Monitoring Serially-Dependent Processes with ...

Monitoring Serially-Dependent Processes with Attribute Data WILLIAM A. STIMSON and CHRISTINA M. MASTRANGELO

University of Virginia, Charlottesville, VA

22903

Traditional methods of statistical process control ( SPC ) assume that observations of a quality character­

istic are uncorrelated. This paper examines the case in which unacceptable variability in a product at one station in a production sequence will cause unacceptable variability in a product at the succeeding station. This serial dependence means that conventional SP( may be ineffective. For example, serially-dependent processes with attribute data may be nonstationary, demonstrating a geometric increase in the probability of occurence of a defect. This increase can be arrested by placing control points in the chain, with respect to the variability of the process at each work station. A control point is an inspection or correction station or both, that will re-establish the product attribute to a desirable level at that point in the sequence. Some of the resulting segments will contain stations with such low probability of defect that a correlated effect over an abbreviated segment may behave according to distributions that can be controlled by conventional SP( methods. Other segments may magnify short trends to create out-of-control conditions. These segments can be controlled by regression control. This paper develops a control system that maintains stability of serially-dependent processes irrespective of the number of stations or basic defect rates of the individual work stations in the sequence.

Markov chains are often used to model a process which generates attribute data. Markov models are powerful in the description of a process, but less tractable in control, due in part to the inevitable need to determine steady-state eigenvalues of large matrices. Conditional probability models ( Budescu (1985)) offer an alternative approach to process mon­ itoring and the preservation of stability.

Introduction

(("'IHANGES in process variability are revealed by the 'l,., monitoring of product characteristics. These

characteristics may be variables or attributes data, and successive observations of these characteristics are assumed to be independent. The independence assumption, however, is often violated. Recently, many authors have studied serial dependence of vari­ ables data, for example, see Kritzman (1989), Mont­ gomery and Mastrangelo (1991), Anderson (1991), Faust (1992), and Wade and Woodall (1993). Some authors have also studied serial dependence of at­ tribute data using Markov models-Budescu (1985), Costamagna, Maestrini, and Sante (1990), and Livny, Melamed, and Tsiolis (1993). This work also exam­ ines serial dependence of attribute data, but from the position that the conditional probability of two suc­ cessive activities is one if a defect is found at the first activity.

In general form, serial dependence of attributes can be expressed by the following. Let an at­ tribute have a value Xt at time t, and let samples of this attribute be taken at equally spaced times, t, t - I, t - 2, . . .. Then the values of such attributes have a conditional probability expressible as

where f (Xt, Xt-I, t) is an arbitrary function of cur­ rent and previous values of lag one and of time. Se­ rial dependence is frequently found in the post-design phases of the production of systems: manufactur­ ing, assembly, and test. This occurs because systems are composed of elements assembled for a synergistic purpose. Consider the following process, in which a manufacturer builds cascade amplifiers. On a single printed circuit board are mounted n single pole-zero

Dr. Stimson is a graduate of the Department of Systems Engineering. He is a Member of ASQC. Dr. Mastrangelo is an Assistant Professor of Engineering in the Department of Systems Engineering. She is a Member of ASQC.

Vol. 28, No.3, July 1996

279

Journal of Quality Technology

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WILLIAM A. STIMSON AND CHRISTINA M. MASTRANGELO

networks, each with a transistor amplifier. One task is the installation and tuning each of the n stages for minimum load to the succeeding stage. Each stage is tuned either correctly or incorrectly. If a stage is mistuned, the subsequent stage will also be mis­ tuned, since it will be tuned to the wrong load. A mistune (or misload) is a defect. The probability of a defect at each stage is conditional upon the tuning of its predecessor as well as its own tuning. At the end of the n stages, the board proceeds to a test sta­ tion, where the bandwidth and gain of the assembled amplifier are tested. An out-of-tolerance bandwidth or gain is defined as a defect. Note that it makes no difference whether the n stages are assembled at one station, or whether one stage is assembled at each of n stations. This example shows a basic characteristic about systems previously mentioned-a system has a syn­ ergistic objective that goes beyond the individual ob­ jectives of its elements, and is therefore dependent upon them. Equally importantly, it often happens that the individual elements become dependent upon one another in meeting the requirements of syner­ gism. It is this interdependence of elements that may show up as serial dependence in a work station se­ quence. Serial dependence may be costly. In the case of the cascade amplifier, if the jth stage is mistuned, then the cost of quality is the cost of retuning not just the jth stage, but all the following stages as well. Again, consider a rocket launcher that is installed on the deck of a fighting ship, then scuff-coat is applied to the deck, and finally the deck is painted. If the launcher is mislocated by even a small amount, all three jobs must be partially undone, then redone. Properties of Serial Dependence Work Station Definition

A work station is defined as a place and time pair in which an activity occurs. Examples of relevant ac­ tivities are work, inspection, measurement, and de­ cision. The result of an activity is measurable as an attribute of time, cost, or quality. Sequences in­ clude: (1) moving a product from station to station in an assembly line; (2) progressing from work item to work item on the critical path of a project; or (3) conducting a series of related tests. Consider the n work stations depicted in Figure 1. The output of each work station provides the input to the following work station. Each work station output is identified

Journal of Qualify Technology

Process Output

P r ocess I nput

�_ \

PROCE SS X2

X1

X3

��-G-� Work S tatlo n1

Work S t a t i on

n

PJ

=

Defect Proportion at Station I

PJ

=

Estimated Defect Proportion at Station I

FIGURE 1. The Generalized Work Station Scenario.

as an attribute, Xj. Associated with each work sta­ tion is a probability of defect, or defect rate, that is intrinsic to that station. It would exist even if the station were in isolated use. For example, let the work station be a stand-alone machine that performs tasks. The defect rate of that machine is the ratio of the number of defective tasks to the total number of tasks performed. This defect rate is denoted as the basic defect rate, Pj. This basic defect rate will not be known for certain; one can only estimate this parameter. However, once the "machines" or work stations are put into the system, one cannot directly make a mea­ surement of the Pj. When the process is in operation, one will measure at station j the correlated defect rate, Pj, rather than the basic, uncorrelated defect rate Pj. It will be shown that there is a way to com­ pute Pj based upon measurements of Pj. The basic, or uncorrelated, defect rate is itself a statistic, but in the interest of avoiding redundant notations, the basic defect rate is denoted Pj, and the correlated defect rate is denoted Pi. Assumptions

In the following development, assumptions are made concerning the behavior of serially-dependent attribute data, and from them is built a probability distribution of this behavior. From this distribution, a model is constructed that describes the produc­ tion process, and from the model, a process control system is developed. Thus, the foundation of this process control scheme lies in the assumptions. Se­ rial dependence of attribute data is a special form of serial dependence, and so must be defined. With re­ spect to attribute data, serial dependence is defined as

Pr(Xj

=

1!Xj-1

=

1)

=

1, V j: 1


Xj 1 and --,Xj ---> Xj 0 will be used. The symbol --. implies "not", and in this context means that no defect was found at the jth activity. Starting with this description of attribute serial dependence, the necessary assumptions are: Assumption 1. Conditional dependence implies that Pr(XjIXj_d 1 where 1 < j ::; n. The condi­ tional dependence of interest is that of serial depen­ dence. Assumption 2. The initial defect rate, PI, is ob­ tainable. We assume that data are available to esti­ mate an initial basic defect rate PI, of the first sta­ tion. If only the overall process defect rate is avail­ able from initial data, then an estimate of the basic defect rate of station 1 may be made from this. Assumption 3. The defect rate for the initial sta­ tion is independent and defined as Pr(XI 1) Pl. Hall (1977) defines the environment as that which lies outside the system. Thus, being the initial ele­ ment of the process, or system, the first work station receives its input from the environment. Its confor­ mance depends only upon itself and is independent of the state, performance, or conformance of ensuing work stations of the process. Therefore, the prob­ ability of defect of Xl is the basic, observed defect rate of station 1, Pl. =

=

=

=

=

=

=

=

=

=

From these three assumptions, the related proba­ bilities are Pr(Xj = 0) Pr(Xj = 0IXj-1

=

qj

=

1)

=

=

=

Development of the Conditional Distribution

From the assumptions stated in the previous sec­ tion, the process distribution can be determined. To develop this distribution, we will assume that each

Vol. 28, No.3, July 1996

work station has the same basic defect rate, Po, where Po Pr(Xjl--,Xj_1 0). In general this would not be true, but the assumption is made in order to deter­ mine the form of the process distribution and not to find accurate solutions of defect rates of the stations. The latter will be done later once the distribution is identified. =

=

The process distribution is established by deter­ mining a conditional probability of defect for each station, Pr(Xjl--.Xj_d Pj, for all j, 1 < j ::; n, where n is the number of stations. This can be done by taking advantage of the assumptions, and by con­ stant iteration of =

Pr(Xj)

Pr(XjIXj-d Pr(Xj-d

=

+

Pr(Xjl--.Xj-d Pr(--.Xj_l).

(1)

Iterating on (1) for n stations, the conditional probability distribution expressed as Pr(Xn), defined as Pn, is Pr(Xn)

=

1 - (1 - p)

n

=

1 - q�

==

fin.

(2)

The conclusion is that the conditional structure of serially -dependent attribute data increases geomet­ rically from an initial basic defect rate, Po PI, at the first station. The individual Pj will reflect both serial dependence and the individual characteristics of each station. From (2), we can compute the cor­ related defect rate at station j + 1 due to serial de­ pendence. =

For control purposes the uncorrelated Pj is also needed, and this can be determined from statisti­ cal data according to the following argument. We will say that the measured probability is Pj. Any measurement of work station defect rate necessarily measures the conditional probability of that station, since it is in the operational sequence. That is,

O.

One further statement is inferred-for a station in control, Pr(Xj 11Xj-1 0) Pj. Use of the term "in control" does not mean that the yth work station is necessarily in control, but that it is subject to its own dynamics and is not conditioned upon the j - 1 work station. =

281

(3) where N is the total count of the process and nj is the number of defects at station j over this iteration. Then starting at station 1 Pr(Xd

PI Pr(X2) = Pr(X2IXI) Pr(Xd + Pr(X21 - -.X I) Pr(--.Xd =

=

(1)(pd + (P2)(iid

where Pr(X21--.Xd

=

P2. Pr(X2) is the conditional

Journal of Quality Technology

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WILLIAM A. STIMSON AND CHRISTINA M. MASTRANGELO

probability of X 2, hence Pr(X 2) == fh. Also, P 2 (PI) + (P2)(ih), which can be solved for the basic defect rate at station 2 =

Similarly, the process can be iterated over n stations with the general expression of (1), therefore (4) Since Pj represents Pr(Xjl-.Xj-d, it is the uncorre­ lated defect rate. Finally, PI PI, by assumption 3. Thus, all Pj can be established on a station-by­ station basis. =

Process Stability

A process subject to serial dependence is not a sta­ tionary process. The expected value for condition­ ally probable defective activities is E[Xj] = 1 - q j. Consider n binary random values, Xn, each with an expected value of Pn, then E[x] =

(1) n . ;;: �(1 - qJ).

For example, when (2) is plotted with Po 0.04, the result, shown in Figure 2, indicates that the prob­ ability of failure approaches certainty, given enough work stations. Because of the increasing likelihood of failure, the historical defect rate of the overall pro-

cess sets a limit on the number of work stations that are feasible for that process. This effect occurs in the presence of serial dependence regardless of the type of process. Consider several examples of this explosive behav­ ior. In a simulation study Stimson (1991) examines the effect of random delays in the repair of a ves­ sel. Along the critical path of the project, successive tasks are dependent on the successful completion of its predecessor. When tasks along the critical path are not completed on time, no slack or float time is available for the activity, and the completion time of the project is subsequently delayed. This is called negative float. Figure 3 shows the state of the repair in terms of the negative float on the critical path. Equating the state of repair to Xj, Figure 3 illus­ trates the unstable effect of serial dependence. The critical path of a project is a Markov pro­ cess, as is the serially-dependent work station model. Livny, Melamed, and Tsiolis (1993) show the effect of autocorrelation of services in an MIMI1 queue is another example of the same instability; this is demonstrated in Figure 4. Both processes are out of control. Serial dependence destabilizes a process. Process Control Methods

=

p .9

I

Processes with serial dependence may be con­ trolled only by arresting or reducing the increasing defect rate. There will always be some variation in defect rate, but in the absence of assignable causes, this variation would be stable. The effect of serial dependence may be reduced in two steps

po· 0.04

1. establish control points to break up the n sta­ tion sequence into m segments, thereby arresting the geometric increase in pj at an acceptable level within each segment; and 2. monitor the segments for statistical control.

10 FIGURE 2.

Control Strategy

20 30 40 50 SERIAL INSPECTION NUMBER Probability of Defect at the

Dependent Work Station.

Journal of Quality Technology

/h

60

N

Serially-

Step 1 will involve inspection and repair at the con­ trol points in order to reduce the correlated defect rate to the basic defect rate inherent at that station. For example, if the control point follows station j, then inspection and repair effectively resets the de­ fect rate at the next station, say j + 1, from PHI to PHI· Step 2, monitoring the segments, may be done in several ways. Segments may be controlled either by

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MONITORING SERIALLY-DEPENDENT PROCESSES WITH ATTRIBUTE DATA

For example, the cost function will maximize the sequence, k, of a series of work stations, k < n, be­ fore accepting the cost of an inspection and repair station. The constraint function ensures that the dis­ tribution (1 - qn ) generated by serial dependence is stochastically dominated by a distribution G[X(n)] that describes an independent process. Thus

200

100

Maximize Subject to

o

o

800

400

1200

1600

Scheduled Man Days FIGURE 3.

A Nonlinear Increase in Negative Float be­

cause of Absolutely Dependent Events.

charting all stations or by charting only selected sta­ tions. If n is a large number for which it is not cost effective to chart all stations ( the general case ) , then only the m control points would need to be charted. In order to be effective in terms of both cost and qual­ ity, the placement of the control points can be deter­ mined by the use of equivalent distributions, which is described in the following paragraphs. Control Point Location

The strategy of control points, developed by Fried­ man (1992), is used to regulate the rate of an indus­ trial process in order to accommodate disruptions and varying in-process inventory. A control point is a place or time in the production line at which production speed can be adjusted. However, adjust­ ments are not free, so that the number of control points must be strategically placed-frequent when necessary, infrequent when possible. Friedman's re­ search provides an analytical way of determining how many points are required and where to place them in the production line. Control points can also be used to regulate the rate of defects. In a work station sequence, control points, by our definition, will occur at an inspec­ tion or a repair work station, or a combination of both. The objective is to maximize the number of work stations between inspections while minimizing the defect probability. These are conflicting objec­ tives because in the presence of serial dependence, the probability of defect increases with the length of the work station sequence. The strategy, then, is to treat one objective as a cost function, and the other as a constraint function.

Vol. 28, No.3, July 1996

k

(5)

G[X(n)]::; ( 1- qn ).

The range of k for which the constraint holds true is the optimum length of the sequence for a given pro­ duction process. At this control point, an inspection is conducted, the product is verified, and, if neces­ sary, repaired. The distribution G[X(n)] can be identified by the equivalence of distributions. Since the right hand side of the constraint of (5) is geometric, an appro­ priate candidate distribution for G[X(n)] is the bi­ nomial, as shown in the following discussion. Independent attribute data are often modeled by the binomial distribution. One measures p, the ra­ tio of defect count to sample count. Control limits are then based on binomial parameters (n, p). A geo­ metric distribution is described by a random variable may be defined as "the number of serially-dependent inspections until the first defect occurs." Stated in terms of a binomial distribution, one might say "the number of independent inspections in which one de­ fect occurs"

M/M/1 WAITING TIME Autocorrelated Service W rJ)