Stream of Variation Modeling and

Analysis of Serial-Parallel

Qiang Huang

Multistage Manufacturing

e-mail: abqgpl@r.postjobfree.com

Department of Industrial and Management

Systems

Systems Engineering,

University of South Florida,

Tampa, Florida 33620

In a Serial-Parallel Multistage Manufacturing System (SP-MMS), identical work-stations

Jianjun Shi are utilized at each stage to meet the productivity and line balance requirements. In such

Department of Industrial and Operations

a system, parts could go through different process routes and some routes may merge at

Engineering,

certain stage(s). Due to the existence of multiple variation streams, it is challenging to

The University of Michigan,

model and analyze variation propagation in a system. This paper extends the state space

Ann Arbor, MI 48109

modeling approach from single process route to the SP-MMS with multiple routes. Several

model dimension reduction techniques are proposed to reduce model complexity. Proper-

ties of these techniques are studied from the perspectives of system representation and

diagnosability. Furthermore, these techniques are applied to analyze system measurement

strategies. DOI: 10.1115/1.1765149

previous methodologies by considering all routes and their inter-

1 Introduction

actions. As such, global optimal solutions are expected for

A multistage manufacturing system MMS involves multiple

SP-MMS.

stages or operations to fabricate a product. Examples of MMSs

Gauging/sensing strategy is a good example to illustrate the

include engine head machining systems or automotive body as- necessities of extending the existing methodologies to SP-MMS.

sembly systems. To meet productivity and line balance require- In Fig. 1, parts would be measured from all six routes to identify

ments, a MMS usually utilizes identical work-stations at each the root causes if the routes are studied separately. Intuitively it

stage. This type of systems is called Serial-Parallel MMS SP- might be suf cient to take measurements, e.g., only from process

MMS . The impact of MMS con guration on system performance routes 1, 3, and 6, because all eight machines in the systems are

has been studied by Koren et al. 1 . In such a system, each part involved in those three routes and root causes might be identi ed

follows the same processing sequence, i.e., sequentially going with given measurements. Systematic approach is preferred not

through every stage once. However, process routes may vary from only for gauging strategy, but also for closely related system

part to part. For instance, in a three-stage serial-parallel machining monitoring and root cause identi cation problems. Previous meth-

system illustrated in Fig. 1, a part could go through process route odologies need to be extended to the case of multiple variation

1, i.e., through the machine tool No. 1 at each stage; or through streams because SP-MMS has been adopted as a common con-

other routes. ( x0 denotes raw workpiece deviation and x( i ) denotes guration in industries 1 .

k

the part deviation after stage k through process route i. As an The focus of this paper is to develop a generic system-level

example, only portion of all potential routes is shown in the methodology to model and analyze multiple variation streams in a

gure. Since part variation streams from different routes are not SP-MMS. Section 2 extends the state space modeling approach to

necessary to be the same, the challenging issues are how to model the SP-MMS. Section 3 discusses the model dimension issues and

and analyze the multiple variation streams in a SP-MMS. proposes u and y reduction techniques to reduce model dimen-

In the eld of Statistical Process Control, control charts have sions. The impacts of u and y reduction techniques on system

been developed to monitor multiple stream processes 2,3 . It representation and system diagnosability are studied in Section 4.

mainly focuses on process change detection, as opposed to root Section 5 analyzes different measurement strategies based on sys-

cause identi cation. Recently, researches have been conducted to tem model and u and y reduction techniques. The conclusion is

model and diagnose single variation stream problem in a MMS. given in Section 6.

Jin and Shi 4 developed state space model to depict variation

propagation in assembly processes. By developing a state transi- 2 Variation Modeling of SP-MMSs

tion model, Mantripragada and Whitney 5 modeled the entire

with Multiple Process Routes

assembly sequence as a set of discrete events to simulate and

predict the propagation of variation in mechanical assemblies. 2.1 Modeling of System Variation Streams. Assume the

Lawless et al. 6 and Agrawal et al. 7 investigated variation total number of process routes is R in an N-stage SP-MMS. De-

transmission in both assembly and machining process by using an viation of part features are represented as a vector x by using

AR 1 model. State space modeling approach was further ex- vectorial surface model 8,13 . For example, the ith part feature Si

tended to model multistage machining processes 8 10 . Root in Fig. 2 can be modeled by a normal vector to Si, i.e.,

cause identi cation has also been studied for single variation (nxi,nyi,nzi), a point on Si, i.e., (pxi,pyi,pzi), and size, e.g., diam-

stream in assembly processes 11 and machining processes 12 . eter of a cylindrical surface. Let x0 represent raw workpiece de-

If no two process routes merge at certain stage s, the previous viation. Denote by x( i ) the part deviation after stage k through

k

work can be directly applied to a SP-MMS by studying every process route i See the example in Fig. 1 . Superscript ( i ) de-

process route separately. If two process routes share at least one notes process route i ( i 1,2, . . ., R ) and subscript k denotes

work-station, i.e., merge at one stage, there is a need to extend stage k ( k 1,2, . . ., N ). The conventions will be followed here-

after. A vector y( i ) denotes deviation of quality characteristics gen-

k

Contributed by the Manufacturing Engineering Division for publication in the

erated after stage k. Note that measurements are not necessary to

JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received

be taken at each stage of process route i. By treating part deviation

June 2003; Revised Feb. 2004. Associate Editor: S. Raman.

AUGUST 2004, Vol. 126 611

Journal of Manufacturing Science and Engineering

Copyright 2004 by ASME

Fig. 1 Process route in a SP-MMS

Fig. 2 Block part

A1. All process routes use the same batch of workpiece. In

x( i ) as a state vector and stage index as time index, the state space

k

another word, raw workpiece deviation x( i ) s follow the same dis-

modeling approach can be applied to model the variation propa- 0

gation for every single route i: tribution as x0, where x0 is negligible if workpiece is of high

quality.

xki Aki 1 xki Bki uki i, k 1, . . ., N ; i 1, . . ., R (1) A2. The machine tools and operations at the same stage are

1 k

identical. Different process routes are expected to perform the

same xturing and cutting operations at stage k. Therefore, the

yki Cki xki i, k 1, . . ., N . (2)

k

system matrices Aki ) 1 B( i ) C( i ) by design are the same for all i.

(

k k

Superscript will be dropped hereafter.

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

k

A3. If routes i and j merge at stage k, the input random vec-

xture and the machine tool at stage k of route i. u( i ) s have the

k

tors uk s for those two routes at that stage are assumed to be the

same dimension for all i ( i 1,2, . . ., R ) at stage k. Input matrix

same, i.e., u( i ) u( j ) .

B( i ) and state transition matrix Aki ) 1 transfer process deviations

( k k

k

A4. The error terms ( i ) ( i ), which represent the normal

and incoming workpiece deviation to state vector x( i ), respec- k k

k

production conditions within designated tooling tolerance, are as-

tively. The physics underlying Aki ) 1 and B( i ) transformations in-

(

k sumed to be the same for every route. The superscript will be

volves xturing and cutting operation at stage k. Since operations

dropped too.

are modeled as kinematic transformations in this study, Aki ) 1 and

(

A remark is given as follows:

(i)

Bk are constant matrices determined only by product and process

R1 These four assumptions are made by considering the fact in

design. Matrix C( i ) maps part deviation x( i ) to y( i ), which char-

k k k a real engine machining plant. By design, all process routes

acterizes the geometric relationship between product characteris- should be identical and well-maintained. Signi cant deviation

tics and part features. ( i ) and ( i ) are error terms. The detailed from design needs to be detected through monitoring mechanism

k k

process-level model derivation can be referred to 4 for assembly not discussed in this paper . Assuming distributions for error

processes and 8 10 for machining processes. terms ( i ) and ( i ) in A4 is more critical for process condition

k k

Following assumptions are made to model multiple variation

monitoring than for the topic of model dimension reduction inves-

streams.

tigated in this paper. When distributions are necessary, assump-

tions need to be investigated based on a given product and manu-

facturing process. For instance, if tool wear is a concern in

production, then a Gaussian random process with correlation

among stages is more reasonable than the assumption of normal

distributions with independent identically distributed

property.

T T (T

Let u( i ) u( i ), u( i ), . . ., uNi ) T be the process deviations from

1 2

T T T

operations 1 to N of route i and y( i ) y1i ), y( i ), . . ., yNi ) T be

( (

2

the deviations of all measured characteristics in route i. Note that

measurement might not be taken at each stage. By following the

procedure in 14,

Fig. 3 Fixture locating scheme and locator deviations

612 Vol. 126, AUGUST 2004 Transactions of the ASME

C1 B1 0 0

C1 1,0

C2 2,1B1 C2 B2 0 C2 2,0

i i

y u 0 x0, where,,

]

] ] ]

0

CN

N,0

CN N,1B1 CN N,2B2 CN B N

1 Ak 2 A j

Ak, k j 1,,

T T TT,, N, and n 1,k Ck .

k, j k k,n n k

1 2

I, k j

Generally, the observed part deviations in a SP-MMS with R second operation. These operations are performed in the machin-

routes can be modeled as: ing system depicted by Fig. 1.

Let ijk denote the deviation of the kth k 1,2, . . .,6 locator in

xture j (j 1,2, . . .,ni) at operation i i 1,2,3 . Assume xture j

y u 0 x0, (3)

is mounted on machine tool j at that operation. Figure 3 illustrates

the xture locating scheme, which is speci ed by geometric di-

T T T

where y y 1 ), y( 2 ), . . ., y( R ) T,

(

mensions Hi, Li1, and Li2 for operation i.

diag, u

(1)T (2)T T

Denote by and the angular and positional

0,, 0,

u, u, . . ., u( R ) T, T T TT, and ij ij

0

0

deviations of machine tool j at operation i. The process,,,

T T TT

.

deviations of route 1 are u( 1 ) (1)

T

111, 112, 11, 11, u2

1

2.2 Block Part Example. A machining system of fabricat- (1)

T T

211, 212, 21, 21, and u3 311, 312, 31, 31

ing block parts is given to illustrate the system model 3 . The part (1)

T

for the three operations. For process route 1, let x0 0, x1

is composed of six surfaces: S1 S6 Fig. 2 . To meet the speci -

x21 ) ( nx3, py3, nx5, py5,0,0) T,

(

( nx3, py3,0,0,0,0) T,

cations for dimensions D1 D3, three operations are selected. The

x31 ) ( nx3, py3, nx5, py5, nx6, py6 ) T,

(

u( 1 )

rst operation is to use datum surfaces S1 and S2 to mill S3 . In

T

1 11, 1 12, 11, 11, 211, 212, 21, 21, 311, 312, 31, 31,

operation 2, S2 and S3 are chosen as datums to mill S5 . The last

y11 )

(

D1, y( 1 ) D2, and y( 1 )

operation is to mill a slot S6 with the same datums used in the D3 . Then

2 3

0 0 0 0

1 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0

x31 x21 u31 1

3

0 0 0 1 0 0 1 1

1 0

1 0 0 0 0 0 L32 L32

0 1 0 0 0 0 L31 L31

H3 1

1

L32 L32

1 1

1 0 0 0 0 0 0 0 0 0

L12 L12

L11 L11

H1 1 0 0 0 0 0 0 0 0

1

L12 L12

1 1 1 1

1 0 1 0 0 0 0 0

L12 L12 L22 L22

u 1 1

(4)

3

L11 L11 L21 L21

H1 1 H2 1 0 0 0 0

1 1

L12 L12 L22 L22

1 1 1 1

1 0 0 0 0 0 1 0

L12 L12 L32 L32

L11 L11 L31 L31

H1 1 0 0 0 0 H3 1

1 1

L12 L12 L32 L32

0 1 0 0 0 0

D1

1

x31 1 1 1

D2 0 1 0 1 0 0

y u, (5)

D3 0 1 0 0 0 1

where

AUGUST 2004, Vol. 126 613

Journal of Manufacturing Science and Engineering

L11 L11

H1 1 0 0 0 0 0 0 0 0

1

L12 L12

L21 L21

0 0 0 0 H2 1 0 0 0 0

1 . (6)

L22 L22

L31 L31

0 0 0 0 0 0 0 0 H3 1

1

L32 L32

Equations 4 6, whose derivation can be referred to only u dimension, but also y dimension. Table 2 lists two special

Huang and Shi 15, only depict the variation propagation of cases, where in the rst case, two process routes coincide with

individual process route. To characterize the six process each other until Stage M and diverge after then. Section 3.2 pro-

T T T

routes in the system, de ne u( 1 ) u( 1 ), u( 1 ), u( 1 ) T, u( 2 ) poses y reduction technique to reduce not only the dimension of u,

1 2 3

T T T T T T

but also the dimension of y. In the second case of Table 2, two

u( 2 ), u( 2 ), u( 2 ) T, . . ., u( 6 ) u( 6 ), u( 6 ), u( 6 ) T, and the cor-

1 2 3 1 2 3

process routes diverge at Stage k and converge at Stage k S.

responding u72 1, y18 1, and 18 72 . As such, system model 3

Besides, the features machined at stage k will be used as datum in

can be obtained to model the six variation streams.

the process segment composed by stages k 1 to k S. Based on

the process knowledge, Section 3.3 discusses reducing the dimen-

sions of u and y.

3 Reducing Model Dimensions

through u and y Reductions 3.1 u Reduction or Column Reduction. If routes i and

j ( i j ) merge at stage k, then u( i ) u( j ) by A3 . Those two iden-

One of the major concerns about the system model 3 is its k k

high dimension. If n k denotes the number of machine tools at tical vectors, i.e., u( i ) and u( j ), can be merged into one sub-vector

k k

stage k, theoretically the total possible number of routes R equals

in u of model 3, i.e., only keeping u( i ) . To describe the dimen-

N k

to k 1 n k . Thus system dimension increases dramatically with

sion reduction of, let ( i ) be the block matrix in correspond-

R. However, when some process routes merge together, system k

ing to u( i ) . Note that ( i ) has the same number of rows

model 3 can be revised and the dimension might be greatly k k

as . The reduced, denoted by u, is obtained by re-

reduced. Before discussing the approaches of model reduction, we

rst classi ed the ways in which process routes merge. placing ( i ) with ( i ) ( j ) and deleting ( j ) in

k k k k

There are three basic ways that two process routes merge to- original .

gether, i.e., coincidence, divergence, and convergence Table 1 .

Per these three basic ways of route merging, Section 3.1 proposes

Example: The process routes 2 and 3 in Fig. 1, for instance,

u reduction technique to reduce the dimension of input vector u.

share machine tools at stages 1 and 2, i.e., u( 2 ) u( 3 ) and u22 )

(

Special combinations of the three basic ways of route merging, 1 1

(3)

together with process information, make it possible to reduce not u2 . Before u reduction,

Table 1 Two routes merge at one stage

614 Vol. 126, AUGUST 2004 Transactions of the ASME

Table 2 Special combinations of route merging

T TT T T T T T TT

y 2, y 3 u12, u22, u32, u13, u23, u33

61 24 1

2T 2T 2T 3T 3T 3T T

diag, 6 24 u1, u2, u3, u1, u2, u3 0

24 1

T T T T

(7) is reduced to ( u12 ), u22 ), u32 ), u( 3 ) ) 16 1 . Correspondingly,

( ( ( T

3

u u

diag 6 24 is reduced to 6 16, where 6 16 is

By conducting u reduction,

L 11 L11

H1 1 0 0 0 0 0 0 0 0 0 0 0 0

1

L12 L12

L21 L21

0 0 0 0 H2 1 0 0 0 0 0 0 0 0

1

L22 L22

L31 L31

0 0 0 0 0 0 0 0 H3 1 0 0 0 0

1

L32 L32

u

.

L11 L11

H1 1 0 0 0 0 0 0 0 0 0 0 0 0

1

L12 L12

L21 L21

0 0 0 0 H2 1 0 0 0 0 0 0 0 0

1

L22 L22

L31 L31

0 0 0 0 0 0 0 0 0 0 0 0 H3 1

1

L32 L32

(8)

R3 Note that when u( i ) u( j ), y( i ) y( j ) is unnecessarily true

Model 7 is thus simpli ed as k k k k

due to the possibility of xki ) 1 xkj ) 1, i.e., the incoming work-

( (

T TT T T T TT

y 2, y 3 u

u12, u22, u32, u33 (9)

0

61 6 16 16 1

pieces might come from different process routes.

Here are some remarks:

3.2 y Reduction or Row Reduction. If routes i and j ( i

R2 The advantage of u reduction is not just reducing the di-

j ) merge together through stage M, i.e., u( i ) u( j ) for k

mensions of system matrices. It is also necessary when the objec- k k

tive of study is to estimate u, i.e., identifying root causes. u re- 1,2, . . .,M, then y( i ) and y( j ) are identical random variables for

k k

duction could increase the accuracy of estimating u by pooling

k 1,2, . . .,M the rst case in Table 2 . In addition to u reduc-

measurement data together for two identical variables. Since u

tion, y can be reduced by eliminating all y( j ) s. The dimension of

reduction leads to reducing column size of, it is also called k

is reduced by eliminating the corresponding rows.

column reduction.

AUGUST 2004, Vol. 126 615

Journal of Manufacturing Science and Engineering

Table 3 u and y reduction for block part machining system

u reduction y reduction

u(1) u(2)

3 3

u(2) u(3) u(4) and u(2) u(3) u(4) y(2) y(3) y(4), y(2) y(3) y(4)

1 1 1 2 2 2 1 1 1 2 2 2

u(5) u(6) and u(2) u(5) u(6) y(5) y(6), y(5) y(6)

1 1 2 2 2 1 1 2 2

u(3) u(5)

3 3

Fig. 4 y Reduction due to common datum u(4) u(6)

3 3

T TT

y 2 ), y( 3 )

(

As a result, 6 1 in 9 can be reduced to

Since variables in y represent quality characteristics to be mea-

(2)T (3)T T (3) (3)

y, y 4 1 by deleting y1 and y2 . By conducting y reduc-

sured, reducing identical random variables in y implies reduction

tion, model 9 is re ned as

of measurements. Measurement reduction is very critical for a

system with multiple variation streams because of the need of T TT T T T TT

y 2, y 3 y

u12, u22, u32, u33

reducing gauging cost. 1

41 4 16 16 1

(10)

Example: For the same example illustrate in u reduction, y( 2 ) 1

y( 3 ) and y( 2 ) y( 3 ) hold because of u( 2 ) u( 3 ) and u( 2 ) u( 3 ) . y

where is

1 2 2 1 1 2 2 4 16

L11 L11

H1 1 0 0 0 0 0 0 0 0 0 0 0 0

1

L12 L12

L21 L21

0 0 0 0 H2 1 0 0 0 0 0 0 0 0

1

L22 L22

y

.

L31 L31

0 0 0 0 0 0 0 0 H3 1 0 0 0 0

1

L32 L32

L31 L31

0 0 0 0 0 0 0 0 0 0 0 0 H3 1

1

L32 L32

(11)

3.3 y Reduction Due to Common Datum in a Process Seg- R4 We can treat y reduction as an extension of u reduction. It

conducts u reduction rst and then delete identical y( j ) s in y and

ment. In machining systems, there is another opportunity to per- k

form y reduction. Suppose the features machined at stage k will be corresponding rows in .

R5 For a model to be in minimal dimension, the maximum

used as datum in the process segment composed by stages k 1 to

dimension of u should be p k 1 n k, where p Dim( u( i ) ), i.e., the

N

k S The second case in Table 2 . Since u( i ) u( j ), uki ) S (

k

k k

dimension of any uk . For instance, Dim( u( i ) ) 4 in block part

(i)

( j) (i) ( j)

uk S, and there is no datum change, yk S and yk S are identi- k

N

example, and p k 1 n k 4 (3 2 3 ) 32. This rule can be used

cal random vectors. The dimensions of u and y can be reduced

to check the accuracy of u reduction procedure.

accordingly.

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 t

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