Optimization of average-run-length properties of control charts using recurrent events

Computers and Industrial Engineering Vol. 25, Nos 1--4, pp. 449--452, 1993 Printed in Great Britain. All rights reserved

0360-8352/9356.00+0.00 Copyright © 1993 Pergamon Press Ltd

Optimization of Average-Run-Length Properties of Control Charts Using Recurrent Events Noel Artiles-Le6n, Ph.D. Associate Director Industrial Engineering Dept. University of Puerto Rico Mayag0ez PR 00681-5000

Roberto Pdrez-Matos, M.E. Industrial Engineering Dept. University of Puerto Rico warning limit; 2 consecutive points above (below) a warning limit; 3 out of 5 points above (below) a warnControl charts are used in many production, research, ing limit; 3 out of 4 points above (below) a warning and development environments. Control limits comput- limit; and 2 out of 3 points above (below) a warning ed from a given standard are used to detect when a limit. The methodology used to compute the ARL is process, which is in control at certain target values of based on the theory of recurrent events and it is the distribution parameters, departs from those values. computationally more efficient than the traditional In 1930, Shewhart recommended the use of 3c limits method for the calculation of ARL's based on Markov as action limits; that is, rectifying action should be chains. taken if the computed value of the statistic being charted is plus or minus three or more standard errors A Short Introduction to Renewal Theory from its target value. More recently, several schemes, and its Application to Control Charts. such as CUSUM and exponentially weighted moving average (EWMA) charts, have been suggested to In this section we provide a brief account of the main control a parameter of a distribution at a target level. A aspects of renewal theory. Renewal theory is conmodification to the basic control chart is the use of cerned with problems relating to processes which, after supplementary tests for runs. A run test calls for every repetition of some given phenomenon (in our corrective action when some samples out of a prede- case, the violation of a control rule), start all over again termined number of samples fall outside a specified with the same initial conditions. A simple example warning line. More precisely, warning lines are drawn arises with the occurrence of an out-of-control signal in in less extreme positions than action lines; the occur- a Shewhart control chart. If quality is constant, the rence of some points above the upper warning line or occurrence of an out-of-control signal can be modeled below the lower warning line is considered as sufficient as a Bernoullian random variable with probability p. evidence for taking corrective action. The waiting time up to the signal has a geometric distribution. The waiting times between successive outEven if elaborate control schemes are used, any of-control signals are mutually independent random scheme will give a signal sometime although the variables having the same geometric distribution. process is operating under control; it is also certain to give a signal sometime after a deterioration in the Denote by fn the probability that a recurrent event, E, process has occurred. Consequently, a quantity of occurs for the first time at time n (at trial n): f, = Prob(waiting time for E = n). fundamental interest is the average run length (ARL) of a given set of control rules. The ARL is the expected It follows that f = ~f, < 1 if the probability of an occurnumber of samples taken before action is taken when rence of E does not tend to certainty as the number of the quality of the process remains constant (not trials increases indefinitely or f = ,T-,fu= 1. Denote by f= necessarily at the target value). Many samples are the difference 1 - f. If f_ = 0, the event is called oersisdesired before receiving an out-of-control signal when tent otherwise it is called trBnsient. the process is in control, and a few samples are desired when the process has departed from target. By convolution, f(r), the probability that the recurrent These are conflicting goals and in practice some kind event E occurs for the rth time at the nth trial, can be of corn promise between these two requirem ents has to expressed as: be accepted. This paper deals with the statistical design of control charts and describes a methodology f n r) = f~ l ~ e ( r - l ) + f 2 ~en -(r-l) + iCn_l_fl(r-l) n-i 2 + .,. to identify control schemes with action and warning limits that, for a fixed in-control ARL, minimize the ARL at a given out-of-control situation as measured by a w h e r e f(r) = 0 for r > n. shift in the process mean. The probability that the recurrent event E occurs at trial In this paper we compare the performance of the n (without taking into account whether it was the first, control rules: 8 consecutive points above (below) a second ..... or whatever) can be computed as: warning limit; 5 consecutive points above (below) a Introduction

449

Proceedings of the 15th Annual Conference on Computers and Industrial Engineering

450

Obviously U. = 0 for n < 1, since we need at least two sample points to obtain a signal. Letting n tend to infinity:

Z'1

and it can be shown that

P~ = l i m U. * Pa lira U. n-÷~

u' = ~ u o /'~=1

=

eo

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or

t]=l

lira U n = If the event is transient, f < 1 and U" = f/(1-f). If the .... l+p a event is persistent, f = 1 and U" = o~. In this case, the expected number of trials for the occurrence of the and the average run length of the chart is given by event (or its expected waiting time) is given by:

T = -'+'1 + 2f2 + 3f3 + "'"

+ nfn

ARL-

+ "'"

and it can be shown that

Example: Consider a p-chart (a control chart to monitor the proportion of defective items) that calls for an outof-control signal if two consecutive samples plot above some fixed limit (see Figure 1). The recurrent event of interest here is "hA" (two consecutive sample points fall above the "a-limit"). The sequence of samples depicted in the figure can be represented as "OOO AO AA".

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Problem Definition and Methodology

1 _ ].im O"n.

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In this paper, T(k,m,a,b) denotes a control rule that calls for an out of control signal if (i) k of the last m standardized sample means fall in the interval (a,b) a
n

Figure 1.: A p-chart with a run rule

For any reasonable control scheme this event is persistent (an out-of-control signal will certainty occur as the number of sample points tends to infinity); consequently, the average run length of the control chart can be computed as the expected waiting time of the event "AA" by ARL

=

(Lira U.) -~"

Given ARLo, r, 5, and the positive integer numbers m 1, m 2..... m,, kl, k2..... k. m~ > kj, i=1,2 ..... r, that define the family of control schemes S : [Tl(kl,ml,al,bl), T2(k2,m2,a2,b2)..... Tr(kr,mr,ar,b,)] find optimal control limits, say a" = [a,', a2", .... a,'], and b" = [bl", 1:)2",.... br] such that, for all a and b satisfying ARL(a,b,0) = ARLo, ARL(a',b',p) _
n-÷m

where U. is the probability that a signal occurs at time n. To compute U., we use the following argument: the occurrence of an A at time n-1 and an A at time n happens if and only if: i) a signal is generated at time n, or ii) a signal is generated at time n-1 and it is followed by an "A". This equivalence implies that (it we let P. = probability that a sample point falls above the a-limit): P~

= U.

+ U._zP a ,

n = 2, 3, 4

In this paper we assume that 5=1 and that control schemes are of the type S = {T1(1,1,c,oo), T2(k,m,w,c ) I w _
One way to solve this problem is by using penalty methods. Instead of minimizing q0, a new objective function, r 0 is minimized. The function r0 is derived from the original function and the constraints in such a way that r0 includes a "penalty" term that increases

ARTXLes-LE6N and P~REz-MATOS: Control Charts

the value of r 0 whenever a constraint is violated with larger violations resulting in a larger "penalty". Therefore, solving problem (P1) is equivalent to solving Min. r(w,c) = ARL(w,c,1) + h I ARL(w,c,0) - ARL01 where h is a positive constant denoting the "penalty". Results In this section, optimal schemes for X charts are presented. The out-of-control condition is defined here as a shift of in the process mean of one standard error. All limits shown in the following tables and figures are given in multiples of ~/-[n. The schemes studied in this work are shown on Table 1. The optimal control schemes found using this methodology significantly reduce the out-of-control ARL when compared to nonoptimal schemes. Our design procedure allows the design of control charts with run tests having greatly improved out-of-control ARL for small shifts, and yet does not degrade the in-control ARL as it would typically happen when run rules are applied to conventional control charts. For the practitioner, tables defining the position of the warning and control limits for several in control ARLs are provided. Table 1. List of schemes analyzed Name Control rules considered T(1,1 ,c,oo),T(2,3,w,c) Scheme A Scheme B T(1,1 ,C,~),T(3,4,w,c) T(1,1 ,c,oo),T(2,2,w,c) Scheme C T(1,1 ,C,oo),T(3,5,w,c) Scheme D Scheme E m(1,1 ,C,oo),T(5,5,w,c) Scheme F T(1,1 ,c,oo),T(8,8,w,c)

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Table 3. O ~timal desi~ln for B = {1"(1,1,c,=,), TI3,4,w,c)} control I In-control Out-ofW~rning Limit ARL Control ARL I' imk 10.307 100.000 3.107 13.969 200.001 3.492 1.288 16.754 300.000 3.709 1.363 19.106 400.000 3.869 1.416 21.188 500.000 3.975 1.456 23.081 600.000 4.078 1.489 24.832 700.000 4.147 1.516 26.470 600.000 4.216 1.539 28.018 71.560 900.000 4.274 29.489 999.999 4.327 1.578 01?-Ot-C0IflIOL AfIIAQt | l l L[10'fl Z6!

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~ble VV~rningiControll,n-ControllOut-ofimit Limit ARL ControlARL 1.642 3.232 100.000 11.436 1.803 3.560 200.000 16.496 1.893 3.737 300.000 20.576 1.956 3.858 400.000 24.144 2.003 3.949 500.000 27.381 2.042 4.022 600.000 30.3~2 2.073 4.083 700.000 33.201 2.101 4.135 800.000 35.975 2.125 4.179 899.999 38.430 2.146 4.222 1000.000 40.884 11.t

451

Table 4 . 0 :)timal design for C = {'1"(1,1 ,c,oo),T(2,2,w,c)} In-Control Out-o~ Warning ~ontrol ARL Control ARL Limit Limit 100.000 12.038 3.092 1.504 17.746 200.001 1.666 3.395 22.402 299.999 1.757 3.562 400.000 26.500 3.676 1.821 30.233 3.764 500.000 1.869 33.705 3.833 599.999 1.908 700.000 36.974 3.894 1.940 40.081 3.943 800.000 1.968 43.054 $.987 900.001 1.993 45.914 4.025 1000.000 2.014 Ol?-Or-col'rlo~ AVZlAO[n l L¢lO'rl 1:

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Figure 2. Graphical representatiOnooOf optimal design for scheme A = {-r(1,1,c, ), T(2,3,w,c)}

Figure 4. Graphical representat~n of optimal design for scheme C = {T(1,1,c, ), T(2,2,w,c)}

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