Operaons Plannng Capacy Plannng Sales and Operaons Plannng Forecasng Capacy plannng Invenory opmzaon How much capacy assgned o each producon un? Realsc capacy esmaes Sraegc level Moderaely long me horzon Larger me buckes Daa no known exacly Poche-Wolsey s model s oo large, NP-hard Decomposon naccurae model Medum-erm plannng horzon, sraegc level Larger me buckes 1 Some defnons Work n Process nvenory beween he sar and end pons of a produc roung s called work n process (WIP) WIP nvenory work n process nvenory FGI nvenory fnal goods nvenory Lead me of a gven roung or lne s he me alloed for producon of a par on ha roung or lne. Lles Law WIP = Throughpu cycle me Maser Producon Schedulng Model (Poche-Wolsey) me horzon = 1,...,n φ un producon cos q fxed producon cos π nvenory cos D demand Decson varables x producon lo sze n perod y bnary varable ndcang producon perod I nvenory a end of perod Moreover C k avalable capacy, resource k me ξ k per un resource consumpon β k fxed resource consumpon 3
Maeral Requremen Plannng Model (Poche-Wolsey) Maeral Requremen Plannng Model (Poche-Wolsey) Mul-em mul-level capacaed lo-szng model Opmze smulaneously producon and purchase of all ems from raw maerals o fnshed producs sasfy exernal demands sasfy nernal demands shor-erm horzon Bll of maerals (BOM) S() se of drec successors of (ems consumng ) r j amoun of em requred o make one em j γ lead-me o produce or delver a lo of.e. x can be delvered a me + γ Model mn m n =1 =1 φ x + q y + π I Subjec o: ( ) I 1 + x γ = D + r j x j + I, j S() x My, m =1 ξ k x + m =1 β k y C k x R +,I R +,y {,1} Large, NP-hard model, dffcul o solve 5 Capacy of Resources Gross capacy Usable capacy Producvy facor Resource Gross Producvy Usable Descrpon Capacy Facor Capacy (hours/day) (hours/day) Machne S1.95 7. Forklf.5. Machne ASS 1.5 13. Machne PPP.95 7. Capacy lm s sof (queung heory) Lead me depends on load Capacy depends on - varaon n arrvng jobs - varaon n lead mes Asmundsson, Sngle-sage mul-produc plannng Smplfed model Subjec o: mn φ X + π I I = I, 1 + X, L D ξ X C X,I R + LP-model (easy o solve),, Seup mes are negleced (fuure work) Consan lead me (producon me) L 7
Inroducon Relaon beween WIP and sysem hroughpu Accurae measuremens of manufacurng capacy s hard o oban (Elmaghraby 1991) Sochasc performance analyss models Queung models capure mporan aspecs (Hopp, Spearman ) Sochasc model of whole producon unrealsc Deermnsc echnques Dvde plannng horzon n dscree buckes Assgn capacy n each bucke Soluon sasfyng aggregaed consran no feasble n pracce Inegrang approaches (Hung, Leachmand 199) Solve LP model for producon plannng Feed no smulaor o esmae lead mes Add cu f no feasble nonlnear clearng funcon. Analyc congeson model for closed producon sysems (Spearman 1991) Queung models of manufacurng sysems (Buzaco, Shanhkumar 1993) Connecon beween WIP, varaon, ulzaon (Medh 1991) WIP = c ρ 1 ρ + ρ c varaon servce me and arrvals, ρ ulzaon of server (WIP+1) + WIP(c ρ = 1) (W IP+1) (c 1) Fgure for dfferen values of c Clearng Funcon Relaonshp wh Lead Tme Theorecal Capacy Throughpu (Capacy) WIP Lead Tme Theorecal Capacy Throughpu Fgure 1 199) Relaonshps beween hroughpu, WIP and lead me (Karmarkar 9 1 Clearng Funcon Capacy lm depends on Work-n-process (W) ξ X f (ξ W ) Asmundsson Sngle Sage Mul-produc Plannng Model Usng Clearng Funcons X number of uns of produc produced me R number of uns or produc released no he sage a he begnnng of me perod W number of uns of produc n WIP nvenory a he end of perod I number of uns of produc n fnshed goods nvenory a he end of perod ξ amoun of resource (machne me) requred o produce one un of em me f (W) clearng funcon represenng resource n perod D he demand for produc n perod If capacy depends on WIP we need o have cos assocaed wh WIP (nernal nvenory) 11 1
CF model (Clearng Funcon) Subjec o: mn φ X + ω W + π I + ρ R W = W, 1 X + R I = I, 1 + X D ξ X f (ξ W ) X,W,I,R R +,,, Problem wh capacy consran X A + X B f(w A +W B ) for wo producs A,B Soluon exss X A >, X B =, W A =, W B > Manan hgh W B f cheap Creae capacy for X A No lnk beween WIP avalable and producon where φ cos of producng un me ω cos of WIP of un me π cos of nvenory un me ρ cos of releasng un me 13 Soluon Z represen an allocaon of expeced hroughpu among dfferen producs X Z f (ξ W ) Z = 1 Depends on oal WIP. Prefer WIP of specfc produc. Assumng ξ W = Z ξ W we ge (?) X Z f (?, ξ W Z ), Z = 1 ACF model (Allocaed Clearng Funcon) Subjec o: mn φ X + ω W + π I + ρ R W = W, 1 X + R I = I, 1 + X D?,, ξ W X Z f ( ) Z Z = 1 X,W,I,R R +, Throughpu (Capacy) Theorecal Capacy WIP 15 1
Ouer approxmaon of clearng funcon Asmundsson, expermenal resuls Ouer lnear funcons α c W + β c for c = 1,...,C ξ W α c W + β c c, hen f(w) = mn c=1,...,c {αc W + β c } Assume ha α 1 > α > α 3 >... > α c = Lnear model mn φ X + ω W + π I + ρ R Subjec o: W = W, 1 X + R I = I, 1 + X D ξ X α c ξ W + Z β c Z = 1,,,,c Z,X,W,I,R R +, Whch s correc ( snce ) { ξ W Z f = Z mn α cξ } W + β c Z c Z = mn c {α c ξ W + β c Z } 17 Sngle-sage sysem, 3 producs Varaon of arrval.5 varaon of servce Max hroughpu 1 ems / perod Mnmze nvenory holdng coss (π 1,π,π 3 ) =(1,,3) Holdng coss of WIP and FGI are same Clearng funcon, lnear approxmaon 1 Asmundsson, expermenal resuls Asmundsson, expermenal resuls Producon Plan Throughpu 1 1 Throughpu 1 1 Demand Demand WIP FGI TH 1 FC PCF Throughpu 1 1 3 5 7 9 11 13 15 17 19 1 3 5 7 9 Tme Perod Fgure Producon plan 1 3 5 7 9 11 13 15 17 19 1 3 5 7 9 31 33 35 37 39 1 3 5 7 9 Perod 1. Lead Tme Fgure 5: Throughpu and Demand of ACF and FC Models. 1. Demand (gray), hroughpu from ACF more smooh han FC 1 WIP and FGI. 1 Perods.. 1 1. Level 1 PCF - WIP PCF - FGI FC - FGI 1 3 5 7 9 11 13 15 17 19 1 3 5 7 9 Tme Perod Fgure 3 Producon lead me 5 Conclusons 1 3 5 7 9 11 13 15 17 19 1 3 5 7 9 31 33 35 37 39 1 3 5 7 9 Perod Fgure : WIP and FGI levels, Cumulave across Iems. WIP hgh when hroughpu hgh, FC model does no capure WIP FC soluon only opmzes FGI holdng coss, fals o capure radeoffs WIP, FGI 19
Asmundsson, expermenal resuls Asmundsson, expermenal resuls. Lead-Tme 1. Lead-Tme vs. Throughpu 7.. Iem 1 Iem 1. Iem 3 Lead-Tme 5. Average Raw Processng.. 3. Lead-Tme... 1.. 5 1 15 5 3 35 5 Perod Fgure 7: Producon Lead-Tme across all Iems... 1 3 5 7 9 1 Throughpu lead me greaer han.1 (raw processng me), vares sgnfcanly Fgure 9: Nonlnear Relaonshp beween hroughpu and lead me 5 Margnal Cos of Capacy PCF FC 15 MMC 1 5 1 3 5 7 9 11 13 15 17 19 1 3 5 7 9 31 33 35 37 39 1 3 5 7 9 Perod Fgure : Margnal Cos of Capacy (MCC). capacy consran acve n ACF, margnal cos posve margnal cos s zero n FC for all perods where FGI s zero 1