statistical process monitoring and optimization

STATISTICAL PROCESS MONITORING AND OPTIMIZATION

edited by

Sung H. Park Seoul National University Seoul, Korea

G. Geoffrey Vining Virginia Polytechnic Institute and State University Blacksburg, Virginia

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MA R C EL

D EKK E R

MARCEL DEKKER, INC .

NEW YORK · BASEL

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A Methodological Approach to the Integration of SPC and EPC in Discrete Manufacturing Processes

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Enrique Del Castillo Pennsylvania State University, University Park, Pennsylvania

Rainer Gi:)b and Elart von Collani University of Wuerzburg, Wuerzburg, Germany

1,

INTRODUCTION

The control or industrial manufacturing processes has long been considered from two differen t points of view. Situisliwl proc('SS co/llml (SPC), which traces baclk to the work of Walter Shewhart in the I920s, was originally developed for discrete manufacturing industries (i ndu stries concerned with the production of di sc rete items). On the other hand , continuous process industri es. chem ical industries for instance. used various forms of adjustment strategies administered by automatic contro llers. TIllS type of process control became known as £'I1gilleerillg process 1'01111'01 (EPC) or (ll/flil/wlie proc('ss conlrol (APC). Sepa rately, both approaches have received enormous interest in the academic literature. Int erest in SPC and EPC integration originated in the 1950s in the chemical industries . Part of this interest is due to the inertial elements in thi s type of production process (e.g., raw materials with drifting properties) that result in autocorrelated quality characteristics of th e end product. Traditional SPC methods assume instead i.i .d. quality characteri stics. and problems of a high number of false alarms and the difficult y in detecting process shift s occur under (pos iti ve) autocorrelation at low lags. If the a utocorrelation structure can be mode led and a co mpensa tory variab le ca n be 77

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f,

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fou nd to modify th e qua lity clwr'lctcris tlc. then a n EPC scheme is put int o place to co mpe nsa te fo r such d rift ing he havio r. Howeve r, a brupt, la rge shi fts in the qua lit y cha racteris ti c ind ica te maj o r railures or errors in the process tha t ca nn ot generall y be co mpensa ted fo r by th e EPC co ntro ller. Fo r thi s rea so n. ma ny a uth o rs have suggested th a t a n SPC cha rt be add ed at the o utp ut of an EPC-co ntro ll ed produc ti o n process to detec t large shifts. Th ere is no cl ea r meth odo logy. howeve r. th a t models such integration effo rt s in a ro rm a l a nd ge nera l wa y. In co ntrast. interes t in SPC · EPC Integ ra tion in d iscrete pa rt manufacturin g is more recent. In thi s type or prod uctio n process. element s th at ind uce au toco rre la ti o n a re not comm on . However. d rift ing behav io r of a process th ,lt " ages" is co mmo n. A Iypical exa mple of thi s is a meta l mac hining process in which Ihe perfo rman ce nr Ih e LU ll ing tool de teri ora tes (in ma ny cases. almo st linearl y) wi lh limc . M;lny years ago, when marke t co mpetil io n was nOI so in tense. specificl lll )nS were wide eno ugh fo r a prod ucti o n process to d rift wilh o ut produ ci ng a la rge proporti o n of no nconfo rmin g prod uct. With increasi ng co mpetiti o n. qualit y spec ifica ti o ns have become mo re rigo ro us. a nd dri fting hehav ior. rather th a n bein g to lera ted. is aCl ive ly co mpensal ed for hy simpl e FPC sc hemes. Aca demic int ereS I in Ih e a rea o r SPC [P C in teg rati o n has occ urred as a na tural rea ctio n to the req uirement s or ind ustri al prac tices . Howeve r, most o r th e a pp roaches suggested duri ng th e d isc uss io n on th is prob lem argued fro m th e point of view of practica l necess iti es a lo ne. Pro po nent s of eith er side ad mit th at ma ny con tro l prob lems in mod ern manufac tu rin g processes ca nn o t be so lved by eilh er SPC or EPC a lo ne. As a co nseq uence, meth ods from eac h fi eld a re reco mm end ed as a uxi li a ry too ls in a scheme origin a ll y deve loped either for SPC o r fo r EPC applica tions alone. None of these a pproac hes have been reall y success ful from a methodological point of view. The models used were origin all y des igned for either proper SPC or EPC appli ca ti o ns but not for an integra ti on of the two . The practica l necessit y of an integ ra ting approach to indu stri al co ntrol problems is obvious, but a ri go ro us ma themati ca l model to re fl ect thi s need is still mi ss in g. As a reacti on to thi s meth odologica l ga p. the prese nt chapter es tablishes a simple model tha t integra tes th e pos iti o ns of SPC a nd EPC.

2.

Integration of SPC and EPC in Manufacturing

Del Castillo et al.

MODELS PROPOSED IN THE LITERATURE FOR SPCEPC INTEGRATION

Alth ough d ive rse auth o rs have di sc ussed the different aim s a nd stra tegies of SPC a nd EPC (e .g .. Ba rn a rd. 1963 : MacG rego r, 1988, 1990; Box a nd Kra mer. 1992: Mo nt go mery et al. . 1994). few speci fi c models have bee n

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pro posed for the int egra ti on of th ese field s. Amo ng th ese models we fi nd algo rithmi c sta tis tica l process co ntro l (AS PC) a nd run-to-run co ntrol procedures. 2.1.

ASPC

Vander Weil et al. (1 992 ) (sec a lso Tucker et aI. , 1993) mod el the observed qu a lit y cha racteri sti c ~ , of a ba tch polymeri za ti o n process at time I as ~,

==

p /, ~ ,,,

1 - OB

+ h-" _I + -I - - . . An essential aspect is Ihe cli stinCli (1 n hctween the IIIWI/-!lIlIp behav ior (behavior without co ntrol actions) and c!osed-!IIIIII behavior (behavior in the presence or co ntrol actions) of such a process . In SPC, detection of an assignable cause and subsequ ent corrective ac tion occu rs only rarely. If it occurs, it amounts to a complete rel/('Im! of the manufacturing process . Hence it is useful to split the entire rroducti o n run int o th e periods (reI/elm! cycles) between two success ive corrective acti o ns (renewals) a nd to co nsider · e~lc h renewal cycle along a se rarate time ax is O. 1,2, ... with corresponding output qualit y cha racteri stics ~o' ~ I ' ~ 2 ' The effect of co ntrol actions is not refl ected in the outrut mode l. In standard EPC, control actions are take n regul a rl y at each time point. Without these perma nent comrensatory ac ti o ns th e process would exhibit a com pletely different behav io r. A model of the process behavior without co nltrol is indispensable for the desi gn and evaluation of control rules. Thus we have the open-loop output quality characteristics ~o' ~; , ~~ , . , . of the process without control (l eft alone) a nd the c1osedloop o utput quality characteristics ~o' ~ I' ~2 ' ... or the process subject to control actions. 3,3.

Process Changes in SPC Models

Statistical process control is designed for manufactu ring rrocesses tha t exhibit di scre te para meter shifts that occur a t rand om time points. Thu s in SPC mod els the most general form of the output process (~:)N is th e sum of a marked point process and a wh ite I/oise component. T'his approach is exrressed by the model

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~:

Integration of SPC and EPC in Manufacturing

= p, + £,

(3)

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ARMA Models

An important famil y of models used to characterize drifting beh av ior occ urring due to autocorrelated data is the family of ARMA(p , II) models (Box and Jenkin s, 1976):

In this formula (P')N" is a marked point process, N,

p, = p'

+

L8

(4)

i

E,; = ¢ 1~;- I + ¢~ ~:- 2 + . .. + ¢"~:- I' - "1 £, - 1 -

i= 1

)' ~f.' _ 2 - . .. -

Aq£ , _ q

+ £, (7)

with a targe t p' , wi th marks (i1. 1i:- . representi ng the si zes or shift s 1.2 .. . and a co untin g process (N,l,.." that gi ves th e numher of shift s in time interva l [0: I) . (£')1'11 0 in Eq. (3) is a whil e nOise process inderendent of (11,),..,, · The whit e noise rropert y is ex p res~ed hy

where (f.')Nois a white noise scqucnce [sec Eq . (5) ]. By introducing th e backk shift o pera lor 6 .r; ='/;-k, EL}. (7) can be written as 1

,I: £[£,]

= o.

for a II

I E

No: k

E ~.J

(5)

A simpl e a nd popular in stance or a ma rk ed point process is one with shift s occ urrin g acco rding to a Poisson process (N')N II ' Mos t inves tigation s on control cha rts use furth er simplifi ca tions. For instance . deterministic abso lut e va lues 18, 1= 6 (6 > 0) of the shifts are frequ ently assumed . Ma ny ap proac hes ass ume a single' shift of a give n abso lute value 6 that occ urs a fter a random (often ass um ed to be expo nenti a lly distributed) time v . In thi s case we have if if

I ::: I

v

(6)

> v

where the rand om va ri able y is th e sign of th e deviation from target with P(y

=

I)

= p,

P( y

= - · 1) =

I - p,

= 1; in

1

or as

where A/6) a nd ¢,,(6) are stabl e po lynom ials in B. Sometimes. nonstationary ARIMA(p . rI. q) mod els or th e form .o r . - -"

,

~, = ( I -

A,,(6)

6)" ¢,,(6)

£,

have bee n used instead to model drifting behavior in co ntinu o us production processes. Deterministic Drift

If the driftilllg behav ior is caused by aging of a tool (see, e.g., Quesenberry , 1988), a simple regression model of the form

P E [0 : I)

In the case of one-sided shifts, we have p = 0 or p sided shifts, it is usually assumed that p = 0.5.

\ 1 r

the case of two-

~;

= T + dl + £,

(8)

is sufficient to model most discrete manufacturing processes. Here, T is a target va lue and dl is a deterministi c time trend . 3.4.

Process Changes in EPC Models

Engineering process control is designed fo r manufacturing processes that ex hibit continuous parameter drifts . Some typica l in stances of open-loop output seq uences' (~ : )N" in EPC models are as follows .

Unit Root Trend

Altern atively, a " un it roo t" process can be used to model lin ea r drifts by using

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Integration of SPC and EPC in Manufacturing

Del Castillo et al.

(9)

(Es )s~ ,) = 0, +

.f)(.)"

L hif., "

85

( 14)

idl

For e;\'I ll1ple, ir A,,(13) = I , then (9) is a random walk wi th drift d that has beha vio r similar to that given by (8) but with variance that increases linearly with tiIll e.

where hi

= '- Ai13i

and where we also identify ( 15)

3,5,

Common Structure of SPC and EPC Models

A n ~ II Y7. ing

the SPC Ill odel s of Sect io n 3.3 and th e EPC Il1IKlcl s of Section point o ut ,I co mmon structure th a t IS helpful In developing an apprll;lc h for a ll Int eg r,ltln g Ill odel. We sl1 - A/d we o btain 6.4.

Effect of Constraints in the Compensatory Variable

An important aspec t in practice, usually not addressed in the literature on SPC- EPC integration , is that the compensa tory variable must usually be cons trained to li e within a certain region of opera tion , i.e., or

or

(42)

Obviously, the arguments in favor of suprlementary application of SPC schemes in the trend model that are put I'orward in Section 6.2 also hold in the case of constrained controllers.

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Effect of Constraints Under the Random Walk Model

Integration of SPC and EPC in Manufacturing

In th c unco nslrain ed case. slIprlementary application or SPC in Ihe ra ndom wa lk mode l makes se nse in o nl y spcc ial cases (sce Section 6.2) . Fro m Eq . (4(,) . II is evi dent Ihal in the co nstr