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MIT2.852

ManufacturingSystemsAnalysis

Lectures1921

Scheduling:Real-TimeControlofManufacturingSystems

StanleyB.GershwinSpring,2007

Copyrightc2007

Stanley

B.

Gershwin.

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DefinitionsEventsmaybecontrollable ornot,andpredictable

ornot.controllable uncontrollable

predictable

loadinga

part

lunch

unpredictable ??? machinefailure

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DefinitionsSchedulingistheselectionoftimesforfuturecontrollableevents.

Ideally,schedulingsystemsshoulddealwithallcontrollableevents,andnotjustproduction.Thatis,theyshouldselecttimesforoperations,

set-upchanges,preventivemaintenance,etc.Theyshouldatleastbeaware ofset-upchanges,

preventivemaintenance,etc.whentheyselecttimesforoperations.


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DefinitionsBecauseofrecurringrandomevents,schedulingis

anon-goingprocess,andnotaone-timecalculation.

Scheduling,orshopfloorcontrol,isthebottomofthescheduling/planninghierarchy.Ittranslatesplansintoevents.

Copyright c 42007StanleyB.Gershwin.

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IssuesinFactoryControlProblemsaredynamic;currentdecisionsinfluence

futurebehaviorandrequirements.Therearelargenumbersofparameters,time-varying

quantities,andpossibledecisions.Sometime-varyingquantitiesarestochastic.Somerelevantinformation(MTTR,MTTF,amountof

inventoryavailable,etc.) isnotknown.Somepossiblecontrolpoliciesareunstable.Copyright c 52007StanleyB.Gershwin.

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Dynamic ExampleProgramming Problem

DiscreteTime,DiscreteState,DeterministicF6

8

A1

B L5

6

22

G5

1064M

9

2D 7

72

4 N

594E

5 I 86

136 2C

H

J

1

4Z

K 3 O

Problem: findtheleastexpensivepathfromAtoZ.Copyright c 62007StanleyB.Gershwin.

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Dynamic ExampleProgramming Problem

Letg(i,j)bethecostoftraversingthelinkfromitoj.Leti(t)bethetthnodeonapathfromAtoZ.Thenthepathcostis

Tg(i(t1), i(t))

t=1whereT isthenumberofnodesonthepath,i(0)=A,andi(T) =Z.T isnotspecified;itispartofthesolution.


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Dynamic ExampleProgramming SolutionApossibleapproachwouldbetoenumerateallpossiblepaths

(possiblesolutions).However,therecanbealotofpossiblesolutions.

Dynamicprogrammingreducesthenumberofpossiblesolutionsthatmustbeconsidered.Goodnews: itoftengreatly reducesthenumberofpossiblesolutions.Badnews: itoftendoesnotreduceitenoughtogiveanexactoptimalsolutionpractically(ie,withlimitedtimeandmemory).Thisisthecurseofdimensionality.

Goodnews: wecanlearnsomethingbycharacterizingtheoptimalsolution,andthatsometimeshelpsingettingananalyticaloptimalsolutionoranapproximation.

Goodnews: ittellsussomethingaboutstochasticproblems.Copyright2007StanleyB.Gershwin.c 8

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Dynamic ExampleProgramming Solution

InsteadofsolvingtheproblemonlyforAastheinitialpoint,wesolveitforall possibleinitialpoints.Foreverynodei,defineJ(i)tobetheoptimalcosttogo fromNodeitoNodeZ(thecostoftheoptimalpathfromitoZ).Wecanwrite

TJ(i) = g(i(t1), i(t))

t=1wherei(0)=i;i(T) =Z;(i(t1), i(t))isalinkforeveryt.Copyright c 92007StanleyB.Gershwin.

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ThenJ(i)satisfiesJ(Z) = 0

and,iftheoptimalpathfromitoZ traverseslink(i,j),J(i) =g(i,j) +J(j).

i j

Z


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SupposethatseverallinksgooutofNodei.

4j

3j

2j

5j

1j

6j

i

Z

Supposethatforeachnodej forwhichalinkexistsfromitoj,theoptimalpathandoptimalcostJ(j)fromj toZ isknown.Copyright2007StanleyB.Gershwin.c 11

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ThentheoptimalpathfromitoZ istheonethatminimizesthesumofthecostsfromitojandfromj toZ.Thatis,

J(i)=min [g(i,j) +J(j)]j

wheretheminimizationisperformedoveralljsuchthatalinkfromitojexists.ThisistheBellmanequation.This

is

a

recursion

or

recursive

equation

because

J()

appears

onbothsides,althoughwithdifferentarguments.J(i)canbecalculatedfromthisifJ(j)isknownforeverynodejsuchthat(i,j)isalink.Copyright c 122007StanleyB.Gershwin.

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Dynamic ExampleProgramming SolutionBellmansPrincipleofOptimality: ifiandjarenodesonanoptimalpathfromAtoZ,thentheportionofthatpathfromAtoZbetweeniandj isanoptimalpathfromitoj.

A

j

i


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Example: AssumethatwehavedeterminedthatJ(O) = 6andJ(J)=11.TocalculateJ(K),

g(K,O) +J(O)J(K)= ming(K,J) +J(J)

3 + 6= min = 9.9+11


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Algorithm1.SetJ(Z) = 0.2.Findsomenodeisuchthat

J(i)hasnotyetbeenfound,andforeachnodej inwhichlink(i,j)exists,J(j)isalready

calculated.AssignJ(i)accordingto

J(i)=min [g(i,j) +J(j)]j3.RepeatStep2untilallnodes,includingA,havecosts

calculated.Copyright c 152007StanleyB.Gershwin.

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K


F

A

B11 L

5

6HC

14

D

EI

O

J

9

G

M

N48

14

13

11

17

12

136

11

Z


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Dynamic ExampleProgramming SolutionTheimportantfeaturesofadynamicprogrammingproblemarethestate(i);thedecision (togotojafteri);

theobjectivefunction T

t=1g(i(t1), i(t))thecost-to-gofunction(J(i));theone-steprecursionequationthatdeterminesJ(i)

(J(i)=minj[g(i,j) +J(j)]); thatthesolutionisdeterminedforeveryi, notjustAandnotjustnodesontheoptimalpath;

thatJ(i)dependsonthenodestobevisitedafteri, notthosebetweenAandi.Theonlythingthatmattersisthepresentstateandthefuture;

thatJ(i)isobtainedbyworkingbackwards.Copyright2007StanleyB.Gershwin.c 17

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Thisproblemwasdiscretetime,discretestate,deterministic.

Otherversions:discretetime,discretestate,stochasticcontinuoustime,discretestate,deterministiccontinuoustime,discretestate,stochasticcontinuoustime,mixedstate,deterministiccontinuoustime,mixedstate,stochastic

instochasticsystems,weoptimizetheexpected cost.Copyright c 182007StanleyB.Gershwin.

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ProgrammingDiscretetime,discretestate

StochasticDynamic

Supposeg(i,j)isarandomvariable;orifyouareatiandyouchoosej,youactuallygotokwith

probabilityp(i,j,k).Thenthecostofasequenceofchoicesisrandom.Theobjectivefunctionis

TE g(i(t1), i(t))

t=1

andwecandefineJ(i) =Emin[g(i,j) +J(j)]

jCopyright2007StanleyB.Gershwin.c 19

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ProgrammingContinuousTime,MixedState

StochasticExampleDynamic

Context: Theplanning/schedulinghierarchyLongterm: factorydesign,capitalexpansion,etc.Mediumterm:demandplanning,staffing,etc.Shortterm:

responsetoshorttermeventspartreleaseanddispatchIn

this

problem,

we

deal

with

the

response

to

short

term

events.

Thefactoryandthedemandaregiventous;wemustcalculateshorttermproductionrates;theseratesarethetargetsthatreleaseanddispatchmustachieve.


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Dynamic


StochasticExamplex1

x2

d1

d2

u (t)1

u (t)2 Type 2

Type 1

r, p

Perfectlyflexiblemachine,twoparttypes.i timeunitsrequiredtomakeTypeiparts,i= 1,2.

Exponentialfailuresandrepairswithratespandr.Constantdemandratesd1,d2.Instantaneousproductionratesui(t), i= 1,2control

variables.Downstreamsurplusesxi(t).Copyright c 212007StanleyB.Gershwin.

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Objective: Minimizethedifferencebetweencumulativeproductionandcumulativedemand.Thesurplussatisfiesxi(t) =Pi(t)Di(t)

t

CumulativeProduction

and Demand production P (t)

surplus x (t)

i

i

idemand D (t) = d ti

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Feasibility:Fortheproblemtobefeasible,itmustbepossibletomake

approximatelydiT TypeipartsinalongtimeperiodoflengthT, i= 1,2.(Whyapproximately?)

ThetimerequiredtomakediT partsisidiT.Duringthisperiod,thetotaluptimeofthemachineie,the

timeavailableforproductionisapproximatelyr/(r+p)T.Therefore,wemusthave1d1T +2d2Tr/(r+p)T,or

2

ridi

r+pi=1


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Ifthisconditionisnotsatisfied,thedemandcannotbemet.Whatwillhappentothesurplus?Thefeasibilityconditionisalsowritten

2di r

i=1 i r+pwherei =1/i.Iftherewereonlyoneparttype,thiswouldber

dr+p

Lookfamiliar?Copyright2007StanleyB.Gershwin.c 24

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Thesurplussatisfiesxi(t) =Pi(t)Di(t)

where

tPi(t) = u

i(s)ds; D

i(t) =d

it

0Therefore

dxi(t)=ui(t)di

dt


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Todefinetheobjectivemoreprecisely,lettherebeafunctiong(x1, x2)suchthatgisconvexg(0,0)=0 lim g(x1, x2) =; lim g(x1, x2) =.

x1 x1 lim g(x1, x2) =; lim g(x1, x2) =.x2 x2


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Examples:g(x1, x2) =A1x12 +A2x22g(x1, x2) =A1|x1|+A2|x2|g(x1, x2) =g1(x1) +g2(x2)where

+ gi(xi) =g(i+)xi +g(i)xi , xi+ =max(xi,0),xi =min(xi,0),

g(i+) >0, g(i) >0.


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DynamicProgramming

Objective:

ContinuousTime,MixedStateStochasticExample

Tmin

E g(x1(t), x2(t))dt

0

x1

g(x ,x )1 2

x2


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Constraints:u1(t)0; u2(t)0

Short-termcapacity:Ifthemachineisdownattimet,

u1(t) =u2(t) = 0


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StochasticExampleAssumethemachineisupforashortperiod[t,t+t].Lett

Dynamic

besmallenoughsothatui isconstant;thatisui(s) =ui(t), s[t,t+t]

Themachinemakesui(t)tpartsoftypei.ThetimerequiredtomakethatnumberofTypeipartsisiui(t)t.Therefore

iui(t)tt 1/i

oriui(t)1

1/1

2

u2

0

iCopyright2007StanleyB.Gershwin.c 30

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Machinestatedynamics: Define(t)tobetherepairstateofthemachineattimet.(t) = 1meansthemachineisup;(t) = 0meansthemachineisdown.

prob((t+t) = 0|(t)=1)=pt+o(t)prob((t+t) =1|(t)=0)=rt+o(t)

Theconstraintsmaybewritteniui(t)(t); ui(t)0

iCopyright2007StanleyB.Gershwin.c 31

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StochasticExampleDynamicDynamicprogrammingproblemformulation: T

minE g(x1(t), x2(t))dtsubjectto: 0

dxi(t)=ui(t)di

dtprob((t+t) = 0|(t)=1)=pt+o(t)prob((t+t) =1|(t)=0)=rt+o(t)

iui(t)(t); ui(t)0i

x(0), (0)specified


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Dynamic ElementsofaDPProblemProgrammingstate: xalltheinformationthatisavailabletodeterminethe

futureevolutionofthesystem.control: utheactionstakenbythedecision-maker.objectivefunction: J thequantitythatmustbeminimized;dynamics: theevolutionofthestateasafunctionofthecontrol

variablesandrandomevents.constraints: thelimitationsonthesetofallowablecontrolsinitialconditions: thevaluesofthestatevariablesatthestart

ofthetimeintervaloverwhichtheproblemisdescribed.Therearealsosometimesterminalconditions suchasinthenetworkexample.


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Dynamic ElementsofaDPSolutionProgrammingcontrolpolicy: u(x(t), t). Astationary ortime-invariant policyisoftheformu(x(t)).

valuefunction: (alsocalledthecost-to-go function)thevalueJ(x,t)oftheobjectivefunctionwhentheoptimalcontrolpolicyisappliedstartingattimet,whentheinitialstateisx(t) =x.


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Bellmans Continuousx,tEquation Deterministic

T

Problem: min g(x(t), u(t))dt+F(x(T))u(t),0tT 0

suchthatdx(t)

=f(x(t), u(t), t)dt

x(0)specified

h(x(t), u(t))0

xRn, uRm, fRn, hRk,andgandF arescalars.Data: T,x(0),andthefunctionsf,g,h,andF.Copyright c 352007StanleyB.Gershwin.

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Bellmans Continuousx,tEquation DeterministicThecost-to-gofunctionis

TJ(x,t)=min g(x(s), u(s))ds+F(x(T))

t

T

J(x(0),0)=min g(x(s), u(s))ds+F(x(T))0 t1 T

= min g(x(t), u(t))dt+ g(x(t), u(t))dt+F(x(T)) .u(t), 0 t1

0tT


C i

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t1 T= min g(x(t), u(t))dt+ min g(x(t), u(t))dt+F(x(T))

u(t),

0 u(t), t1

0tt1 t1tT t1 = min g(x(t), u(t))dt+J(x(t1), t1) .

u(t), 00tt1


C ti

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where T

J(x(t1), t1)= min g(x(t), u(t))dt+F(x(T))u(t),t1tT t1

suchthatdx(t)

=f(x(t), u(t), t)dt

x(t1)specifiedh(x(t), u(t))0


C ti t

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Breakup[t1, T]into[t1, t1 +t][t1 +t,T] :t1+tJ(x(t1), t1)=min g(x(t), u(t))dt

u(t1) t1+J(x(t

1 +t), t

1 +t)}

wheretissmallenoughsothatwecanapproximatex(t)andu(t)withconstantx(t1)andu(t1),duringtheinterval.Then,approximately,J(x(t1), t1)=min g(x(t1), u(t1))t+J(x(t1 +t), t1 +t)

u(t1)


C ti t

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Or,J(x(t1), t1)=min g(x(t1), u(t1))t+J(x(t1), t1)+

u(t1)J J

(x(t1), t1)(x(t1 +t)x(t1)) + (x(t1), t1)tx tNotethat

dxx(t1 +t) =x(t1) + t=x(t1) +f(x(t1), u(t1), t1)tdt


Continuous t

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ThereforeJ(x,t1) =J(x,t1)

J J+min g(x,u)t+ (x,t1)f(x,u,t1)t+ (x,t1)t

u x twherex=x(t1);u=u(t1) =u(x(t1), t1).Then(droppingthetsubscript)

J J (x,t)=min g(x,u) + (x,t)f(x,u,t)t u x


Continuous x t

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Bellmans Continuousx,tEquation DeterministicThisistheBellmanequation. Itisthecounterpartoftherecursionequationforthenetworkexample.IfwehadaguessofJ(x,t)(forallxandt)wecouldconfirmitbyperformingtheminimization.

IfweknewJ(x,t)forallxandt,wecoulddetermineubyperformingtheminimization.Ucouldthenbewritten

Ju=U x, , t .

xThiswouldbeafeedbacklaw.

TheBellmanequationisusuallyimpossibletosolveanalyticallyornumerically.Therearesomeimportantspecialcasesthatcanbesolvedanalytically.Copyright2007StanleyB.Gershwin.c 42

Continuous x t

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Bellmans Continuousx,tEquation ExampleBang-BangControl

min |x|dt0

subjecttodx

=udtx(0)specified1u1


B Continuous x t

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Bellmans Continuousx,tEquation ExampleTheBellmanequationis

J J (x,t)= min |x|+ (x,t)u .t xu,

1u1J(x,t) =J(x)isasolutionbecausethetimehorizonisinfiniteandtdoesnotappearexplicitlyintheproblemdata(ie,g(x) =|x|isnotafunctionoft.Therefore

dJ0= min |x|+ (x)u .

dxu,1u1

J(0)=0becauseifx(0)=0wecanchooseu(t) =0forallt.Thenx(t) = 0foralltandtheintegralis0.Thereisnopossiblechoiceofu(t)thatwillmaketheintegrallessthan0,sothisistheminimum.Copyright2007StanleyB.Gershwin.c 44

B ll Continuous x t

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Bellmans Continuousx,tEquation Example

Theminimumisachievedwhen

u=

Why?

1 if dJ(x)>0dx

1 if dJ(x)

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ConsiderthesetofxwheredJ/dx(x)0andu=1,dJ

(x) =|x|dx


B ll Continuous x t

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Tocompletethesolution,wemustdeterminewheredJ/dx>0,0forallx 0because|x|>0sotheintegralof|x(t)|mustbepositive.=SinceJ(x)> J(0)forallx 0,wemusthave=

dJ(x)0dx


B ll Continuous x, t

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ThereforedJ

(x)>=xdxso

1J = x22

and

1 if x 0


Continuous x, t,Discrete Bellmans

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Continuousx,t,DiscreteStochastic

T

BellmansEquationJ(x(0), (0),0)=minE g(x(t), u(t))dt+F(x(T))

u 0

suchthatdx(t)

=f(x,,u,t)dt

prob [(t+t) = = =ijtforalli,j,ii|(t) j] =jx(0), (0)specified

h(x(t), (t), u(t))0Copyright2007StanleyB.Gershwin.c 49

Bellmans Continuous x, t,Discrete

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BellmansEquation

Continuousx,t,DiscreteStochastic

GettingtheBellmanequationinthiscaseismorecomplicatedbecausechangesbylargeamountswhenitchanges.LetH()besomefunctionof.Weneedtocalculate

EH((t

+

t))

=

E

{H((t

+

t))

|(t)}

=

H(j)prob {(t+t) =j |(t)}

j


Bellmans Continuousx,t,Discrete

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Bellman sEquation

Co t uous , , sc ete

Stochastic

= H(j)j(t)t+H((t))1 j(t)t+o(t)j j=(t)=(t)

= H(j)j(t)t+H((t)) 1 +(t)(t)t +o(t)j=(t)

E{H((t+t))|(t)}=H((t))+

H(j)j(t)t+o(t)

jWeusethisinthederivationoftheBellmanequation.



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Bellman sEquation

, ,

Stochastic

TJ(x(t), (t), t)= min E g(x(s), u(s))ds+F(x(T))

u(s), tts

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Bellman sEquation

t+t= min E g(x(s), u(s))ds

, ,

Stochastic

u(s),

t0st+t

T

+

min

E g(x(s), u(s))ds+

F

(x(T

))

u(s), t+t

t+tsT



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Bellman sEquation Stochastic

= minu(s), E

t+tt g(x(s),u(s))ds

tst+t

+J(x(t+t), (t+t), t+t)

Next,

we

expand

the

second

term

in

a

Taylor

series

about

x(t).

Weleave(t+t)alone,fornow.



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J(x(t), (t), t) =minE g(x(t), u(t))t+J(x(t), (t+t), t) +u(t)

J J(x(t), (t+t), t)x(t) + (x(t), (t+t), t)t +o(t)

x twhere

x(t) =x(t+t)x(t) =f(x(t), (t), u(t), t)t+o(t)



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UsingtheexpansionofEH((t+t)),J(x(t), (t), t)=min g(x(t), u(t))t

u(t)

+J(x(t), (t), t) + J(x(t),j,t)j(t)tj

J J

+ (x(t), (t), t)x(t) + (x(t), (t), t)t +o(t)x t

Wecancleanupnotationbyreplacingx(t)withx,(t)with,andu(t)withu.Copyright2007StanleyB.Gershwin.c 56


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J(x,,t)=minu

g(x,u)t+J(x,,t)+

jJ(x,j,t)jt

J

J

+ (x,,t)x+ (x,,t)t +o(t)x t WecansubtractJ(x,,t)frombothsidesandusetheexpressionforxtoget...



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0=min g(x,u)t+ J(x,j,t)jtu j

J J +

x(x,

,

t)f(x,

,

u,

t)t

+

t(x,,t)t+o(t)

or,



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J

(x,

,

t) =

J(x,

j,

t)j+

t j

Jmin

g(x,

u) +

(x,

,

t)f(x,

,

u,

t)

u x

Badnews: usuallyimpossibletosolve;Goodnews: insight.



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Anapproximation:whenT islargeandf isnotafunctionoft,typicaltrajectorieslooklikethis:

x

t



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e a s

Equation StochasticThatis,inthelongrun,xapproachesasteady-stateprobabilitydistribution.

Let

J

be

the

expected

value

of

g(x,

u),

where

u

is

theoptimalcontrol.Supposewestartedtheproblemwithx(0)arandomvariablewhoseprobabilitydistributionisthesteady-statedistribution.Then,forlargeT,

T

EJ =minuE 0 g(x(t), u(t))dt+F(x(T))JT



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Equation StochasticForx(0)and(0)specified

J(x(0), (0),0)JT +W(x(0), (0))or,moregenerally,forx(t) =xand(t) =specified,

J(x,,t)J(Tt) +W(x,)


FlexibleManufacturing

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ManufacturingSystemControlSinglemachine,multipleparttypes.x,u,dareN-dimensionalvectors.

TminE g(x(t))dt

subjectto: 0dxi(t)

=ui(t)di, i= 1,...,Ndt

prob((t+t) = 0|(t)=1)=pt+o(t)prob((t+t) =1|(t)=0)=rt+o(t)

iui(t)(t); ui(t)0i

x(0), (0)specified



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ManufacturingSystemControl

Define() ={u| iiui}.Then,for= 0,1,J (x,,t) = J(x,j,t)j+t

j

Jmin g(x) + (x,,t)(ud)u() x



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ApproximatingJ withJ(Tt) +W(x,)gives:J = (J(Tt) +W(x,j))j+

jW

min g(x) + (x,,t)(ud)u() xRecallthat

j = 0...j



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soJ = W(x,j)j+

jW

min g(x) + (x,,t)(ud)u() xfor= 0,1



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Thisisactuallytwoequations,onefor= 0,onefor= 1.WJ =g(x) +W(x,1)rW(x,0)r (x,0)d,x

for= 0,WJ =g(x) +W(x,0)pW(x,1)p+ min (x,1)(ud)

u(1) xfor= 1.


Flexible Single-part-typecaseManufacturing

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ManufacturingSystemControl Technically,notflexible!

Now,xanduarescalars,and(1)=[0,1/]=[0, ]

dWJ =g(x) +W(x,1)rW(x,0)r (x,0)d,

dxfor= 0,dW

J =g(x) +W(x,0)pW(x,1)p+ min (x,1)(ud)0u dxfor= 1.



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Seebook,Sections2.6.2and9.3;seeProbabilityslides#91120.

When= 0, u= 0.When= 1,ifdW 0, u= 0.dx



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W(x,)hasbeenshowntobeconvexinx. IftheminimumofW

(x,

1)

occurs

at

x

=

Z

and

W

(x,

1)

is

differentiable

for

all

x,

thendW 0x > Zdx

Therefore,ifx < Z, u=,ifx=Z, uunspecified,ifx > Z, u= 0.Copyright c 702007StanleyB.Gershwin.


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Manufacturing

SystemControlSurplus,orinventory/backlog:Productionpolicy: ChooseZ(thehedgingpoint)Then,if= 1,

ifx < Z, u=,ifx=Z, u=d,ifx > Z, u=0;

if= 0, u

= 0.

HowdowechooseZ?

dx(t)=u(t)d

dtCumulative

Production and Demand production

d t + Z

hedging point Z

surplus x(t)

demand dt

t



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g

SystemControl DeterminationofZ

ZJ =Eg(x) =g(Z)P(Z,1)+ g(x) [f(x,0)+f(x,1)]dx

inwhichP andf formthesteady-stateprobabilitydistributionofx.WechooseZ tominimizeJ. P andf aregivenby

f(x,0)=Aebx

f(x,1)

=

A

debx

d

P(Z,1)=ApdebZ



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g

SystemControl DeterminationofZwhere

r p

b=

d d

andAischosensothat

Z[f(x,0)+f(x,1)]dx+P(Z,1)=1

Aftersomemanipulation,A= bp(d) ebZ

db(

d) +

pand

db(d)P(Z,1)=

db(d) +pCopyright2007StanleyB.Gershwin.c 73


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g

SystemControl DeterminationofZSinceg(x) =g+x+ +gx ,

ifZ0,then ZJ =gZP(Z,1) gx[f(x,0)+f(x,1)]dx;

ifZ >0,

0

J =g+ZP(Z,1) gx[f(x,0)+f(x,1)]dx

Z+ g+x[f(x,0)+f(x,1)]dx.

0



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g

SystemControl DeterminationofZTominimizeJ:

ln Kb(1+g)

ifg+Kb(g+ +g)

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g

SystemControl DeterminationofZThatis,wechooseZsuchthat

ebZ =min 1, Kb g+ +gg+or

bZ 1 g+e =max 1,Kb g+ +g



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SystemControl DeterminationofZ

0prob(x0) = (f(x,0)+f(x,1))dx

0d=A 1 + ebxdx

d d 1 =A 1 + =A

d b b(d)bp(d) bZ = e

db(d) +p b(d)p bZ= e

db(d) +pCopyright2007StanleyB.Gershwin.c 77


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SystemControl DeterminationofZOr,

prob(x0)=

pmax 1,

1 g+db(d) +p Kb g+ +g

Itcanbeshownthatp

Kb= p+bd(d)Therefore

prob(x0)=Kbmax 1, 1 g+Kb g+ +gp g+

=max ,p+bd(d) g+ +g



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SystemControl DeterminationofZThatis,if

p>

g+,thenZ = 0andp+bd(d) g+ +g

prob(x0)= p ;p+bd(d)

if p < g+ ,thenZ >0andp+bd(d) g+ +g

prob(x0)= g+ .g+ +gThislooksalotlikethesolutionofthenewsboyproblem.Copyright2007StanleyB.Gershwin.c 79


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Zvs.dSystemControlBasevalues:g+ = 1,g =10d=.7,= 1.,r=.09,p=.01.

10090807060

Z 50403020100

d0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9



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SystemControl Zvs.g+Basevalues:g+ = 1,g =10d=.7,= 1.,r=.09,p=.01.

70605040

0 0.5 1 1.5 2 2.5 3g+

Z3020

100

3.5



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SystemControl Zvs.gBasevalues:g+ = 1,g =10d=.7,= 1.,r=.09,p=.01.

14

12

10

8

Z

6

4

2

00 1 2 3 4 5 6 7 8 9 10 11

g



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SystemControl Zvs.pBasevalues:g+ = 1,g =10d=.7,= 1.,r=.09,p=.01.

140012001000800

Z600400

2000

0.005 0.01 0.015 0.02 0.025 0.03 0.035p0 0.04


Flexible Two-part-typecaseManufacturing

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SystemControlx1

x2

d1

d2

u (t)1

u (t)2 Type 2

Type 1

r, p

u11/1

1/

2

u2

0

Capacityset(1)whenmachineisup.Copyright2007StanleyB.Gershwin.c 84


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SystemControl

Wemustfindu(x,)tosatisfyW

min (x,,t) u

u() xPartialsolutionofLP:IfW/x1 >0andW/x2 >0,u1 =u2 = 0.IfW/x1

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1

0

2

SystemControlCase:ExactsolutionifZ = (Z1, Z2) = 0x

2

x1

1 2u = u = 0

dx

dt

2

2u = 0

u = 01

1u =

1

2

u =

u1

1/

1/2

u2

0

u1

1/1

1/2

u2

1/1

1/

u2

0 u1



S C l

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1

2

0

SystemControlCase:ApproximatesolutionifZ >0x

2

1 2u = u = 0

dx

dt

2

2u = 0

u = 01

1

u =1

2

u =

u1

1/

1/2

u2

0

1/1

1/

u2

0

u1

1/1

1/2

u2

x1

u1


Flexible Two-part-typecaseS t C t l

Manufacturing

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45

SystemControlTwoparts,multiplemachineswithoutbuffers:

e12

4

61e

e

34e

23e

3

12

x2

Z

6x

1

556e

6

1 2

3

4

5

u2

e34

12e

u1

Ie23

e45

e56

d

61e


Flexible Two-part-typecaseManufacturingS t C t l

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SystemControlProposedapproximatesolutionformultiple-part,

singlemachinesystem:Rankordertheparttypes,andbringthemtotheirhedgingpointsinthatorder.


Flexible Single-part-typecaseS t C t l

ManufacturingS rpl s and tokens

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SystemControl SurplusandtokensOperatingMachineMaccordingtothehedgingpointpolicyisequivalenttooperatingthisassemblysystem

according

to

a

finite

B

bufferpolicy.D

M

S

FG


Flexible Single-part-typecaseS t C t l

ManufacturingSurplus and tokens

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SystemControl SurplusandtokensDisademandgenerator.

Whenever a demand arrives, DsendsatokentoB.

S isasynchronizationmachine. S is perfectly reliable and in

finitelyfast.

M

D

S

FG

B

F Gisafinitefinishedgoodsbuffer.

Bis

an

infinite

backlog

buffer.


Flexible Single-part-typecaseManufacturingSystem Control Material/token policies

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SystemControl Material/tokenpoliciesOperator

AnoperationcannottakeMachineplace

unless

there

is

a

Part Part tokenavailable.Operation

Consumable Waste

TokensauthorizeToken Token

production.Thesepoliciescanoftenbeimplementedeither withfinite

bufferspace,orafinitenumberoftokens.Mixturesarealsopossible.

Bufferspacecouldbeshelfspace,orfloorspaceindicatedwithpaintortape.


Multi-stage Proposedpolicysystems

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systems

TocontrolM B M B M

1 1 2 2 3

addaninformationflowsystem:B

1M B M

2 2 3M

1

S S2 3

D

S1

BB1

SB2

BB2

SB3

BB3

SB1


Multi-stage Proposedpolicysystems

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systemsB

1M B M

2 2 3M

1

S S2 3

D

S1

BB1

SB2

BB2

SB3

BB3

SB1

Bi arematerial buffersandarefinite.SBi aresurplus buffersandarefinite.BBi arebacklog buffersandareinfinite.ThesizesofBi andSBi arecontrolparameters.Problem: predictingtheperformanceofthissystem.Copyright c 942007StanleyB.Gershwin.

Multi-stagesystems

ThreeViewsofScheduling

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systemsThreekindsofschedulingpolicies,whicharesometimes

exactly

the

same.

Surplus-based: makedecisionsbasedonhow

muchproductionexceeddemand.Time-based: makedecisionsbasedonhowearlyor

lateaproductis.Token-based: makedecisionsbasedonpresence

orabsenceoftokens.Copyright c 952007StanleyB.Gershwin.

Multi-stage ObjectiveofSchedulingsystems

Surplusand

time

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Surplus and time

and Demand

earliness

production P(t)

demand D(t)

surplus/backlog x(t)

Objectiveistokeepcumulativeproductionclosetocumulativedemand.

CumulativeProduction

Surplus-basedpolicieslookatverticaldifferencesbetweenthegraphs.

Time-basedpolicieslookatthehorizontal

t

differences.Copyright2007StanleyB.Gershwin.c 96

Multi-stage Otherpoliciessystems

CONWIPkanban

and

hybrid

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CONWIP, kanban, and hybrid

CONWIP: finitepopulation,infinitebufferskanban: infinitepopulation,finitebuffershybrid: finitepopulation,finitebuffers



CONWIP,kanban,

and

hybrid

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CONWIPSupply Demand

Token flow

Demandislessthancapacity.How

does

the

number

of

tokens

affect

performance

(production

rate,inventory)?



CONWIP,kanban,

and

hybrid

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0.85

0.855

0.86

P


0.835

0.84

0.845

0.865

0.87

0.875

0 20 40 60 80 100 120

0

5

10

20

25

30

AverageBufferLevel

n1n2n3

15

0 20 40 60

Population Population

80 100 120

cCopyright2007StanleyB.Gershwin. 99


Basestock

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Basestock

Demand



FIFO

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First-In,FirstOut.Simpleconceptually,butyouhavetokeeptrackofarrivaltimes.Leavesoutmuchimportantinformation:duedate,valueofpart,currentsurplus/backlog

state,etc.



EDD

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Earliestduedate.Easytoimplement.Doesnotconsiderworkremainingontheitem,value

oftheitem,etc..



SRPT

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ShortestRemainingProcessingTimeWheneverthereisachoiceofparts,loadtheonewithleastremainingworkbeforeitisfinished.Variations: includewaitingtimewiththeworktime.

Useexpectedtimeifitisrandom.



Criticalratio

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Widelyused,butmanyvariations.Oneversion:Processingtimeremaininguntilcompletion

DefineCR= Duedate- CurrenttimeChoosethejobwiththehighestratio(provideditispositive).Ifajobislate,theratiowillbenegative,orthedenominator

willbezero,andthatjobshouldbegivenhighestpriority.Ifthereismorethanonelatejob,schedulethelatejobsin

SRPTorder.



LeastSlack

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Thispolicyconsidersapartsduedate.Defineslack =duedate- remainingworktimeWhenthereisachoice,selectthepartwiththeleast

slack.Variationsinvolvedifferentwaysofestimating

remainingtime.



Drum-Buffer-Rope

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DuetoEliGoldratt.

Basedon

the

idea

that

every

system

has

a

bottleneck.

Drum: thecommonproductionratethatthesystemoperates

at,whichistherateofflowofthebottleneck.

Buffer:

DBRestablishes

a

CONWIP

policy

between

the

entranceofthesystemandthebottleneck.ThebufferistheCONWIPpopulation.

Rope: thelimitonthedifferenceinproductionbetweendifferentstagesinthesystem.

But:Whatifbottleneckisnotwell-defined?Copyright c 1062007StanleyB.Gershwin.

Conclusions

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Manypoliciesandapproaches.Nosimplestatementtellingwhichisbetter.Policiesarenotallwell-definedintheliteratureorinpractice.Myopinion:

Thisisbecausepoliciesarenotderived fromfirstprinciples.Instead,theyaretestedandcompared.Currently,wehavelittleintuitiontoguidepolicydevelopment

andchoice.


MIT OpenCourseWarehttp://ocw.mit.edu

2.852 Manufacturing Systems Analysis

Spring 2010
http://ocw.mit.edu/http://ocw.mit.edu/

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Spring 2010

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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