Control of computer chip semi-conductor wafer fabs

Transcription

1 Conrol of compuer chip semi-conducor wafer fabs Cos: x9 $ Reurn: years Cycle ime: 6 weeks WIP: 6, wafers, 8x6 $ Challenge: Conrol he queues a ~5 work seps Conrol of Manufacuring Sysems:! Sochasic racking of Fluid Soluion! and Robus Fluid Soluion! Simplex Algorihm o Solve Coninuous Linear Programs Eurandom Workshop on Robus Opimizaion in Applied Probabiliy Eindhoven, ovember 5 Gideon Weiss Universiy of Haifa Join work wih: Ana Kopzon, Yoni azarahy, Evgeny Shindin Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Conrol of compuer chip semi-conducor wafer fabs Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Conrol of compuer chip semi-conducor wafer fabs x9 Cos: $ Reurn: years Cycle ime: 6 weeks WIP: 6, wafers, 8x6 $ Challenge: Conrol he queues a ~5 work seps Conrol 6 jobs, sales value 8x6$, over 6 weeks Scheduling approach: Use combinaorial opimizaion o find opimal schedule Inracable because of size, Useless because of noise Queueing approach: Use Markov decision heory o opimize seady sae Inracable because of size, Useless because no seady sae Bridging approach: Queueing model, o accommodae noise Finie horizon, for scheduling he curren WIP Inracable, approximaion for large sysem. Opimize a Fluid Model wo approaches for he sochasics Model sochasic noise around exac fluid Solve a robus fluid model Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5

2 wo Fluid Approaches Modeling Sep : Muli class queueing neworks MCQ Manufacuring Plan Wafer Fab Model as Muliclass Queueing ework Finie Horizon Problem (Rolling Horizon model predicive) MCQ problem: conrol (Q(),()), over << Dynamics Q k () = Q k () S k ( k ()) + Φ k 'k (S k ' ( k ' ())) k K k ' Fluid Model + Sochasic Deviaions Robus Fluid Model Bersimas, asrabadi, Paschalidis 5 Separaed Coninuous Linear Program Solved by Simplex ype Algorihm Weiss 8 rack Fluid Soluion Discree Sochasic azarahy, Weiss Asympoic Opimaliy? Maximal hroughpu? Objecive Capaciy min γ k k ( ) + c k Q k ()d k k () k (s) s k:s(k)=i re-enran line Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Modeling Sep : Fluid nework approximaion o MCQ SCLP Fluid problem is a separaed coninuous linear program V * = min ( γ u() + c q() )d q() fluid levels u() processing raes s.. q() = q() Ru(s)ds, R Rouing marix Mu(), M consiuen mean imes marix u(),q(), [, ] Robus fluid formulaion SCLP Robus fluid problem is a again an SCLP (Bersimas, asrabadi, Paschalidis 5) V Robus = min ( γ u() + c q() )d s.. q() = q() Ru(s)ds, u() M C Γ M! α(), I J β() u(),α(),β(),q(), [, ] Robus processing imes m k m k () = m k + z k ()!m k m k +!m k z k (), z k () Γ i machine i Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 7

3 Derivaion of fluid formulaion (Bersimas, asrabadi, Paschalidis 5) Opimal fluid soluion (finie horizon) V Robus = min ( γ u() + c q() )d s.. q() = q() Ru(s)ds, u() M C Γ M! α(), I J β() u(),α(),β(),q(), [, ] Replace Mu() By max u U M ()u() consiuen mean ime marix Rouing R = m Sochasic: [ M ] i,k = k if s(k) = i + oise else Robus : m [ M ()] i,k = k + z k ()!m k if s(k) = i else!! Soluion of SCLP - Cenralized planning Primal Z i (u,) = max ( m k + z k ()!m k )u k () s(k)=i z k () Γ i s(k)=i z k () Dual Z i (u,) = minγ i β i () + α k () + m k u k () s(k)=i β i () + α k ()!m k u k (), k : s(k) = i β i (),α k () Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 8 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 9 Opimal fluid soluion (finie horizon) Soluion of SCLP - Cenralized planning racking he fluid soluion IVQs Model deviaions from fluid soluion as MCQ wih infinie virual queues IVQ Pariions ime horizon o a finie number of inervals piecewise consan processing raes (conrols) coninuous piecewise linear fluid level (saes) Use disribued conrol o sabilize deviaions heorem: Policy is asympoically opimal - as sysem becomes large Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5

4 Muli class queueing neworks (Harrison 88, Dai 94, ) MCQ wih infinie virual queues (Massey 84, W 6) Muli-class queueing neworks wih infinie virual queues! IVQ Q k () = Q k () + A k () S k ( k ()) + Φ k 'k (S k ' ( k ' ())) k ' Κ Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Muli class queueing neworks (Harrison 88, Dai 94, ) MCQ wih infinie virual queues (Massey 84, W 6) IVQ in he fluid soluion Opimal Fluid Soluion In each inerval, pariion buffers 6 6 K = {5, 7, 9} k K : Q k () = Q k () + A k () S k ( k ()) + Φ k 'k (S k ' ( k ' ())) k ' Κ k K : Q k () = Q k () + α k S k ( k ()) Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 K = {,,,, 4, 6, 8} 5,7,9 are empy 7 fully uilizes machine 9 and 6 fully uilize machine Machine pumps ou of 8 bu is no fully uilized. Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5

5 Maximum pressure policy (assiulas, Solyar, Dai & Lin) Disribued conrol policy for MCQ: Observe queue of each class and is downsream classes Calculae pressure of class Serve Max Pressure class a each machine racking he fluid soluion wih IVQ and Max Pressure! racking heorem (Dai and Lin 5): Max pressure achieves sabiliy for any sysem wih offered load < rae sabiliy if offered load = heorem: Same resul holds for MCQ wih IVQs We model deviaions from fluid soluion as MCQ wih IVQs Sabiliy implies: deviaions are negligible if sysem large Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 7 Asympoic opimaliy (azarahy, W 9) heorem: Le Q() be he queue lengh process of a finie horizon MCQ. Le Q () be scalings of Q(), wih Q ()=Q(), and wih -fold speed Of processing. Le q() be he opimal fluid soluion and le V f be is Objecive value. (i) Le V denoe he objecive values of Q () for any general policy. hen: (ii) Under max pressure racking of he fluid soluion and lim inf V V f Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 8 a.s. lim Q() = q() a.s. uniformly on < < lim V = V f a.s. Finie horizon conrol of a processing sysem! Model as a muli-class queueing nework MCQ! Fluid nework approximaion o MCQ Sochasic vs. Robus! Soluion of fluid problem! separaed coninuous linear program SCLP! racking he opimal fluid soluion! MCQ wih infinie virual queues IVQ! - minimal daa raes and roues,! - cenralized planning for fluid,! - disribued execuion for racking! - rolling horizon implemenaion! Sochasic: hroughpu Opimal for large! Robus: Lose some hroughpu sable for smaller Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 9

6 Separaed coninuous linear programming! Symmeric dualiy & complemenary slackness max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() u(), x() b < < primal saes x k () primal conrols u j () SCLP Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Symmeric dualiy & complemenary slackness max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() b primal saes x k () primal conrols u j () Symmeric dualiy & complemenary slackness max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() b primal saes x k () primal conrols u j () u(), x() < < u(), x() < < min (( α + ( ) a ) p() + b q() ) d s.. G p(s)ds + H q() γ + c F p() d dual saes dual conrols q j ( ) p k ( ) min (( α + ( ) a ) p() + b q() ) d s.. G p(s)ds + H q() γ + c F p() d dual saes dual conrols q j ( ) p k ( ) Dual runs in reversed ime p(),q() < < p(),q() < < Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5

7 Symmeric dualiy & complemenary slackness max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() b primal saes x k () primal conrols u j () Piecewise consan conrols and linear saes Soluion feaures: pariion of ime horizon = < < < = piecewise consan conrols u() coninuous piecewise linear x() eed o know (primal and dual): u(), x() < < Inerval lenghs: au n = n - n- Linear Inerval Equaions min (( α + ( ) a ) p() + b q() ) d s.. G p(s)ds + H q() γ + c F p() d dual saes dual conrols q j ( ) p k ( ) Dual runs in reversed ime Values of primal and dual conrols, u(), p(-): u n j, p n k Slopes of primal and dual saes,!x(),!q( ) :!x k n,!q j n Raes LP p(),q() < < x k ()p k ( )d = q j ( )u j ()d = primal dual feasible complemenary slack soluions are opimal Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Inial sae values x() q(): x k, q j Boundary LP Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Complemenary slack opimal soluions!x k p k x!x k k Boundary and Rae LPs max (( γ + ( ) c )u() + d x() ) d s.. G u(s)ds + Fx() α + a H u() u(), x() b < < min (( α + ( ) a ) p() + b q() ) d s.. G p(s)ds + H q() γ + c F p() d p(),q() < < = = = = Boundary max d x s.. Fx α min b q s.. H 'q γ u j!q j!q j q j x q Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 7

8 Boundary and Rae LPs d max (( γ + ( ) c )u() + d x() ) d d d s.. G u(s)ds d + Fx() α + a H u() b d min (( α + ( ) a ) p() + b q() ) d d d s.. G p(s)ds + H q() γ + c d F p() d Equaions for inerval lengh!x k x!x k k p k x k ( ) = u(), x() < < p(),q() < < Boundary max d x s.. Fx α min b q s.. H 'q γ = = = = Raes x max c'u + d!x s.. Gu + F!x a Hu b min a' p + b!q s.. G ' p + H '!q c F ' p q d u j q j ( ) =!q j!q j q j Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 8 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 9 Soluion of a simple SCLP example Solving he example 6 max (8 )u () x () ( ) d s.. u (s)ds + x () + u () u, x Primal and dual problems max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Parameric soluion Insead of =6 we solve for parameric, saring a and increasing o We solve simulaneously he primal problem and is dual. Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5

9 Solving he example Boundary LP max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example Raes LP > max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q max x s.. + x = x min s.. q + q = 4 q max u!x s.. u +! +!x = u + u = min p +!q s.. p! +!q = p p! U, u Z,!x,u P! U, p Z, p,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 Solving he example max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P! U, p Z, p,!q P! U, u Z,!x,u P! U, p Z, p,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5

10 Solving he example max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P! U, p Z, p,!q P! U, u Z,!x,u P! U, p Z, p,!q P!x U, u,!x,u P p Z, p,!,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 7 Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P! U, p Z, p,!q P! U, u,!x,u P q p Z, p,!,!q P! U, u Z,!x,u P! U, p Z, p,!q P! U, u,!x,u P q p Z, p,!,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 8 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 9

11 Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P! U, p Z, p,!q P! U, u,!x,u P q p Z, p,!,!q P! U, u Z,!x,u P! U, p Z, p,!q P! U, u,!x,u P q p Z, p,!,!q P Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P 4! U, p Z, p,!q P!x 4 U, u,!x,u P p Z, p,!,!q P! U, u Z,!x,u P 4! U, p Z, p,!q P!x 4 U, u,!x,u P p Z, p,!,!q P q q Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 4

12 Solving he example max 4 + ( ) (( )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Solving he example max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q! U, u Z,!x,u P 5! U, p Z, p,!q P!x 5 U, u,!x,u P p Z, p,!,!q P! U, u Z,!x,u P 5 6! U, p Z, p,!q P!x 5 U, u,!x,u P 6 p Z, p,!,!q P!,!x,u,u P p,!, p,!q P q q q Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 44 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 45 Complee soluion: p max (( 4 + ( ) )u () x ()) d s.. u (s)ds + () + x () = + u () + u () = u, x min ((+ )p () + q ()) d s.. p (s)ds () + q () = 4 + p () p () p,q Analog from parameric LP Lemke algorihm Solving LP by parameric Lemke algorihm: max ( ( λ) + λc)' x ( ) x Ax ( λ) + λb max c' x Ax b x (b,c) 6 p u p u u p u 6 (, ) q (! b,!c) Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 46 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 47

13 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 48 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 49 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5

14 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 5 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 54 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 55

15 Solving he fluid problem SCLP algorihm demo Solving he fluid problem SCLP algorihm demo Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 56 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 57 Solving he fluid problem SCLP algorihm demo Summary: We discuss a mehod for finie horizon conrol of large manufacuring plans We large scale sochasic vs. Robus approach Sochasic approach is asympoically opimal, for large scale operaions Robus approach sub-opimal bu sable on smaller scale Calculaes cenralized fluid soluion for overall conrol Uses de-cenralized local conrol o rack opimal fluid soluion Uses minimum amoun of daa, readily available Rolling horizon implemenaion Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 58 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 59

16 References () A Simplex based algorihm o solve Separaed Coninuous Linear Programs Gideon Weiss, Mah Programming Series A, 5:5-98, 8. () ear opimal conrol of queueing neworks over a finie ime horizon Yoni azarahy and Gideon Weiss, Annals of OR 7:-49, 9 () A Push-Pull nework wih infinie supply of work Ana Kopzon, Yoni azarahy and Gideon Weiss, QUESA 6:75-, 9 (4) Posiive Harris recurrence and diffusion scale analysis of a Push-Pull queueing nework wih infinie supply of work Yoni azarahy and Gideon Weiss, Performance Evaluaion, 67:-7, (5) Robus Fluid Processing eworks Dimiris Bersimas, Ebrahim asrabadi and Ioannis Paschalidis, IEEE rans Auomaic Conrol 6():75-78, 5. hank you! (6) A new algorihm for sae-consrained separaed coninuous linear programs, X. Luo and Dimiris Bersimas, SIAM J Conrol & Opimizaion, 7:77, 998. Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Where are we now? racking he fluid soluion Example nework µ Sacked Queue level represenaion: µ µ Sacked Queue Levels Q Q Finished Jobs Q rajecory of a single job Resource is he boleneck µ > µ + µ How o conrol queues, Schedule resource, for opimal draining ime Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 6

17 Fluid formulaion Fluid policies: LBFS, min makespan, opimal min () + q () + q ()d s. : () = () µ u (s)ds q () = q () + µ u (s)ds µ u (s)ds q () = q () + µ u (s)ds µ u (s)ds µ µ µ LBFS V=76 min Makespan V=6 u () + u () u () u,q [, ] Opimal V=5 µ µ µ his is a Separaed Coninuous Linear Program (SCLP) Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 64 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 65 ime inervals of he opimal fluid soluion τ = (,4) τ = (4,8) τ = (8,4) τ 4 = (4,4) Allocaions u = u = u =.5 u =.5 u = u = u = u = u = u = u =.75 u =.5 Calculaing he pressure for neworks wih fixed rouing Pressure for buffer k based on k and k+ downsream curren k + K k + K k K µ Q µ k ( Qk Qk + ) k k n Κ = { k qk ( ) =, τ n} Κ n = {k q k () >, τ n } α k = u k µ k k K µ k (α k D k Q k + ) µ k (α k D k ) Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 66 Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 67

18 Using max pressure o conrol MCQ wih IVQ:! racking he fluid soluion Gideon Weiss, Universiy of Haifa, Fluid Model Soluion for Manufacuring, 5 68