Indural Engneerng, 4, 99, pp [6]. Kar, A. K. Bhuna, and M. Ma, Invenory of mul-deerorang em old from wo hop under ngle managemen wh conran on pace and nvemen, Compuer and Operaon Reearch, 8,, pp. 3-. [7] J.D. enard, and B. Roy, Mul-em nvenory conrol: A mul-crera vew, European Journal of Operaonal Reearch 87, 995, pp [] M.J. Roenbla, Mul-em nvenory yem wh budgeary conran: A comparon beween he lagrangan and he fxed cycle approach, Inernaonal Journal of Producon Reearch 9, 4, 98, [] M.J. Roenbla, and U. G. Rohblum, On he ngle reource capacy problem for mul-em nvenory yem, Operaon Reearch, 38, 99, pp [] G. W. ewar, Inroducon o marx compuaon, Academc Pre, 973 [3] A. Yaemn, and Erenguc,.., Mul-em nvenory model wh coordnaed replenhmen : A urvey, Inernaonal Journal of Operaon and Producon Managemen, 8,, 988, pp Iue, Volume 3, 9 4