A Mathematical Framework for Statistical Process Control

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A Mathematical Framework for Statistical Process Control E. v. Collani, Wiirzburg, Germany

I , I ntroduction

Only one year after SHEWHART's [10] first paper on control charts had appeared in the Bell System Technical Journal, BECKER, PLAUT & RUNGE [1] published their monograph on Statistical Process Control (SPC) in 1927. In their monograph they develop most of the mathematical tools necessary for judging and comparing lots and processes, but don't mention SHEWHART's control chart as a simple and effective graphical method for performing process monitoring. Since then almost 70 years passed resulting in a simply indeterminable body of literature. But, unfortunately, the majority of publications in the field of SPC concentrates on methodological and technical problems, thus omitting a thorough fundamental discussion of the problem itself. The result is that Statistical Process Control resembles a more or less unorganized pile of technical tools derived from different approaches. Practitioners are invited to just select one of these tools, and often feel overwhelmed. Even experts in Statistical Process Control must often surrender faced with the ever increasing number of different alternatives for one and the same situation. This survey aims in clarifying the confused state in Statistical Process Control by working out its foundations and introducing a natural order within the existing methodology. Statistical Process Control aims in controlling production processes with respect to quality by applying statistical methods. There are two catchwords which have to be explained: control and quality. The term quality is explained in the next chapter. The remaining chapters are devoted to the concept control, which is described by W . A. SHEWHART [ll] in the following way:

A Phenomenon will be said to be controlled when, through the use of past experiences, we can predict, at least within limits, how the phenomenon may be expected to vary in the future. Here it is understood that prediction within limits means that we can state, at least approximately, the probability that the observed phenomenon will fall within the given limits. Accordingly, application of Statistical Process Control should enable the user to predict within limits the quality produced by the process leading immediately to the questions how to describe product quality on the one hand, and process quality on the other hand.

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2. Mathematical Model for Quality 2.1

Item Quality

Any production process ends not in itself but aims in producing some product, and it is easily seen that any produced product can be divided into discrete items. Either it is produced piecemail, or it can be divided into rational pieces defining the items. The items on the other hand are produced to fu lfill some purpose, and often the degree of their fitness for use is identified with their quality. Let n be the set af all possible realizations of items. Then it is assumed that the fitness for use, i.e. quality of any item wEn can be determined completely by means of a number of well-defined item features

Y = (Yj(w), ... , Yr(W)) In particular, the vector Y may be used to rate each item with respect to its fitness for use, i.e. its quality. This quality rating procedure defines a function X on all possible realizations of (Yj(w), ... , Yr(w) ) called item quality indicator:

X : JR.r

---+

I

c JR.

(1)

where I is the set of all possible values of X . Two items are of the same quality, if their item quality indicators adopt the same value. Moreover, without loss of generality, it is assumed here that small values of X are assigned to items of high quality, and large values of X to items of low quality. Obviously, X enables the comparison of any two items with respect to quality : Let

Wl

and

W2

be two items, then:

(2) where "~" means better quality, "=" means equal quality, and "---" means

¢} ¢} ¢}

II(O(l») > II(0(2») II( 0(1») = II( 0(2») II( 0(1») < II( 0(2»)

(12)

better quality, ";:" means equal quality, and "-(" means worse quality.

The relation between the one-dimensional process quality indicator II and the s-dimensional process parameter 0 is similar as the relation between the one-dimensional item quality indicator X and the r-dimensional vector of item features Y . Hence, analogous to the quality classes SiEI with respect to the items introduced in (3), it is possible to introduce quality classes 8 aEA with respect to the process:

8 a ={0III(0)=a}

(13)

The quality classes {8 a }aEA constitute a partition of the parameter space 8 into subsets of, in general, more than one element

8 = ua 8 a where each 0 E 8 a determmes same process quality, given by the process quality indicator II = a.

Example:

Pi'

Let be the normal probability measure implying that 0 = (J.L,0"2). Moreover consider the twosided conforming/nonconforming case with a lower specification limit L and an upper specification limit U, and let W(X) = X. Then II(O) = Eo[W] = Eo [X] = : p E (0,1) is the probability of producing a nonconforming item. The quality classes 8 p are given by

8p =

1 ~ =l-p} {(J.L,0"2) I 1,Lu V2irO"e2a

for

pE(O,l).

Remarks:



It is clear from the above example that selection of the probability measure {Pi'} is of eminent significance. The usual practice of uncritically assuming normality can definitely not be recommended, as it may lead to dramatic errors.



The process quality indicator II(O) enables the rational selection of a target value 00 E 8 for the process. Assume that large values of II indicate good quality of the process and small values bad quality, then

00 •

is target value

{=?

II(Oo)

~

II(O)

for any

0E 8

The process quality indicator II (O) measures in a natural way the process capability to meeting expectations with respect to a certain well defined criterion . Hence II seems to be sui table to serve as a process capability index.

75 3. Contro l P o licies In Statistical Process Control the process is monitored and an alarm is released whenever an unfavourable value of the process quality indicator II is predicted, which, of course, includes false predictions, i.e. false alarms. From this the three fundamental problems of Statistical Process Control follow, namely the determination of •

control intensity,



control accuracy,



decision rule.

An instruction answering these three questions is called control policy. The statistician's task is to develop and make available appropriate control policies to be implemented into industrial practice. Appropriate means that the policies mu'st be 1. simple, and 2.

effective.

Simple, because it should be possible to implement them into industrial practice without major problems. Effective means that they should be adjusted to the process at hand in order to draw maximum profit. The difficulty which led to the development of Statistical Process Control is the simple fact that generally it is impossible to observe the process parameter {£I j } or the process quality indicator {IIj} directly, but only indirectly by monitoring the realizations of the item quality indicator {Xj}, or realizations of the underlying item feature {Yj} = {(Yj,I, ... , Yj,m)} , or by monitoring a random variable {Zj} being correlated with {Xj} or {Yj }, and drawing inference on II j from the observations by statistical methods.

3.1

Intensity and Accuracy

Any control policy works basically in the same way. To answer the questions with respect to intensity and accuracy, the sequence {X j } j =l ,... or {Y j }j=l, .. , respectively, is divided into subsequences {Md of length Nk E lNo, with

Mf =

{X1+,\,k-l N.' .... ,X'\'k

N}

(14)

M,r = {YH'\'k-1N ., .. .. ,Y'\'k N}

(15)

L...J=l

J

L...,=l

J

or L."J=l

J

UJ=l

J

No item of the subsequences {M 2k -d k=l, ... is tested, each item of the subsequences {M2d k=I, .. tested. In other words no testing and 100% testing alternate. There are principally two types of sequences {Md:

IS

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Type 1: The length



Type 2: The length variable.

of M k , for k = 1,2, .. . , is an nonnegative integer.

Nk

of M

Nk

for k = 1,2, ... , is a nonnegative, integer-valued random

k,

In view of the demand for simplicity the practical significance of Type 2 policies is limited. Type 1 policies, on the other hand, are widely used in industrial practice. The following special cases are distinguished in literature: •

N 2k - 1

= 0,

N2k

> 0 for k

= 1,2, .. :

In this case each produced item is tested, the corresponding control policy is called screening policy. •

N 2k - 1

= N > 0,

N2k

> 0 for k = 1,2, .. :

In this case the interval .beween consecutive sampling actions is constant, the policy is called periodic sampling policy. •

N 2k - 1

01 constant,

N2k

> 0 for k

= 1,2, .. :

Such policies are called VSI control policy, i.e control policy with variable sampling intervals . •

N 2k - 1

> 0,

N2k

= n ~ 1,

k = 1,2, .. .

In this case the number of consecutively tested items, i.e. the sample size, is constant, hence it is called fixed sample size control policy •

N 2k - 1

=

N, N2k

= 0,

k = 1,2, ...

In this case sampling is dispensed and at regular time intervals process inspections are performed. This case is known as routine inspection policy. Remarks: •

Clearly, the sequence {N2k - 1 } k-l 2 determines the times of control and hence, control intensity. The sequence {N2d k=1,2, .. giv-;;s' the sample sizes, and therefore control accuracy. The two sequences determine completely the monitoring actions to be performed.



The simplest control policies with respect to industrial implementation are routine inspection policies and periodic control policies with fixed sample size, respectively.

3.2

The Decision Function

Decisions are made at the end of each subsequence M 2k , k = 1,2, .. . , which corresponds to the end of production of item No. (L:;~l N j ) • Therefore, the timepoints at which production of the

(I:;~l N j )-th item for k

= 1,2, ... is completed, are called decision points.

Generally in Statistical Process Control only two decision alternatives are considered: alarm leadi ng to some subsequent intervention, and no alarm leaving the process alone. Unfortunately, two-valued decision functions don't take into account situations where, after an alarm has been released, one has to decide upon the type of intervention to be performed, e.g. replacement, renewal, repair, etc.

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Let d denote a decision function, then

d= {01

¢o} ¢o}

no

alarm alarm

(16)

Similar to the two types of monitoring sequences, there are two types of decision functions used: 1.

independ ent decision functions

2.

d ependent decision functions

For independent decision functions the decision at decision point (E;~l N j ) is based exclusively on the actual sample, given either by

{ X1+,,2.-1 UJ=l

N.' .... ,X"'. N'} 1

~j=l

J

or

., .... ,Y,,2. { Y 1+,,2.-1 Wl=l NJ L.",=l

N'} J

For dependent decision functions any decision depends on all past observations, I.e. on either the sequence

X 1+ N"

... ,

X N, + N2 , ... , X 1 +,,2k-l N . ' L.."J = l

.... ,

J

X",.

l...Jj=l

N· J

or on the sequence

Y1+N ... , Y N,+N" "

... ,

Y1+,,2.-1 L.,..,_l

N ·' .... , J

Y",. N. Lj_l J

From the viewpoint of simplicity independent decision functions are to be preferred . In industrial quality control independent as well as dependent decision functions are use. The Western Electric runs rules are an example for the latter ones. For any situation there are several possible decision functions which could be used. Hence, the question of quality of decision functions arises implying the need for a rating procedure by means of which two given decision functions may be compared . In literature there are two different approaches which correspond to the two different types of decision function: •

The Operating Characteristing Function £ Let d be a independent decision function then the operating characteristic function £ of d is defined as the conditional probability of no alarm under the condition that the process parameter adopts the value B:

(17)

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The A verage Run Length ARL Let d be any decision function, and R.:J.(B) be the number of decisions until the first alarm is released under the condition that the process parameter adopts the value B. Then the average run length ARL is defined as the following conditional expectation:

(18) Either of these approaches for evaluating and comparing decision functions Control have in common at least two major deficiencies: 1.

III

Statistical Process

C and ARL rank decision functions according to statistical properties, but not according to their benefits with respect to the overall objective.

2. C and ARL are based on the observation that a good decision function should give no alarm if the process parameter is satisfactory, and it should give an alarm in the opposite case. Therefore, an appropriate determination of "satisfactory" and "non-satisfactory" process parameters is most crucial, but generally neglected when different decison fun ctions are compared by means of the operating characteristic function Cd(B) or the average run length ARLd(O) . As a trivial consequence, decisions functions ranked and selected by means of the operating characteristic function C or the average run length ARL may neither the best nor even good in certain circumstances. Generally the same decision function d is used in any of the decision points. But, of course, one can generalize the concept and have different decision function to be used, e.g. depending on the history of the process. In this case the subsequent decision functions form a sequence {d k h=1,2, .. '

4. The Design of Control Policies

In the last section, it was mentioned that selecting the decision function by means of statistical criteria may lead to an unsatisfactory result. Here, some principles are very briefly indicated how a control policy given by the following quantities should be designed: •

The sequence {N2k -d k=1 ,2, .. which can be interpreted as the subsequent control intervals.



The sequence {N2d k=l,2, .. which can be interpreted as the sequence of subsequent sample sizes.



The sequence {dd k=l,2, .. of subsequent decision functions .

The design should be appropriate in order to draw maximum benefit from applying the control policy. Consequently the applied design criterium must necessarily take into account the following two points: 1.

the ultimate purpose of having the process run, and

2.

existing and non-existing information about the technical and statistical properties of the process .

Clearly, the purpose or aim of the process is necessary in order to define and determine the benefits of a control policy, and existing information should be used to adjust the policy as well as possible to the given situation. Moreover, non-existing information must be overcome by suitable measures, for instance by worst case considerations.

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4 .1

Traditional Control Policies

Looking at procedures proposed in relevant literature or recommended by national and international organisations for quality control, it is easily verified that many of them don't follow the above described lines. Therefore, one may conclude that in many cases inappropriate control policies are used in industry. A control policy might be optimal in some statistical sense, but this doesn't imply that it is good with respect to the proper purpose of the process . The following features characterize the traditional approach and reveal the necessity for some changes: 1.

A majority of relevant publications deal exclusively with the isolated question of selecting and designing the decision sequence {d k }.

2.

The recommendation for a certain decison function and its design is often made on the basis of some marginal improvements with J;espect to some statistical properties, e.g. the probabilities of Type I and Type II errors.

3.

Complexity of procedures on the one hand and simplicity on the other hand are hardly taken into account.

4.

The question how to select control intensity and accuracy, respectively, is neglected dreadfully. Often they are assumed to be fixed, or some rules of thumb are proposed to the practitioner.

As to the last point, assume that the available decision functions have similar properties. Then it is obvious that the selection problem with respect to the decision function is of rather minor significance when compared with the selection problem of the control intensity and control accuracy, respectively. For instance, the number of fal se alarms as well as the delay time in detecting some disturbances depend rather on the frequency and the accuracy of the monitoring actions than on the subsequent decision function .

4.2

Control Strategies

As already mentioned the two sequences which determine intensity and accuracy of a control policy, determine several distinct control strategies, i.e. sets of similar control policies: •

Screening Strategy for

N 2k - 1

= 0,

N2k

> 0,

k = 1,2, ...



Sampling Strategy for

N 2 k-l

> 0,

N2k

> 0,

k = 1,2, ...



Inspection Strategy for

N 2k - 1

> 0,

N2k

= 0,

k

= 1,2, ...

Of course, this list could be easily extended by classifying strategies with respect to the sampling mode, decision function, types of interventions, etc. In any case, there are policies which include proper sampling actions, other dispense with sampling in favor of routine inspections, and others screen the production process. Generally, each of these different strategies can be applied in any given situation. Hence, investigations aiming in deriving an appropriate control policy must necessarily consider policies from any possible strategy. But, searching relevant literature for papers taking into account more than one strategy yields only a poor result.

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5.

Objective FUnction

The key for solving many of the above mentioned problems is the development of appropriate objective functions for optimization. Clearly comparing different procedures is impossible without an appropriate objective function, which must reflect the overall purpose the procedure in question is applied for. As a matter of fact, the objective functions used in traditional Statistical Process Control express only "partial" aims and not the overall purpose. For illustration take a widely used objective function namely the average fraction inspected AF I :

AFI -- l'1m

"k

L.",=l 2k

N

2k

k~oo Lj=1 Nk

which, of course, adopts its absolute minimum and therefore its optimum for any policy with N2k

= 0,

for

k

= 1,2, ..

For omitting the trivial and undesired result more or less arbitrary side conditions have to be imposed, making it almost impossible to evaluate the resulting policy appropriately. There are only few examples of objective functions in Statistical Process Control attemp ting to reflect the overall purpose, i.e. the objective of the process. The best known led to the so-called Economic Approach in Statistical Process Control. Investigations started in the 50th particularly by Duncan's (1956) pioneering ' paper on the economic design of control charts. Since then , this approach has been adopted by a growing number of researchers. Very recently, Ho & Case have provided an excellent literature review on the economic design of control charts for the past decade. But, besides Montgomery's book (1991), there are only very few textbooks on Statistical Process Control in which the economic design has been included. Moreover, there is no industrial standard based on the economic approach, and therefore, it is more or less unknown in industrial practice.

6. Conclusions The short message of this paper is that there is no sound theoretical foundation of Statisti cal Process Control. It is widely considered as a mere field of application of statistical methodology and presents itself as a rather unarranged pile of different methods. In order to change Statistical Process Control to a proper stochastical theory of its own, it is necessary to define the special processes of interest on the one hand, and to develop relevant types of objective functions to be used for different purposes on the other h?-nd. Both concepts, i.e. process model and objective function, depend to a wide extend on quality indicators as introduced in Chapter 2. Any rating procedure for control policies must be based on a suitable process quality indicator, which itself is derived by means of an item quality indicator. The rather unsatisfactory state of the art of Statistical Process Control, results in a large number of extremely interesting problems, which have not been considered adequately yet. Some of them are listed below: •

Development of a theory of quality indicators for different objects of interest, including items and processes with by-products concerning specifications and process capability indices, respectively.

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Development of suitable process models and objective functions for optimization with respect to complete control policies taking into account any admissible control startegy.



Investigations about the abuse of the normality assumption and development of alternatives including non-parametric methods.



Extension of the usually used two-valued decision functions to m-valued ones with m > 2.

The theory should make available tools, basically allowing to determine an appropriate control policy based on the ultimate purpose for having the process operate, and the set of admissible control policies: 1.

selection of a suitable item quality indicator

2.

selection of a suitable process quality indicator

3. deriving an appropriate objective Function 4.

determining an optimal control policy by optimization.

Acknowledgement Research partly supported by the Commission of the European Communities, DGlII, Industry, under contract COPERNIKUS CP93: 12074.

References [IJ

BECKER, R., PLAUT, H. and RUNGE, 1. (1927) : Anwendungen der mathematischen Statistik auf Probleme der Massenfabrikation. Berlin: Julius Springer.

[2J

COLLANI, E. v. (1989): The Economic Design of Control Charts. Stuttgart: Stuttgart.

[3J

DUNCAN A.J. (1956): The Economic Design of X-Charts 'Used to Maintain Current Control of a Process, JASA 51, 228-242.

[4J

HO, C. and CASE, K.E. (1994): Economic Design of Control Charts: A Literature Review for 1981-1991, Jounal of Quality Technology 26, 39-53.

[5J

LORENZEN, T. J. and VANCE, L.C. (1986): The Economic Design of Control Charts: A Unified Approach, Technometrics 28, 3-10.

[6J

MARCUCCI, M. (1985): Monitoring Multinomial Processes, Journal of Quality Technology 17, 86-91.

[7J

MONTGOMERY, D.C. (1980): The Economic Design of Control Charts: A Review and Literature Survey, Journal of Quality Technology 12, 75-87.

[8J

MONTGOMERY, D.C. (1991): Introduction to Statistical Quality Control. New York: Wiley.

[9J

MONTGOMERY, D.C. (1992): The Use of Statistical Process Control and Design of Experiments in Product and Process Improvement, lIE Transactions 24, 4-17 .

[10J

SHEWHART, W.A. (1926): Quality Control Charts, Bell System Technical Journal, 593-603.

82

[ll]

SHEWHART, W.A . (1931) : Economic Control of Quality of Manufactured Product . New York: Van Nostrand.

[12]

SVOBODA, L. (1991): Economic Design of Control Charts: A Review and Literature Survey (1979-1989). In: Statistical Process Control in Manufacturing. Eds. J.B . Keats & D.C. Montgomery. New York: Marcel Dekker.

[13]

WOODALL, W. H., and FALTIN, F . W. (1995) : An Overview and Perspective on Control Charting. In: Statistical Applications in Process Control and Experimental Design. Eds. J.B . Keats & D.C. Montgomery. New York: Marcel Dekker.