Computational Stochastic Multistage Manufacturing Systems

Computational Stochastic Multistage Manufacturing Systems withStrikes and Other Adverse Random Events

F. B. Hanson∗

Laboratory for Advanced Computing, University of Illinois at Chicago851 Morgan St., M/C 249, Chicago, IL 60607-7045, USA

[email protected]://www.math.uic.edu/

˜

hanson/

J. J. Westman

Department of Mathematics, University of CaliforniaBox 951555 Los Angeles, CA 90095-1555, USA

[email protected]://www.math.ucla.edu/

˜

jwestman/

Keywords: Stochastic optimal control, dynamic program-ming, manufacturing systems, Poisson process.

Abstract

Multistage manufacturing systems (MMS) are models for theassembly of consumable goods. In the simple case, a lin-ear assembly line of workstations, components, or value, areadded to the product. Some examples assembly line prod-ucts are automobiles or printed circuit boards. Productionscheduling typically takes in to account workstation repair,failure, and defective pieces as stochastic events, effectingthe workstation production rates. The supply routing prob-lem of raw materials is not usually taken into account. How-ever, in this treatment, the effects of strikes and natural disas-ters, which may affect the routing of raw materials, are con-sidered for the MMS. Numerical results illustrate the optimalcontrol of MMS undergoing strikes, as well as workstationrepair and failure.

1. Introduction

In this paper, multistage manufacturing systems (MMS) areconsidered for the assembly of a single consumable good.The sequence of stages necessary to complete the finishedgood is represented as a linear chain of stages at which asubcomponent or value is added. Each stage consists of anumber of workstations which are assumed to be identicalin all respects and operate at the same level. The worksta-tions are subject to repair and failure. The control model forthe production scheduling problem needs to account for these

∗Work supported in part by the National Science Foundation GrantsDMS-96-26692 and DMS-99-73231.

stochastic events in order to insure that the production goalis met. The discipline assumed for the MMS is that ofJustin Timeor Stock less Productionwhich does not require thatlarge inventories of raw materials be kept on hand (see Hall[6]). Bielecki and Kumar [2] show that a zero inventory pol-icy is optimal for a manufacturing system that is subject touncertainties, this further justifies theJust in Timemanufac-turing discipline.

The goal of the control problem is to account for allstochastic events, such as workstation repair and failure,strikes, and natural disasters, so that the production goal isachieved in a specified way. The cost functional is usedto impose penalties for shortfall or surpluses in productionwhile maintaining a minimum control effort discipline. Inthis model, strikes and natural disasters can affect the MMSdirectly or the way in which raw materials enter the MMS,and therefore can limit the throughput of the MMS if thereis not sufficient raw materials which is therouting problem.This consideration is of great importance since strikes andnatural disasters can have very significant impact on the fi-nancial well being of a company. These concepts which arefeatures of the model presented here are illustrated by theUnited Auto Workers strikes against General Motors [1, 8],the strike of the United Parcel Service [10, 11], and naturaldisasters such as earthquakes and floods, for example.

The model considered here is an extension of the pro-duction scheduling model by Westman and Hanson [18]which utilizes state dependent Poisson processes [16] tomodel the rare events of workstation repair and failure aswell as strikes and natural disasters. In [18], all catastrophicevents, strikes and natural disasters, that effect the MMS arelumped together in one term. This allows the model to bea lower dimension, but presents a problem with tracking thecatastrophic events and there resolution. In this treatment,

each of the catastrophic events is represented by its own statevariable so that complete tracking of the event history canbe accurately represented. Additionally, this model includestemporal consideration for the effects of the strikes and nat-ural disasters on the demand rate, which is viewed as a con-stant in [18]. This feature of the model is necessary in orderto redefine the way in which the production goal is to be metdependent on whether a strike occurs or not. The productionscheduling in this modelanticipatescatastrophic events andcompensates for them, however this may not be enough (de-pendent on the length of the strike) or too much if a strikedoes not occur. Therefore the demand rate may need to bedynamically adjusted.

For completeness theLQGP problem(see [13]) usingextensions for state dependent Poisson noises (see [16, 18])is presented in Section 2.. The model for the productionscheduling of a MMS presented here forms aLQGP problemutilizing state dependent Poisson noises and a rebalancing ofthe demand rate is given in Section 3. In Section 4. two ex-amples of state path realizations are presented that show theeffects of a strike in conjunction with demand rate rebalanc-ing.

2. LQGP Problem Formulation

For completeness we present the canonical form for theLQGP problem that originally appears in Westman and Han-son [13], for the case with state independent Poisson noise,and [16] for state dependent Poisson noise. Additionally,considerations for modeling a physical system are presentedas well, as well as formal solution to the LQGP problem.

The linear dynamical system for the LQGP problem isgoverned by the stochastic differential equation (SDE) sub-ject to Gaussian and state dependent Poisson noise distur-bances is given by

dX(t) = [A(t)X(t) + B(t)U(t) + C(t)]dt

+ G(t)dW(t) + [H1(t) ·X(t)]dP1(X(t), t)+ [H2(t) ·U(t)]dP2(X(t), t)

+ H3(t)dP3(X(t), t), (1)

for general Markov processes in continuous time, withm×1state vectorX(t), n × 1 control vectorU(t), r × 1 Gaussiannoise vector dW(t), andq`×1 space-time Poisson noise vec-tors dP`(X(t), t), for ` = 1 to 3. The dimensions of therespective coefficient matrices are:A(t) is m × m, B(t) ism × n, C(t) is m × 1, G(t) is m × r, while theH`(t) aredimensioned, so that

[H1(t) · x]=

[

∑

k

H1ijk(t)xk

]

m×q1

, (2)

[H2(t) · u]=

[

∑

k

H2ijk(t)uk

]

m×q2

, (3)

and

H3(t) = [H3ij(t)]m×q3 . (4)

Note that the space-time Poisson terms are formulated tomaintain the linear nature of the dynamics, but the first twoare actually bilinear in eitherX or U anddP` for ` = 1 or2, respectively.

The state dependent Poisson noisecan be viewed as asequence of events that is represented by itsith couple

{Ti(X(Ti)), Mi(X(Ti))}, (5)

for i = 1 to k, whereTi(X(Ti)) is the time for the oc-currence of theith jump with state dependent mark ampli-tudeMi(X(Ti). This representation of the Poisson processprovides more realism and flexibility for a wider range ofstochastic control applications since the arrival times and am-plitudes may depend of the state of the system. Additionally,this formulation allows for simpler dynamical system mod-eling of complex random phenomena.

The state dependent vector valued marked Poissonnoises are related to the Poisson random measure (see Gih-man and Skorohod [5] or Hanson [7]) and are defined as

dP`(X(t), t) = [dP`,i(X(t), t)]q`×1

=

[

∫

Z`,i

zP`,i(dz,X(t), dt)

]

q`×1

(6)

for ` = 1 to 3 which consists ofq` independent differen-tials of space-time Poisson processes that are functions of thestate,X(t), wherez is the Poisson jump amplitude randomvariable or the mark of thedP`,i(X(t), t) Poisson processwhere` = 1 to 3 andi = 1 to q`. The mean or expectation isgiven by

Mean[dP`(X(t), t)]

= Λ`(X(t), t)dt∫

Z`

zφ`(z,X(t), t)dz

≡ Λ`(X(t), t)Z`(X(t), t)dt, (7)

whereΛ`(X(t), t) is the diagonal matrix representation ofthe state dependent Poisson ratesλ`,i(X(t), t) for ` = 1 to3 andi = 1 to q`, Z`(X(t), t) is the mean of the jump am-plitude mark vector andφ`,i(z,X(t), t) is the density of the(`, i)th amplitude mark component. Assuming component-wise independence,dP`(X(t), t) has covariance given by

Covar[dP`(X(t), t), dP>` (X(t), t)]

= Λ`(∗)dt∫

Z`

(z− Z`(∗))(z− Z`(∗))>φ`(z, ∗)dz

≡ Λ`(∗)σ`(∗)dt, (8)

with, for instance,σ`(∗) = σ`(X(t), t) = [σ`,i,jδi,j ]q`×q`

denoting the diagonalized covariance of the amplitude markdistribution fordP`(X(t), t). Again, the mark vector is notassumed to have a zero mean, i.e.,Z` 6= 0, permitting addi-tional modeling complexity. Note, that for discrete distribu-tions the above integrals need to be replaced by the appropri-ate sums.

The Gaussian white noise term,dW(t), consists of rindependent, standard Wiener processesdWi(t), for i = 1 tor. These Gaussian components have zero infinitesimal mean,

Mean[dW(t)]=0r×1 (9)

and diagonal covariance,

Covar[dW(t), dWT (t)] = Irdt. (10)

It is further assumed that all of the individual componentterms of the Gaussian noise are independent of all of thePoisson processes,

Covar[dW(t), dPT` (t)] = 0r×q` , (11)

for all `.Thejth jump of the{`, i}th space-time Poisson process

at timet`,i,j with amplitudeM`,i,j causes the following jumpfrom t−`,i,j to t+`,i,j in the state:

[X](t`,i,j)=

[H1(t`,i,j)X(t−`,i,j)]iM`,i,j , ` = 1[H2(t`,i,j)U(t−`,i,j)]iM`,i,j , `=2

[H3(t`,i,j)]iM`,i,j , ` = 3

. (12)

¿From the above statistical properties of the stochastic pro-cesses,dW anddP`, it follows that the first two conditionalinfinitesimal moments of the state, fundamental for modelingapplications, are

Mean[dX(t) | X(t) = x,U(t) = u]

=[

A(t)x + B(t)u + C(t) + [H1(t)x](Λ1Z1)(x, t)

+ [H2(t)u](Λ2Z2)(x, t) + H3(t)(Λ3Z3)(x, t)]

dt (13)

and the conditional infinitesimal covariance,

Covar[dX(t) | X(t) = x,U(t) = u]

=[

(GGT )(t) + [H1(t)x](Λ1σ1)(x, t)[H1(t)x]>

+ [H2(t)u](Λ2σ2)(x, t)[H2(t)u]>

+ H3(t)(Λ3σ3)(x, t)H>3 (t)

]

dt. (14)

The quadratic performance index or cost functional that isemployed is quadratic with respect to the state and controlcosts, is given by thetime-to-goor cost-to-gofunctionalform:

V [X,U, t] =12(X>SX)(tf ) +

∫ tf

t

C(X(τ),U(τ), τ)dτ (15)

with

C(x,u, t) =12

[

x>Q(t)x,+u>R(t)u]

(16)

where the time horizon is(t, tf ), with S(tf ) ≡ Sf isthe quadratic final cost coefficient matrix andC(x,u, t) isquadratic instantaneous cost function. The final cost, knownas thesalvage cost, is given by the quadratic form,

x>Sfx = Sf : xx> = Trace[Sfxx>]. (17)

In order to minimize (15) requires that the quadratic controlcost coefficientR(t) is assumed to be a positive definiten×

n array, while the quadratic state control coefficientQ(t) isassumed to be a positive semi-definitem × m array. ThecoefficientsR(t) andQ(t) are assumed to be symmetric forsimplicity. The LQGP problem is defined by (1, 15).

The stochastic dynamic programming approach is usedto solve the control problem. So a functional, theoptimal,expected cost, is defined as:

v(x, t) ≡ Minu[t,tf )

[

MeanP,W[t,tf )

[V | X(t) = x,U(t) = u]

]

, (18)

where the restrictions on the state and control are that theybelong to the admissible classes for the state,Dx, and con-trol, Du, respectively. A final condition on the optimal, ex-pected value, is determined from the final orsalvagecostusing (18) withV [X,U, tf ] in (15) and is given by

v(x, tf ) =12x>Sfx, for x ∈ Dx. (19)

Upon applying the principle of optimality to the opti-mal, expected performance index, (18, 15) and the chain rulefor Markov stochastic processes in continuous time for theLQGP problem yields

0 =∂v∂t

(x, t) + Minu

[(A(t)x + B(t)u

+ C(t))T ∇x[v](x, t) +12

(

GGT )

(t) : ∇x[∇>x [v]](x, t)

+12x>Q(t)x +

12u>R(t)u

+q1

∑

k=1

λ1,k(x, t)∫

Z1,k

[v(x + [H1(t) · x]kz, t)

− v(x, t)] φ1,k(z,x, t)dz

+q2

∑

k=1

λ2,k(x, t)∫

Z2,k

[v(x + [H2(t) · u]kz, t)

− v(x, t)] φ2,k(z,x, t), t)dz

+q3

∑

k=1

λ3,k(x, t)∫

Z3,k

[v(x + H3,k(t)z, t)

− v(x, t)] φ3,k(z,x, t)dz] , (20)

where the notation below defines the column arrays used inthe Poisson terms,

[H1(t) · x]k ≡

m∑

j=1

H1,i,k,j(t)xj

m×1

,

[H2(t) · u]k ≡

n∑

j=1

H2,i,k,j(t)uj

m×1

,

H3,k(t) ≡ [H3,i,k(t)]m×1, (21)

and where the double dot product is defined by

A : B =∑

i

∑

j

Ai,jBi,j = Trace[AB>]. (22)

The backward partial differential equation (PDE) (20) isknown as the Hamilton-Jacobi-Bellman (HJB) equation and

is subject to the final condition. The argument of the mini-mum is the optimal control,u∗(x, t); if there are no controlconstraints the optimal control is known as the regular con-trol, ureg(x, t).

To solve (20) subject to the final condition, for theLQGP problem a modification of the formal state decomposi-tion of the solution for the usual LQG problem (for the usualLQG, see Bryson and Ho [3], Dorato et al. [4], or Lewis [9])is assumed:

v(x, t) =12xT S(t)x + DT (t)x + E(t)

+12

∫ tf

t

(

GGT )

(τ) : S(τ)dτ. (23)

The final condition is satisfied, provided that

S(tf ) = Sf , D(tf ) = 0, and E(tf ) = 0. (24)

The ansatz (23) would not, in general, be true for thestate dependent case, but would be applicable if the Pois-son noise is locally state independent, while globally statedependent. That is, the state domain is decomposed into sub-domains,Dx =

⋃

iDxi , where the arrival rates and momentsfor all the Poisson processes are constant in the regionDxi

and can be expressed as

Λ(X(t), t) = Λi(t)Z(X(t), t) = Zi(t)σ(X(t), t) = σi(t)

, for X(t) ∈ Dxi , (25)

for all subdomainsi. If there are any explicit dependence onX(t) then the resulting system would then form a LQGP/Uproblem (for more details see Westman and Hanson [14, 15,16, 17]).

Assuming the ansatz (23) holds the regular, uncon-strained optimal control,u∗ = ureg, is given by

ureg(t) = − R−1(t) BT (t) [S(t)x + D(t)] . (26)

Assuming regular control, the coefficients for the optimal ex-pected performance (23) are given by

0m×m = S(t) +[

AT S + SA + Q]

(t)

+ ˜Γ1(t)−[

S B R−1BT S

]

(t), (27)

0m×1 = D(t)+[

(

A + (Λ1Z1)T HT1

)TD

]

(t)

+[

S(

C + H3Λ3Z3)

−S B R−1BT D

]

(t), (28)

0 = E(t) +[

(

C + H3Λ3Z3)T

D]

(t)

+12

[

(

HT3 SH3

)

: Λ3ZZ3 −DTB R−1

BT D]

(t), (29)

where

Γ1(t) ≡[(

[HT1 ]iS[H1]j : Λ1ZZ1

)

(t)]

m×m

+ 2[

(

Λ1Z1)T

HT1 S

]

(t), (30)

Γ2(t) ≡[(

[HT2 ]iS[H2]j : Λ2ZZ2

)

(t)]

n×n, (31)

and

ZZ`(t) ≡ σ`(t) +(

Z`ZT`

)

(t)

=[

σ`,iδi,j + Z`,iZ`,j]

q`×q`(32)

for ` = 1 to 3 with

R(t) ≡ R(t) + ˜Γ2(t), (33)

B(t) ≡ B(t) + ((Λ2Z2)T HT2 )(t), (34)

and

˜Γ` ≡ (Γ` + ΓT` ). (35)

Since the matrixR is positive definite,R−1 exists and then sodoes R−1. Note (27) appears to have Riccati-like quadraticform, but in general is highly nonlinear through theS de-pendence ofR and if H` = [H`,i,j,k]m×q`×m` , thenHT

` =[H`,j,i,k]q`×m×m` .

Due to uni-directional coupling of these matrix differ-ential equations, it is assumed that the nonlinear matrix dif-ferential equation (27) forS(t) is solved first and the resultfor S(t) is substituted into equation (28) forD(t), whichis then solved, and then both results forS(t) andD(t) aresubstituted into equation (29) for the state-control indepen-dent termE(t). SinceS(t) is a symmetric matrix by be-ing defined with a quadratic form, only a triangle part ofS(t) need be solved, orn · (n + 1)/2 component equations.Thus, for the whole coefficient set{S(t),D(t), E(t)}, onlyn·(n+1)/2+n+1 component equations need to be solve, sothat for largen the count isO(n2/2), asymptotically, whichis the same order of effort in getting the triangular part ofS(t).

3. LQGP Problem Formulation for MMS

Consider a MMS that produces the single consumable com-modity. The MMS consists ofk stages that form a linearsequence that is used to assemble the finished product. Themechanisms by which the input,loading stage, of raw mate-rials and the delivery of finished products,unloading stage,are not considered as stages in the MMS. However, state de-pendent Poisson noises are used to model catastrophic eventsthat affect the delivery of raw materials to the MMS. At timet in the manufacturing planning horizon for stagei, there areni(t) operational workstations. For each stagei, all work-stations are assumed to be identical and produce goods at thesame rateci(t) with a capacity of producingMi parts per unittime. For each stagek of the manufacturing system the stateof the MMS is given by the number of operational worksta-tions, the surplus aggregate level, and the 3 state indicatorsfor the effects of primary and secondary strikes and naturaldisasters on the MMS, respectively. Therefore the dimensionof the state of the system is a5k × 1 vector.

A primary strike is any strike that directly affects theassembly of the consumable good in such a way that whenthey occur the MMS is shut down and no goods are produced.A secondary strike is any strike that reduces the number ofgoods that can be produced, but does not disable the MMS.The impact of strikes and natural disasters on stagei, sij(t),evolves according to the purely stochastic equation,

dsij(t) = −dPSij(sij(t), t) + dPR

ij(sij(t), t). (36)

The term−dPSij(sij(t), t) is used to model the effects of the

start of an event anddPRij(sij(t), t) is used to model the reso-

lution of an event, where the events arej = 1 primary strike,j = 2 secondary strike, andj = 3 natural disasters. The ar-rival rate for strikes is usually deterministic in the sense thatnormally there is a fixed date, sayts, for the termination ofa labor agreement, which if not resolved can lead to a strike.For all of these stochastic processes, the arrival rate is themean time between the occurrences of such events and theamplitude or mark density function is modeled to representthe expected value and covariance for the event to occur. Thevalues for state indicators are bounded by

0 ≤ sij(t) ≤ smaxij , (37)

wheresmaxij ≤ 1 is the maximum impact the event can have

on the MMS. The various events are considered to be additiveand a functional is used to represent the net effect given by:

si(t) = Min[1, si1(t) + si2(t) + si3(t)]. (38)

If si(t) = 0, then there is no effect on the MMS. Ifsi(t) = 1then no production takes place.

Each workstation is subject to failure and can be re-paired. The mean time between failures and the repair dura-tion is exponentially distributed. The evolution of the numberof active workstations is bounded by

0 ≤ ni(t) ≤ Ni, (39)

for all time, process usingstate dependent Poisson noisesgiven by

dni(t) = dPR(ni(t), t)− dPF (ni(t), t), (40)

wheredPR(n(t), t) anddPF (n(t), t) are used to model therepair and failure processes, respectively, which only de-pends on the current number of active workstations. Thisprocess forms a birth and death process or a random walk onthe interval (39). The number of active workstations,ni(t),determines the arrival rates and mean mark amplitudes forfailure and repair events respectively given by

1/λFi =

{

0, ni = 01/λF

i , 1 ≤ ni ≤ Ni

}

, (41)

ZFi =

{

0, ni = 0∑ni

j=1 jPrj , 1 ≤ ni ≤ Ni

}

, (42)

1/λRi =

{

1/λRi , 0 ≤ ni < Ni

0, ni = Ni

}

, (43)

and

ZRi =

{∑Ni−ni

j=1 jPrNi−j , 0 ≤ ni < Ni

0, ni = Ni

}

, (44)

with

Pri = Pr[ni(t) = i] (45)

which represents the probability of havingi operationalworkstations at timet.

The surplus aggregate level represents the surplus (ifpositive) or shortfall (if negative) of the production of piecesthat have successfully completedi stages of the manufactur-ing process is given by

dai(t) = [Mici(t)ni(t) + ui(t)− di(t)] dt + gi(t)dWi(t)

−3

∑

j=1

Hij(t)dPSij(sij(t), t). (46)

The change in the surplus aggregate level,dai(t), is de-termined by the number of pieces that have successfullycompleted i stages of the manufacturing process (i.e.,Mini(t)ci(t)dt), that are not defective, and are not consumedby stagei + 1 (i.e., di(t)dt), and by the status of the work-stations. The production rateci(t) needs to be physicallyrealizable with respect to the number of operational worksta-tions and to the impact of strikes and natural disasters. Thetermui(t)dt is used to adjust the production rate where thecontrol ui(t) is expressed as the number of pieces per unittime. The term,gi(t)dWi(t), is used to model the randomfluctuations in the number of pieces produced, for exampledefective pieces. The demand term,di(t)dt, is the consump-tion of the pieces produced by stagei by stagei + 1. Thedemand needs to be adjusted to compensate for over or un-der production as a result of strikes or natural disasters tomeet the fixed production goal. The last term uses Poissonprocesses to represent the effects of strikes and natural disas-ters where the coefficients,Hij(t), are the expected value forthe shortfall in the number of pieces produced.

The surplus aggregate level,ai(t), for stagei is depen-dent on the number of operational workstations,ni(t). Thebirth and death process for the number of operational work-stations is anembedded Markov chain(see Taylor and Karlin[12], for instance), for the surplus aggregate level. Thus thebirth and death process is used to describe the sojourn timesfor the discontinuous jumps in the surplus aggregate leveldue to the effects of workstation repair or failure. Hence,the surplus aggregate level is a piecewise continuous processwhose discontinuous jumps are determined by the stochasticprocess for the number of operational workstations. Addi-tionally, the processes for strikes and natural disasters alsoinduce discrete large jumps in the surplus aggregate levelthrough the sum in the last term of (46).

The demand ratedi(t) is the number of parts needed perunit time to insure that the manufacturing process is a contin-uous flow of work, so that the desired number of completed

pieces are produced. The demand rate must also take intoaccount, based on past history, a minimal buffer level suf-ficient to compensate for defective pieces as well as work-station failures, and to insure that the proper start-up surplusaggregate levels are present for the next planning horizon. Inorder for the MMS to be well posed, it is required that

0 ≤ di(t) ≤ MiNi (47)

per unit time so that the production goal of the MMS is at-tainable.

In this formulation of the production scheduling, a re-balancing of the demand rate is used to compensate for theeffects of strikes and natural disasters. LetPG denote thetotal production goal for the manufacturing planning horizonT > 0. The simple average demand rate which is used in thispaper would initially be given as:

di(0) =PGT

, (48)

for i = 1 to k.Since the problem formulation presented here antici-

pates the strikes and natural disasters additional pieces areproduced to compensate for these shortfalls in production.Here a rebalancing of demand should be done to adjust forthese effects. For simplicity we focus only on the primarystrike. A primary strike may occur at timets which coincideswith the termination of a labor agreement. Assume that theresolution of the strike if one occurs is at timetsr = ts + δs

whereδs is the actual duration of the strike. LetPPi(t) bethe cumulative number of nondefective pieces produced atstagei during the production interval[0, t] for t < T . De-pending on whether or not a strike occurs a rebalancing ofthe demand rate would be given by

di(t) =PG− PPi(t)

T − t, (49)

where

t ={

ts, No Striketsr, Strike

}

. (50)

This use of rebalancing of demand can also be used to con-sume pieces that remain in surplus due to workstation fail-ure. Suppose a failure occurs at stagei > 1, then a surplus ofpieces may accrue at stagei which would need to be con-sumed by stagesi throughk (the remaining stages of theMMS). The plant manager would need to decide the disci-pline for doing this. For example, the manager may choose toconsume the pieces over the remaining manufacturing hori-zon as demonstrated above. Note, the production demandmay be time dependent and therefore would require modi-fications to meet the objectives of the planning horizon, aswould be the case of cyclic or seasonal demand of commodi-ties.

The cost function used is the standardtime-to-goorcost-to-goform (15, that is motivated by azero inventoryor

Just in Timemanufacturing discipline (see Hall [6] and Bi-elecki and Kumar [2]) while utilizing minimum control ef-fort. In this formulation, the salvage cost,S(tf ), is used toimpose a penalty on surplus or shortfall of production at theend of the planning horizon. The termQ(t) is used to penal-ize shortfall and surplus production during the planning hori-zon, this term is used to maintain a strict regimen on whenthe consumable goods are to be produced. The termR(t) isused to enforce a minimum control effort penalty.

To solve this problem, assume the regular control (26)and solve the nonlinear system of ordinary differential equa-tions (27,28,29). This allows the calculation of the produc-tion rates used by the plant manager of the MMS . The pro-duction rate,ci(t) is a utilization, that is the fraction of timebusy. The physically realizable production rate is boundedby

0 ≤ ci(t) ≤ cmaxi (t), (51)

where

cmaxi (t) =

{

si(t), i = 1min[1,MPRi], 1 < i ≤ k

}

, (52)

with

MPRi =si−1(t)ci−1(t)ni−1(t)Mi−1

ni(t)Mi, (53)

which is the unconstrained maximum physical productionrate for stagei. The maximum production rate,cmax

i (t), isthe minimum value of the physical production rate,1.00 orfull utilization, and production limitations that arise due toa shortfall of production from the previous stage due to ei-ther machine failure, strikes, or natural disasters, wheresi(t)is the total impact of strikes and natural disasters on stageigiven by

si(t) = Max

1−3

∑

j=1

sij(t), 0

. (54)

The strike and natural disaster influence is used to limit theamount of pieces that can be produced by a given stage andis bounded by0 ≤ si(t) ≤ 1, such thatsi(t) = 0 means thatno production can occur.

In this formulation the production rate is a parameter ofthe dynamic system and is adjusted by the control decision.The regular controlled production level,

cregi (t) =

{

0, ni(t) = 0

ci(t) + uregi (t)

Mini(t), ni(t) > 0

}

, (55)

which anticipates for the stochastic effects of workstation re-pair and failure, defective parts, strikes or natural disasters.Note, that with the assumption of regular control, the surplusaggregate level will always be forced to be zero, thereforethe regular controlled production level may not be physicallyrealizable. In the case of a primary strike,ci(t)=0 for all

i, the regular controlled production level which is the sameas the regular control would be the number of pieces thatneeds to be produced to force the surplus aggregate level tozero, which clearly is not physically realizable. The con-strained controlled production level,c∗i (t), is the restrictionof the regular controlled production level to be physically re-alizable and is given by

c∗i (t) = min[cregi (t), cmax

i (t)], (56)

wherecmaxi (t) is given in (51). The constrained controlled

production rate is used as the production rate for the work-stations in the state equation for the surplus aggregate level(46).

4. Numerical Example of LQGP MMS

Here we present two path realizations for this model. Con-sider a MMS withk = 3 stages with a planning horizon ofT = 100 days and a production goal ofPG = 57, 500 pieceswhich means the initial demand rate for all stages is given bydi(t) = 575 pieces per day. Let the initial surplus aggre-gate level for all stages be zero, the total number of work-stations,Ni, for each stage be 3, 5, and 4, respectively, theGaussian random fluctuations of production is assumed ab-sent (gi(t) = 0 for i = 1 to 3), and that secondary strikes andnatural disasters are not considered, for simplicity the strikeimpact state variable will be referred to assi(t). The statevariables the MMS are

x(t)=

n(t)

a(t)

s(t)

9×1

, (57)

where each component is a3 × 1 vector. A single primarystrike can occur at the beginning of day 63 (i.e.,ts = 63)of the planning horizon with an expected time of 14 days toresolve itself (i.e.,Mean[δs] = 14), that is the arrival ratesfor the strike are given by

1/λS(si(t), t) ={

(63− t) days, t < 630 days, t ≥ 63

}

, (58)

and

1/λR(si(t), t) ={

0 days, si(t) = 114 days, si(t) < 1

}

(59)

with an expected impact or shortfall of575 ∗ 14 = 8050pieces. The effects of a primary strike and its resolution onthe MMS will disable or enable production for all stages.

The operational characteristics for the workstations aresummarized in Table 4.

Let ΦRk,i,j andΦF

k,i,j denote the discrete mark transitionprobabilities for the repair and failure, respectively, ofj − 1

Production Mean Time Mean TimeStage Capacity,Mi between Failure to Repair

i (pieces/day) 1/λFi (days) 1/λR

i (days)

1 238 85.0 2.502 143 75.0 1.503 178 90.0 1.75

Table 1: Operational workstation parameters.

workstations for stagek when there arei operational work-stations, with transition matrices given by

ΦR1 =

0.00 0.95 0.050.00 1.00 0.001.00 0.00 0.00

, (60)

ΦF1 =

1.00 0.00 0.000.00 1.00 0.000.00 0.90 0.10

, (61)

ΦR2 =

0.00 0.90 0.07 0.02 0.010.00 0.92 0.07 0.01 0.000.00 0.93 0.07 0.00 0.000.00 1.00 0.00 0.00 0.001.00 0.00 0.00 0.00 0.00

, (62)

ΦF2 =

1.00 0.00 0.00 0.00 0.000.00 1.00 0.00 0.00 0.000.00 0.95 0.05 0.00 0.000.00 0.94 0.05 0.01 0.000.00 0.92 0.05 0.02 0.01

, (63)

ΦR3 =

0.00 0.96 0.03 0.010.00 0.97 0.03 0.000.00 1.00 0.00 0.001.00 0.00 0.00 0.00

, (64)

and

ΦF3 =

1.00 0.00 0.00 0.000.00 1.00 0.00 0.000.00 0.95 0.05 0.000.00 0.90 0.07 0.03

. (65)

The cost functional used is (15) where the coefficient matri-ces are given by

S(tf ) =

03×3 03×3 03×3

03×3 Sf 03×303×3 03×3 03×3

, (66)

Sf =

1.2 0.0 0.00.0 1.9 0.00.0 0.0 2.6

, (67)

Q(t) =

03×3 03×3 03×3

03×3 Q2 03×303×3 03×3 03×3

, (68)

Q2 =

0.9 0.0 0.00.0 1.6 0.00.0 0.0 2.3

, (69)

and

R(t) =

12000 0 00 12000 00 0 12000

. (70)

By comparing the coefficients of (1) with the state equa-tions for the MMS (40,46,36) the deterministic coefficientsare given by

A(t) =

03×3 03×3 03×3

diag[M]diag[c(t)] 03×3 03×3

03×3 03×3 03×3

, (71)

B(t)=

03×3

I3×3

03×3

, (72)

and

C(t) =

03×1

−d(t)03×1

, (73)

wherediag[M] = [Miδi,j ]k×k is the diagonal matrix repre-sentation of the vectorM and with the onlynonzerostochas-tic process and corresponding coefficient matrix given by

dP3(X(t), t) =

dPR(n(t), t)

dPF (n(t), t)

dPS(s(t), t)

dPR(s(t), t)

, (74)

and

H3(t) =

1 0 0 −1 0 0 0 00 1 0 0 −1 0 0 00 0 1 0 0 −1 0 00 0 0 0 0 0 −8050 00 0 0 0 0 0 −8050 00 0 0 0 0 0 −8050 00 0 0 0 0 0 −1 10 0 0 0 0 0 −1 10 0 0 0 0 0 −1 1

. (75)

Using the above numerical values and assuming the regularcontrol the temporal dependent coefficientsS(t), dD(t), andE(t) can be determined from (27,28,29). With the tempo-ral coefficients known the regular control can be determinedfrom (26) for any state value. Finally, the regular control and

value for the state can be used to determine the MMS oper-ating parameters for the regular controlled production rate,cregi (t), and constrained controlled production rate,c∗i (t).

Figures 1, 2, and 3 show the results for the case whena strike occurs with a rebalancing of the demand rate at theend of the strike which leads to a higher demand rate for theremaining manufacturing horizon. The percent relative errorfor this sample path for the final stage 3 which is the outputof the MMS is0.1406%. Figures 4, 5, and 6 are for the

575

600

625

0 20 40 60 80 100

Piec

es p

er D

ay

�

Time into Planning Horizon (Days)

Production Demand Rate

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Prod

uctio

n R

ate

(Util

izat

ion)

�


Controlled Production Rates

0

1

2

3

0 20 40 60 80 100

Stat

e V

alue

�


Sample Path Realization

Workstations Stage 1Strike

-2

2

6

10

14

18

0 20 40 60 80 100

Perc

ent R

elat

ive

Err

or

�


Percent Relative Error

Figure 1: State sample path realization for active worksta-tions, production rate for stage 1, demand rate, and percentrelative error of throughput of stage 1.

575

600

625

0 20 40 60 80 100

Piec

es p

er D

ay

�



0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Prod

uctio

n R

ate

(Util

izat

ion)

�



0

1

2

3

4

5

0 20 40 60 80 100

Stat

e V

alue

�




-2

2

6

10

14

18

0 20 40 60 80 100

Perc

ent R

elat

ive

Err

or

�




case when the strike does not occur and rebalancing of thedemand rate occurs atts = 63 days which leads to a reduceddemand rate for the remaining manufacturing horizon. Thepercent relative error for this sample path for the final stage 3which is the output of the MMS is−0.006426%. The re-sults presented here do not reflect the need to rebalance thedemand rate for workstation repairs and failures.

575

600

625

0 20 40 60 80 100

Piec

es p

er D

ay

�



0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Prod

uctio

n R

ate

(Util

izat

ion)

�



0

1

2

3

4

0 20 40 60 80 100

Stat

e V

alue

�




-2

2

6

10

14

18

0 20 40 60 80 100

Perc

ent R

elat

ive

Err

or�




450

500

550

600

0 20 40 60 80 100

Piec

es p

er D

ay

�



0.6

0.7

0.8

0.9

1

0 20 40 60 80 100

Prod

uctio

n R

ate

(Util

izat

ion)

�



0

1

2

3

0 20 40 60 80 100

Stat

e V

alue

�




0

3

6

9

12

15

18

0 20 40 60 80 100

Perc

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elat

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Err

or

�




5. Conclusions

A sudden labor strike or natural disaster can have catas-trophic consequences that are much more serious than por-trayed by the typical continuous state model, in addition tothe jumps due to the random failure and repair of multistagemanufacturing system (MMS) workstations. The model pre-sented in this paper can be used to account for all of theserandom events, alter the demand rate to meet the produc-tion goal, and to determine the production rates of the work-stations in order to minimize adverse financial effects. Ourcomputational procedures lead to systematic approximationsto the MMS model formulated here for strikes and other ran-dom catastrophic events.

450

500

550

600

0 20 40 60 80 100

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0.6

0.7

0.8

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0 20 40 60 80 100

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0.6

0.7

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References

[1] Auster B.B. and Cohen W.,Rallying the Rank and File,Online U.S. News, 1996.{URL: http://www.usnews.com/usnews/issue/1labor.htm}

[2] Bielecki T. and Kumar P.R.,Optimality of Zero-Inventory Policies for Unreliable Manufacturing Sys-tems, Operations Research, Vol.36, pp.532-541, 1988.

[3] Bryson A.E. and Ho Y.,Applied Optimal Control, Ginn,Waltham, 1975.

[4] Dorato P., Abdallah C. and Cerone V.,Linear-Quadratic Control: An Introduction, Prentice-Hall, En-glewood Cliffs, NJ, 1995.

[5] Gihman I.I. and Skorohod A.V.,Stochastic DifferentialEquations, Springer-Verlag, New York, 1972.

[6] Hall R.W., Zero Inventories, Dow Jones-Irwin, Home-wood, IL, 1983.

[7] Hanson F.B.,Techniques in Computational StochasticDynamic Programming, Digital and Control SystemTechniques and Applications, edited by C.T. Leondes,Academic Press, New York, pp.103-162, 1996.

[8] Holstein W.A., War of the Roses, Online U.S. News,July 20, 1998.{URL: http://www.usnews.com/usnews/issue/980720/20gm.htm}

[9] Lewis F.L.,Optimal Estimation with an Introduction toStochastic Control Theory, Wiley, New York, 1986.

[10] Package Deal, Online News Hour, August 19, 1997.{URL: http://www.pbs.org/newshour/bb/business/july-dec97/ups8-19a.html}

[11] Return to Sender, Online News Hour, August 4, 1997.{URL: http://www.pbs.org/newshour/bb/business/july-dec97/ups8-4a.html}

[12] Taylor H.M. and Karlin S.,An Introduction to Stochas-tic Modeling, Academic Press, San Diego, 1984.

[13] Westman and J.J. Hanson F.B.,The LQGP Prob-lem: A Manufacturing Application, Proceedings of the1997 American Control Conference, Vol.1, pp.566-570, 1997.

[14] Westman J.J. and Hanson F.B.,The NLQGP Prob-lem: Application to a Multistage Manufacturing Sys-tem, Proceedings of the 1998 American Control Con-ference, Vol.2, pp 1104-1108, 1998.

[15] Westman J.J. and Hanson F.B.,Computational Methodfor Nonlinear Stochastic Optimal Control, Proceedingsof the 1999 American Control Conference, pp 2798-2802, 1999.

[16] Westman J.J. and Hanson F.B.,State Dependent JumpModels in Optimal Control, Proceedings of the 1999Conference on Decision and Control, pp.2378-2384,December 1999.

[17] Westman J.J. and Hanson F.B.,Nonlinear State Dynam-ics: Computational Methods and Manufacturing Ex-ample, International Journal of Control, to appear, 17pages in galleys, 12 November 1999.

[18] Westman J.J. and Hanson F.B.,MMS ProductionScheduling Subject to Strikes in Random Environments,Proceedings of the 2000 American Control Conference,to appear, 5 pages in galleys, June 2000.

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Computational Stochastic Multistage Manufacturing Systems