ar

cerbeen developed to monitor multiple stream processes @2,3#. Itmainly focuses on process change detection, as opposed to rootcause identification. Recently, researches have been conducted tomodel and diagnose single variation stream problem in a MMS.Jin and Shi @4# developed state space model to depict variationpropagation in assembly processes. By developing a state transi-tion model, Mantripragada and Whitney @5# modeled the entireassembly sequence as a set of discrete events to simulate andpredict the propagation of variation in mechanical assemblies.Lawless et al. @6# and Agrawal et al. @7# investigated variationtransmission in both assembly and machining process by using anAR~1! model. State space modeling approach was further ex-tended to model multistage machining processes @810#. Rootcause identification has also been studied for single variationstream in assembly processes @11# and machining processes @12#.If no two process routes merge at certain stage~s!, the previouswork can be directly applied to a SP-MMS by studying everyprocess route separately. If two process routes share at least onework-station, i.e., merge at one stage, there is a need to extend

representation and system diagnosability are studied in Section 4.Section 5 analyzes different measurement strategies based on sys-tem model and u and y reduction techniques. The conclusion isgiven in Section 6.

2 Variation Modeling of SP-MMSswith Multiple Process Routes

2.1 Modeling of System Variation Streams. Assume thetotal number of process routes is R in an N-stage SP-MMS. De-viation of part features are represented as a vector x by usingvectorial surface model @8,13#. For example, the ith part feature Siin Fig. 2 can be modeled by a normal vector to Si , i.e.,(nxi ,nyi ,nzi), a point on Si , i.e., (pxi ,pyi ,pzi), and size, e.g., diam-eter of a cylindrical surface. Let x0 represent raw workpiece de-viation. Denote by xk

(i) the part deviation after stage k throughprocess route i ~See the example in Fig. 1!. Superscript (i) de-notes process route i (i51,2, . . . ,R) and subscript k denotesstage k (k51,2, . . . ,N). The conventions will be followed here-after. A vector yk

(i) denotes deviation of quality characteristics gen-erated after stage k. Note that measurements are not necessary tobe taken at each stage of process route i. By treating part deviation

Contributed by the Manufacturing Engineering Division for publication in theJOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedJune 2003; Revised Feb. 2004. Associate Editor: S. Raman.

Journal of Manufacturing Science and Engineering AUGUST 2004, Vol. 126 611Copyright 2004 by ASMEQiang Huange-mail: huangq@eng.usf.edu

Department of Industrial and ManagementSystems Engineering,

University of South Florida,Tampa, Florida 33620

Jianjun ShiDepartment of Industrial and Operations

Engineering,The University of Michigan,

Ann Arbor, MI 48109

Stream oAnalysisMultistagSystemsIn a Serial-Parallel Mare utilized at each sta system, parts couldcertain stage(s). Duemodel and analyze vamodeling approach fromodel dimension reduties of these techniqudiagnosability. Furthestrategies. @DOI: 10.1

1 IntroductionA multistage manufacturing system ~MMS! involves multiple

stages or operations to fabricate a product. Examples of MMSsinclude engine head machining systems or automotive body as-sembly systems. To meet productivity and line balance require-ments, a MMS usually utilizes identical work-stations at eachstage. This type of systems is called Serial-Parallel MMS ~SP-MMS!. The impact of MMS configuration on system performancehas been studied by Koren et al. @1#. In such a system, each partfollows the same processing sequence, i.e., sequentially goingthrough every stage once. However, process routes may vary frompart to part. For instance, in a three-stage serial-parallel machiningsystem illustrated in Fig. 1, a part could go through process route1, i.e., through the machine tool No. 1 at each stage; or throughother routes. (x0 denotes raw workpiece deviation and xk(i) denotesthe part deviation after stage k through process route i.! As anexample, only portion of all potential routes is shown in thefigure. Since part variation streams from different routes are notnecessary to be the same, the challenging issues are how to modeland analyze the multiple variation streams in a SP-MMS.

In the field of Statistical Process Control, control charts havef Variation Modeling andof Serial-Parallele Manufacturing

ultistage Manufacturing System (SP-MMS), identical work-stationsge to meet the productivity and line balance requirements. In such

go through different process routes and some routes may merge atto the existence of multiple variation streams, it is challenging toiation propagation in a system. This paper extends the state spacem single process route to the SP-MMS with multiple routes. Severaltion techniques are proposed to reduce model complexity. Proper-s are studied from the perspectives of system representation andmore, these techniques are applied to analyze system measurement

115/1.1765149#

previous methodologies by considering all routes and their inter-actions. As such, global optimal solutions are expected forSP-MMS.

Gauging/sensing strategy is a good example to illustrate thenecessities of extending the existing methodologies to SP-MMS.In Fig. 1, parts would be measured from all six routes to identifythe root causes if the routes are studied separately. Intuitively itmight be sufficient to take measurements, e.g., only from processroutes 1, 3, and 6, because all eight machines in the systems areinvolved in those three routes and root causes might be identifiedwith given measurements. Systematic approach is preferred notonly for gauging strategy, but also for closely related systemmonitoring and root cause identification problems. Previous meth-odologies need to be extended to the case of multiple variationstreams because SP-MMS has been adopted as a common con-figuration in industries @1#.

The focus of this paper is to develop a generic system-levelmethodology to model and analyze multiple variation streams in aSP-MMS. Section 2 extends the state space modeling approach tothe SP-MMS. Section 3 discusses the model dimension issues andproposes u and y reduction techniques to reduce model dimen-sions. The impacts of u and y reduction techniques on system

same dimension for all i (i51,2, . . . ,R) at stage k. Input matrixBk

(i) and state transition matrix A(i) transfer process deviationsandtivelyvolveare m

Bk(i) a

desigactertics aproceproce

Fostrea

tors uks for those two routes at that stage are assumed to be thesame, i.e., uk

(i)5uk( j)

.

rmalas-

l be

ct inutestionismrror

itionves-mp-anu-

intion

rmaluted

romT bethatthe

l

F

612 MEk21incoming workpiece deviation to state vector xk

(i), respec-

. The physics underlying Ak21(i) and Bk

(i) transformations in-s fixturing and cutting operation at stage k. Since operationsodeled as kinematic transformations in this study, Ak21

(i) andre constant matrices determined only by product and processn. Matrix Ck(i) maps part deviation xk(i) to yk(i) , which char-izes the geometric relationship between product characteris-nd part features. jk

(i) and hk(i) are error terms. The detailed

ss-level model derivation can be referred to @4# for assemblysses and @810# for machining processes.llowing assumptions are made to model multiple variationms.

A4. The error terms @jk(i)hk

(i)# , which represent the noproduction conditions within designated tooling tolerance, aresumed to be the same for every route. The superscript wildropped too.

A remark is given as follows:R1 These four assumptions are made by considering the fa

a real engine machining plant. By design, all process roshould be identical and well-maintained. Significant deviafrom design needs to be detected through monitoring mechan~not discussed in this paper!. Assuming distributions for eterms jk

(i) and hk(i) in A4 is more critical for process cond

monitoring than for the topic of model dimension reduction intigated in this paper. When distributions are necessary, assutions need to be investigated based on a given product and mfacturing process. For instance, if tool wear is a concernproduction, then a Gaussian random process with correlaamong stages is more reasonable than the assumption of nodistributions with independent identically distribproperty.

Let u

(i)5@u1(i)T

,u2(i)T

, . . . ,uN(i)T#T be the process deviations f

operations 1 to N of route i and y

(i)5@y1(i)T

,y2(i)T

, . . . ,yN(i)T#

the deviations of all measured characteristics in route i. Notemeasurement might not be taken at each stage. By followingprocedure in @14#,ig. 3 Fixture locating scheme and locator deviations

Vol. 126, AUGUST 2004 Transactions of the ASxk(i) as a state vector and stage index as time index, the state space

modeling approach can be applied to model the variation propa-gation for every single route i:

xk~ i !5Ak21~

i ! xk21~ i ! 1Bk

~ i !uk~ i !1jk

~ i !, k51, . . . ,N;i51, . . . ,R (1)

yk~ i !5Ck~ i !xk~ i !1hk~ i ! , $k%,$1, . . . ,N%. (2)

where input vector uk(i) represents process deviations from the

fixture and the machine tool at stage k of route i. uk(i)

s have the

Fig. 1 Process ro

Fig. 2 BA1. All process routes use the same batch of workpiece. Inanother word, raw workpiece deviation x0

(i)s follow the same dis-

tribution as x0 , where x0 is negligible if workpiece is of highquality.

A2. The machine tools and operations at the same stage areidentical. Different process routes are expected to perform thesame fixturing and cutting operations at stage k. Therefore, thesystem matrices @Ak21

(i) Bk(i)Ck(i)# by design are the same for all i.

Superscript will be dropped hereafter.A3. If routes i and j merge at stage k, the input random vec-

ute in a SP-MMS

ock part

C1B1 0 fl 0 C1F1,0

S D 2 1 02S 11 L11L12D L11L12 H1 1 0 0 0 0 S 11 L31L32D L31L32 2H3 21

y

~1 !5S DD1DD2DD3

D 5S 0 1 0 0 0 00 1 0 21 0 00 1 0 0 0 21

D x3~1 !1~1 !5Gu~1 !1~1 ! , (5)1 0 0 0 0 00 1 0 0 0 0

1 L32 L3S 11 L31L32D L31L32

52

1L12

1L12

21 0 0 0

2S 11 L11L12D L11L12 H1 1 0 02

1L12

1L12

21 01

L222

1L22

2S 11 L11L12D L11L12 H1 1 S 11 L21L22D L21L222

1L12

1L12

21 0 0 0where

Journal of Manufacturing Science and Engineering2

2H3 212

0 0 0 0 0 0

0 0 0 0 0 0

1 0 0 0 0 0

2H2 21 0 0 0 0

0 01

L322

1L32

1 0' u~1 !1j3~1 ! (4)y

~ i !5Gu

~ i !1G0x01 , where G5F C2F2,1B1 C2B2 fl 0] ] ]

CNFN ,1B1 CNFN ,2B2 fl CNBNG , G05F C2F2,0]

CNFN ,0G ,

Fk , j5H Ak21Ak22flAj , k> j11I, k5 j , 5@1T ,2T ,fl ,NT #T, and k5Sn51,kCkFk ,njn1hk .

Generally, the observed part deviations in a SP-MMS with Rroutes can be modeled as:

y5Gu1G0x01, (3)

where y5@y

(1)T,y

(2)T, . . . ,y

(R)T#T, G5diag(G,G

, . . . ,G

), u

5@u

(1)T,u

(2)T, . . . ,u

(R)T#T, G05@G0T

,G0T

,fl ,G0T #T, and 5@

T,

T,fl ,

T#T.

2.2 Block Part Example. A machining system of fabricat-ing block parts is given to illustrate the system model ~3!. The partis composed of six surfaces: S12S6 ~Fig. 2!. To meet the specifi-cations for dimensions D12D3 , three operations are selected. Thefirst operation is to use datum surfaces S1 and S2 to mill S3 . Inoperation 2, S2 and S3 are chosen as datums to mill S5 . The lastoperation is to mill a slot S6 with the same datums used in the

second operation. These operations are performed in the machin-ing system depicted by Fig. 1.

Let t ijk denote the deviation of the kth ~k51,2, . . . ,6! locator infixture j (j51,2, . . . ,ni) at operation i ~i51,2,3!. Assume fixture jis mounted on machine tool j at that operation. Figure 3 illustratesthe fixture locating scheme, which is specified by geometric di-mensions Hi, Li1 , and Li2 for operation i.

Denote by g ij and d ij the angular and positionaldeviations of machine tool j at operation i. The processdeviations of route 1 are u1

(1)5@t111,t112,g11,d11#T, u2

(1)

5@t211,t212,d21,g21#T, and u3

(1)5@t311,t312,d31,g31#T

for the three operations. For process route 1, let x050T, x1(1)5(Dnx3 ,Dpy3,0,0,0,0)T, x2(1)5(Dnx3 ,Dpy3 ,Dnx5 , Dpy5,0,0)T,x3

(1) 5 (Dnx3 ,Dpy3 ,Dnx5 ,Dpy5 ,Dnx6 ,Dpy6)T, u(1)5@t111,t112,g11,d11 ,t211 ,t212 ,d21 ,g21 ,t311 ,t312 ,d31 ,g31#

T,

y1(1)5DD1 , y2

(1)5DD2 , and y3(1)5DD3 . Then

x3~1 !5

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 0

x2~1 !1

0 0 0 00 0 0 00 0 0 00 0 0 01 1

u3~1 !1j3

~1 !AUGUST 2004, Vol. 126 613

G

2S 11 L11D L11 H1 1 0 0 0 0 0 0 0 0. (6)

Equations ~4!Huang and Shiindividual proceroutes in the s5@u1

(2)T,u2

(2)T,u3

(2

responding u7231can be obtained t

3 Reducing Mthrough u and

One of the mahigh dimension.stage k, theoreticato Pk51

N nk . Thus system dimension increases dramatically withR. However, when some process routes merge together, systemmodel ~3! can be revised and the dimension might be greatly

lists two speciales coincide withSection 3.2 pro-dimension of u,of Table 2, twoat Stage k1S.

used as datum ink1S. Based on

ucing the dimen-

If routes i andThose two iden-o one sub-vector

in u of model ~3!, i.e., only keeping uk . To describe the dimen-sion reduction of G, let (G)k(i) be the block matrix in Gcorrespond-ing to uk

(i). Note that (G)k(i) has the same number of rows

u614 Vol. 126, AUGUST 2004 Transactions of the ASMEreduced. Before discussing the approaches of model reduction, wefirst classified the ways in which process routes merge.

There are three basic ways that two process routes merge to-gether, i.e., coincidence, divergence, and convergence ~Table 1!.Per these three basic ways of route merging, Section 3.1 proposesu reduction technique to reduce the dimension of input vector u.

Special combinations of the three basic ways of route merging,together with process information, make it possible to reduce not

Table 1 Two routesas G. The reduced G, denoted by G , is obtained by re-placing (G)k(i) with (G)k(i)1(G)k( j) and deleting (G)k( j) inoriginal G.

Example: The process routes 2 and 3 in Fig. 1, for instance,share machine tools at stages 1 and 2, i.e., u1

(2)5u1(3) and u2

(2)

5u2(3)

. Before u reduction,

merge at one stage5S L12 L120 0 0 0 S 11 L21L22D L21L22 2H2 21 0 0 0 00 0 0 0 0 0 0 0 S 11 L31L32D L31L32 2H3 21

D~6!, whose derivation can be referred to@15#, only depict the variation propagation ofss route. To characterize the six processystem, define u

(1)5@u1(1)T

,u2(1)T

,u3(1)T#T, u

(2))T#T, . . . ,u

(6)5@u1(6)T

,u2(6)T

,u3(6)T#T, and the cor-

, y1831 , and G18372 . As such, system model ~3!o model the six variation streams.

odel Dimensionsy Reductionsjor concerns about the system model ~3! is itsIf nk denotes the number of machine tools atlly the total possible number of routes R equals

only u dimension, but also y dimension. Table 2cases, where in the first case, two process routeach other until Stage M and diverge after then.poses y reduction technique to reduce not only thebut also the dimension of y. In the second caseprocess routes diverge at Stage k and convergeBesides, the features machined at stage k will bethe process segment composed by stages k11 tothe process knowledge, Section 3.3 discusses redsions of u and y.

3.1 u Reduction or G Column Reduction.j (i, j) merge at stage k, then uk(i)5uk( j) ~by A3!.tical vectors, i.e., uk

(i) and uk( j)

, can be merged int(i)

Gu52 11 H 1 0 0 0 0 0 0 0 0 0 0 0 0

2S 11 L11L D L11 H1 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0

0 2H3 21

' .(8)

Model ~7! is thus

@y

~2 !T,y

~3 !T#

Here are some reR2 The advan

mensions of systtive of study is tduction could inmeasurement dareduction leads tcolumn reduction

necessarily trueincoming work-

routes i and j (ik(i)5uk

( j) for kom variables fortion to u reduc-he dimension ofws.

Table 2 Special combinations of route merging

Journal of Man12 L12

0 0 0 S 11 L21L22D L21L22 2H2 21 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 S 11 L31L32D L31L32

simplified as

631T 5G6316

u ~u1~2 !T

,u2~2 !T

,u3~2 !T

,u3~3 !T!1631

T 10 (9)marks:tage of u reduction is not just reducing the di-em matrices. It is also necessary when the objec-o estimate u, i.e., identifying root causes. u re-crease the accuracy of estimating u by poolingta together for two identical variables. Since uo reducing column size of G, it is also called G.

R3 Note that when uk(i)5uk( j) , yk(i)5yk( j) is undue to the possibility of xk21

(i) xk21( j)

, i.e., thepieces might come from different process routes.

3.2 y Reduction or G Row Reduction. If, j) merge together through stage M, i.e., u51,2, . . . ,M, then yk

(i) and yk( j) are identical rand

k51,2, . . . ,M ~the first case in Table 2!. In addition, y can be reduced by eliminating all yk

( j)s. T

G is reduced by eliminating the corresponding roS L12D L12 10 0 0 0 S 11 L21L22D L21L22 2H2 210 0 0 0 0 0 0 0ufacturing Science and Engineering0 0 0 0 0 0 0 0

S 11 L31L32D L31L32 2H3 21 0 0 0 0@y

~2 !T,y

~3 !T#631T

5diag~G,G

!6324~u1

~2 !T,u2

~2 !T,u3

~2 !T,u1

~3 !T,u2

~3 !T,u3

~3 !T!2431T 10

(7)By conducting u reduction,

~u1~2 !T

,u2~2 !T

,u3~2 !T

,u1~3 !T

,u2~3 !T

,u3~3 !T!2431

T

is reduced to (u1(2)T,u2

(2)T,u3

(2)T,u3

(3)T)1631T . Correspondingly,diag(G

,G

)6324 is reduced to G6316u , where G6316u is

L11 L11AUGUST 2004, Vol. 126 615

the same machine tool at Stage 1. The machining operations at

Table 3 u and y reduction for block part machining systemStage 2 do not affect y3(i) and y3

( j) because of no datum change.Therefore, two vectors y3

(i) and y3( j) can be reduced into one, e.g.,

y3(i)

. The corresponding rows in G can also be eliminated.By implementing the approaches presented in Sections 3.1, 3.2,

and 3.3, dimensions of system model ~3! can be reduced. Theconcept of minimal dimension of a model is proposed.

Definition 3.1 A system model has minimal dimension if no uor y reduction can be further performed on it.

Example: By using u and y reductions ~Table 3!, the systemmodel of block part machining system can be refined. The u7231is reduced almost by half to u3231 , and y1831 is reduced by one-third to y1231 . Correspondingly G18372 is significantly reduced toG12332 . The refined model has minimal dimension.

Here are two remarks regarding y reduction and minimal modeldimension:

616 Vol. 126, AUGUST 20044 Impacts of u and y Reductionon System Representationand System Diagnosability

The developed system model is expected to describe the varia-tion streams of all process routes in the system. The refined modelshould carry the same amount of information as ~3!. Secondly, thereduction procedures should not worsen the condition of identify-ing root cases. The former requirement is related to system repre-sentation issue, while the latter is about diagnosability, i.e., theability to identify root causes. For a SP-MMS to be diagnosable,input vector u in ~3! should be estimable with given measurementstrategy y. The system diagnosability can be determined by study-ing the rank of matrix GTG @11,16#. The impacts of u and y re-duction on system representation and diagnosability are addressedin this section.

Claim 4.1 The system model with minimal dimension containsthe product variation streams of all process routes in the system.

Transactions of the ASME(11)

3.3 y Reduction Due to Common Datum in a Process Seg-ment. In machining systems, there is another opportunity to per-form y reduction. Suppose the features machined at stage k will beused as datum in the process segment composed by stages k11 tok1S ~The second case in Table 2!. Since uk

(i)5uk( j)

, uk1S(i)

5uk1S( j)

, and there is no datum change, yk1S(i) and yk1S

( j) are identi-cal random vectors. The dimensions of u and y can be reducedaccordingly.

Example: For the block part example, datum surface S3 is ma-chined in Stage 1 and later used in Stages 2 and 3. For the twoprocess routes shown in Fig. 4, they use the datum machined by

R4 We can treat y reduction as an extension of u reduction. Itconducts u reduction first and then delete identical yk

( j)s in y and

corresponding rows in G.R5 For a model to be in minimal dimension, the maximum

dimension of u should be p(k51N nk , where p5Dim(uk(i)), i.e., the

dimension of any uk(i)

. For instance, Dim(uk(i))54 in block partexample, and p(k51

N nk54(31213)532. This rule can be usedto check the accuracy of u reduction procedure.Since variables in y represent quality characteristics to be mea-sured, reducing identical random variables in y implies reductionof measurements. Measurement reduction is very critical for asystem with multiple variation streams because of the need ofreducing gauging cost.

Example: For the same example illustrate in u reduction, y1(2)

5y1(3) and y2

(2)5y2(3) hold because of u1

(2)5u1(3) and u2

(2)5u2(3)

.

Gy512S 11 L11L12D L11L12 H1 1 0 0 0 0

0 0 0 0 S 11 L21L22D L21L22 2H2 210 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Fig. 4 y Reduction due to common datumAs a result, @y

(2)T,y

(3)T#631T in ~9! can be reduced to

@y

(2)T,y

(3)T#431T by deleting y1

(3) and y2(3)

. By conducting y reduc-tion, model ~9! is refined as

@y

~2 !T,y

~3 !T#431T 5G4316

y ~u1~2 !T

,u2~2 !T

,u3~2 !T

,u3~3 !T!1631

T 11(10)

where G4316y is

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

S 11 L31L32D L31L32 2H3 21 0 0 0 00 0 0 0 S 11 L31L32D L31L32 2H3 21

2 .

u reduction y reduction

u3(1)5u3

(2)

u1(2)5u1

(3)5u1(4) and u2(2)5u2(3)5u2(4) y1(2)5y1(3)5y1(4) , y2(2)5y2(3)5y2(4)

u1(5)5u1

(6) and u2(2)5u2(5)5u2(6) y1(5)5y1(6) , y2(5)5y2(6)u3

(3)5u3(5)

u3(4)5u3

(6)

The techniques of u and y reductions aim at eliminating theredundant variables, whose procedures guarantee the claim.

~See R2!. If only one of uk(i) and uk

( j), or none of them could be

u ( j)

Definition 4.1 The rank deficiency index Id is defined as

Id5Dim~GTG!2Rank~GTG!, (12)

which represents rank needed to make u fully estimable ordiagnosable.

Since the Id index represents the amount of information lackingfor the system to be diagnosable, a larger Id would suggest that thesystem is less diagnosable. Based on this criterion, followingtheorem holds:

Theorem 4.1 The system diagnosability would not decrease byperforming u and y reduction procedures on system model ~3!.

Proof: To prove the theorem, we need to prove that the changeof Id should be less or equal than 0 after u and y reduction, i.e.,DId

eliminate the variables in both input vector and output vector. Assuch, system model with minimal dimension could be obtained todescribe all variation streams. As proved in the paper, u and yreduction techniques do not affect the model to capture the varia-

Table 4 Comparison of measurement strategies T1T4

Amount of Measurement Sufficiency

T1 y(1) , y(2) , y(3) , y(4) , y(5) , y(6) Yes for MP&IRCsufficiency of information to monitor all process streams ~MP!and identify the root causes ~IRC!. As shown in Table 4, T1 re-quires the largest amount of measurement effort. Although it issufficient, it is not economical. T2 ignores the variation streams indifferent process routes and randomly select parts from each ma-chine tool. This strategy is not effective for statistical analysisbecause the collected data might come from different distribu-tions. For instance, the part measured at machine 2 of stage 2could be y2

(2) or y2(5)

, which represent different variation streams.The data could be insufficient for both process monitoring androot cause diagnosis due to the randomness. Therefore, strategyT2 is least preferable. The amount of measurement for T3 is lessthan T1 because of u and y reduction procedures. It is optimal inthe sense of all the variation streams can be captured based on thegiven measurement. Comparing T3 with T4, T4 requires less mea-surement and it is still sufficient to estimate root causes u. How-ever, T4 is not sufficient to describe all the variation streams in thesystem.

Definition 5.1 Type I Set, denoted as SI, is defined as the mini-mal set of measurement y that is sufficient to describe the varia-tion streams in a SP-MMS.

The set of measurement in strategy T3 by definition belongs toSI. By Claim 4.1, SI equals to the y in the system model withminimal dimension.

Definition 5.2 Type II Set, denoted as SII, is defined as theminimal set of measurement y that is sufficient to estimate u in thesystem model with minimal dimension.

The set of measurement in strategy T4 by definition belongs toSII. The relationship between above two measurement sets is es-tablished by Theorem 5.1.

Theorem 5.1 If D50, then SII#SI.Proof: Since every single machine tool is utilized in manufac-

turing in a SP-MMS, SII by definition contains the minimal num-ber of process routes which go through all the machine tools. ByClaim 4.1, the process routes in SII compose of a subset of theroutes in SI, i.e., SII#SI. j

The result suggests that less measurement is required to diag-nose root causes than to monitor all variation streams in aSP-MMS.

If D0, the conclusion does not hold because SII requires moreinformation for the system to be diagnosable. Then SI and SII arenot comparable.

6 ConclusionThis paper developed a generic system-level methodology to

model the multiple variation streams in a SP-MMS. The statespace modeling approach for single process route was extended tothe SP-MMS. To identify the redundant variables caused by pro-cess coupling, u and y reduction techniques were proposed to

Stage 1: y1(1) , y1(2)(5y1(3)5y1(4)), y1(5)(5y1(6))T2 Stage 2: y2(1) , y2(2)(5y2(3)5y2(4))/y2(5)(5y2(6)) No for MP&IRC

Stage 3: y3(1)/y3(2) , y3(3)/y3(5) , y3(4)/y3(6)

y1(1)

, y1(2)(5y1(3)5y1(4)), y1(5)(5y1(6))

T3 y2(1) , y2(2)(5y2(3)5y2(4)), y2(5)(5y2(6)), Yes for MP&IRCy3

(1), y3

(2), y3

(3), y3

(5), y3

(4), y3

(6)

y1(1)

, y1(2)(5y1(3)5y1(4)), y1(5)(5y1(6))

T4 y2(1)

, y2(2)(5y2(3)5y2(4)), y2(5)(5y2(6)), Yes for IRC

y3(1)

, y3(3)

, y3(6) but No for MP

618 Vol. 126, AUGUST 2004tion streams. Based on rank deficiency index, u and y reductionwas proved not to increase Id , i.e., the system diagnosabilitywould not decrease. It was also proved that the rank deficiency ofsystem model with minimal dimension is bounded.

These results are applied to evaluate different system measure-ment strategies. It was identified that less measurement is requiredto diagnose root causes than to monitor all variation streams in aSP-MMS, if single process route is fully diagnosable. Two opti-mal sets are defined and their relationship was established whenthe process route is diagnosable.

The methodology development is demonstrated by using simpleexamples. Efforts have been made to implement the modelingtechnique in an engine machining plant with success. Future re-search efforts can be devoted to study system root cause diagnosisand optimal measurement strategy for a SP-MMS. Parameter es-timation approaches, such as least square estimation, can bereadily applied therein, because the relationship between measure-ment y and root causes u has been established through a linearmodel.

AcknowledgmentThis work is partially supported by NSF Engineering Research

Center for Reconfigurable Machining Systems ~NSF GrantEEC95-92125! at the University of Michigan and by the Univer-sity of South Florida, Internal Award Program. The authors arealso thankful for the discussion with Prof. Zhifang Zhang at theUniversity of Michigan.

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