ECONOMIC ADJUSTMENT DESIGNS FOR X CONTROL CHARTS

ELART VON COLLAN! Institut fUr Angewandte Mathe11UltiJc und StatistiJc, University of Wurzburg, WUrzburg, Germany

ERWIN M. SANIGA Department of Business Administration, University of Deloware, Newarlc, Delaware 19716

CHRISTOPH WEIGAND Deutsche BankAG, Frankfurt 50, Ger11Ulny

We develop an economic model of a process whose quality can be affected by an assignable cause resulting in a shift of the mean of the distribution of output or by incorrect adjustment of the process when it is operating according to its capability. An i-con­trol chart is used to signal the assignable cause. The model is used to determine the parameters of the control chart that maximize the long term-profitability of the process .

• Deming [4 (p. 125)] explains that there are two kinds of mistakes the production worker can make on the job. They are to overadj~t a process or to underadjust a process. He goes on to explain that the control chart provides "a rational and economic guide to minimize loss from both mistakes."

Precise methods to design control charts that mini­mize the cost or maximize the profit of a process have been proposed by a number of authors. These methods yield control chart designs known as economic designs. A review of this literature is available in Montgomery [6] or Vance [9]. A survey on more recent results in this area is given in von Collani [3]. Economic design involves the optimization of econom­ic models of the production process that explicitly con­sider the costs of underadjustment along with a number of other costs. That is, these models assume that an assignable cause will be detected with a particular probability, or that an opportunity cost will accrue to the process otherwise.

In economic design, ·the search for an assignable cause is assumed to be perfect. That is, when the con­trol chart signals an assignable cause, a search is under­taken and the process is adju~ted or the cause is removed. Further, if the signal is false, the process is not adjusted but allowed to continue in its current state. Economic models assign a cost to searching for an assignable cause when the alarm is false but do not allow for the process to undergo a transformation in this state.

In practice, a common problem in statistical process control is process overadjustment. Processes may be overadjusted due to incorrect use of the control chart by the operator or because the only information about the process state available is due to sampling. In the lat­ter situation, a control chart signal outside the control limits is associated with process adjustment. If the sig­nal was a false alarm, the process will be adjusted incorrectly. As Woodall [10] notes, the effect of this overadjustment is an increase in the variability of the process over that obtainable under the true natural vari­ability of the process. This increase in variability and

corresponding loss of quality can be quite marked. Figure 1 depicts an actual example taken from a large U. S. food processing firm by one of the authors. It shows the distribution of the product characteristic before and after an overadjustment problem was elimi­nated.

Figure 1 a. Product output with overadjustment

Figure 1 b. Product output with no overadjustment

November 1994, lIE Transactions, Volume 26, Number 6 0740-817X/94/S3.00x.OO © 1994 "lIE" 37

In this paper, we consider this problem from an eco­nomic perspective. In particular, we develop an eco­nomic model that considers the effects of process overadjustment and underadjustment. The problem of overadjustment has not been addressed in the econom­ic design literature; moreover, as we wrote earlier, it is common in practice and therefore of importance. Our model will allow for the determination of the control chart design that will maximize the profitability of the process or, equivalently, as Deming [4 (p. 127)] writes, minimize "the economic loss from both mistakes" of overadjustment and underadjustment. We shall also derive a relatively simple solution procedure to the problem and present some numerical results.

In the next section the economic process model is developed. A method of finding the control chart design that maximizes the profitability of this process is also given. Some numerical results and an example are given in the third section. Finally, a brief summary is provided in the last section.

The Economic Process Model

Suppose we have a production process which can operate in one of two states. While in state I, the process is operating at the desired level of quality or is in control. The system is in state II when an assignable cause occurs, causing the process to be out-of-control. The process can also operate in state II if it is incor­rectly adjusted when it is in state I. We monitor the quality of the process with an x chart. Specifically, we take a sample of n units of output every h hours of pro­duction time and adjust the process if the sample mean falls outside the control limits of the x chart. The con­trol limits are a function of the control limit parameter k. Our objective is to find the set of parameters n, h, and Ie, or design of the control chart, such that the aver­age long-term profit of the process is maximized. Note that in this analysis we are using the control chart to signal the need for adjustment in the key dimension of the product. And we mention that, unlike previously developed economic models, our model allows for the possibility of incorrect adjustment.

Suppose the product's quality can be represented by one key dimension, say X. When the process is in state I, or in control, X - N (p., 02). A single assignable cause of poor quality can affect the process and when this assignable cause occurs, there is a shift in the dis­tribution of X to X - N (~ + 00, u2) with probability w and to X - N (~ - 8u, u2) with probability 1 - w. This is state II. Note that we assume no shift in the process variance. We assume the time of production at which the shift occurs has the exponential distribution with parameter A. This assumption implies that the shift is in the form of a shock to the system and not a shift due to gradual system deterioration. We note that

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this is the most commonly used distribution in eco­nomic control chart design (see, e.g., Montgomery [6]). We also assume that the process cannot correct itself; i.e., once the process has entered state II, a transition to state I can occur only if the process is adjusted.

An adjustment to the process is performed if the sample mean falls outside the control limits, respec­tively LCL and UCL, where

LCL = ~ - kurV~

UCL = ~ + kurV7. (1)

This adjustment can take one of two forms since state II consists of two substates. That is, when the shift is such that X - N (~ - 8u, UZ), the process is adjust­ed upward; when the shift is such that X - N (~ + 00, u l

), the process is adjusted downward. The adjustment policy can be summarized by the following rule:

Adjust up iff x < LCL

Adjust down iff x > UCL

Do not adjust iff LCL ~ x ~ UCL (2)

It can be seen that the use of this decision rule can lead to errors in adjustment following a false alarm, or a Type I error. That is, if the signal is a false alarm and the process is adjusted up, the process distribution shifts from X - N (~,a2) to X - N (~ + OO,UZ). If the alarm is false and the process is adjusted down, the process distribution shifts from X - N (~,UZ) to - N (~ - 8u, UZ). We assume that if the process is incor­rectly adjusted to either of the substates in state II, no further incorrect adjustment will occur. This is a sim­plifying assumption that is easily justified because it is easily shown that the probability of such an adjustment would be extremely small. We also assume that the time to sample and plot x is small and hence can be neglected in the model. Following the usual convention in economic modeling, we assume that a transition from state I to state II during sampling is not possible.

Let a be the probability that the process is incor­rectly adjusted when it is in state I, an error we call deadjustment. This type I error probability, or a, can be expressed as:

a = 1 - P(LCL ~ X ~ UCL I X - N [~, UZ])

= 2<1>( - k) (3)

where <I> denotes the standard normal distribution func­tion. Let ~ represent the probability of not adjusting the process when it is in state II. This type II error proba­bility can be written as

lIE Transactions, November 1994

13 = w [P(LCL :::::; i :::::; VCL Ix - N (IJ. + 00, tT)]

+ (1 - w) [P (LCL :::::; i :::::; VCL I X - N(IJ. - 00, 0-2)]

= w <l> (k - &Vn) + (1 - w) [1 - <l> (&Vn-k)]

= w <l> (k - &Vn) + (1 - w) <l> (k - &Vn)

= <l>(k - &Vn) (4)

Define gI and gn to be the average profit per unit accruing to the firm if the process is operating in state I and state II respectively. Suppose gI > 0 and gn < gl· Let a* be the variable sampling cost per unit, v be the production rate in units produced per hour and c* be the actual cost of adjusting the process.

We assume the sampling cost is proportional to the number of units sampled, or is equal to a*n. Note that we assume there is no fixed cost of sampling, an assumption not usually made in economic design of control charts. The justification for the latter assump­tion is that von Collani [2] has shown that the optimal design is more or less unrelated to the fixed cost of sampling provided these fixed costs are not too large.

If the process is in state II, there is a benefit to the firm of adjusting the process to state I. This adjustment may be regarded as a renewal of the process. The expected benefit, or b* , represents the expected mar­ginal profit resulting from the transition into state I reduced by the actual cost of an adjustment. In equation form,

b* = E(t) v(gl - gIl) - c*

= v(gI - gn)/'A - c* (5)

where t is the length of time the process is in state I. In our model we allow the process to be incorrectly

adjusted, or deadjusted, if it is operating in state I. This is coincident with a Type I error. In this case there is a reduction in benefit following a preceding adjustment of the process. If td is the actual length of time the process is in state I, then under the condition that the process is deadjusted, td can take on the values h, 2h, 3h, . .. only. Since the length of time in State I is dis­tributed exponentially, it follows that

(6)

The expectation of td is of interest as it characterizes the behavior of the adjustment policy: .. '"

E[td] = L ih P(td = ih) + L E[tdl ih < td < i =1 i =O

November 1994, lIE Transactions

(i + 1)h] P (ih < td < (i + 1) ih)

'" = L ih (1 - ay - 1 ae-LlJo + i=1 ..

L(ih+(1-e-"'(1 +M)Y(M..1-e-IA»X1-a'j e-M(1-e'-') i =O

<" - I =~ (7)

If e* is the expected cost of a deadjustment, then it follows that

= V(gI - gn)/'A + c *

= b* + 2c* (8)

Our objective is to find the values of h, n, and k, or the design of the control chart, such that the long-term profit accruing to the firm from the production process is maximized. Let G(m) be the profit obtained from the first m units produced. The average long-term profit per unit produced can then be expressed as

lim G(m)/m m-oo (9)

If we define a renewal cycle as the time between two successive adjustments with G being the profit obtained in one renewal cycle and N being the number of units produced during this renewal cycle, we can show with probability 1 that

lim G(m)/m = E[G]/E[N] (10) m-oo

Our objective is to maximize this ratio, which we express as 'TT'* , where

'TT'* (h, n, k) = E[G]/E [N] (11)

The design h*, n* , k* that maximizes 'TT'* is the eco­nomically optimal design.

If we define AI as the number of sampling actions during the in-control period of one renewal cycle, An as the number of sampling actions during the out-of­control period of one renewal cycle andAF as the num­ber of false alarms or deadjustments during one renewal cycle (whereAF = (0,1», then

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IX>

44il = LIIP(td = ih)+p(ih < td(i + 1Yz)] i=1

IX>

= Li(l - a) ;- 'ae-· ... +(l-aye-.... (l-e -.... ) i=1

IX>

= [a+(l-a) (l-e-M] e-M L i[(l-a)e-.... J - I

i=1

= 11(~ - 1 + a)

E/[AII] =11(1- Il)

and

Since

and

E[N] = E[Ar + All] hv,

we have, after some algebra,

1T'*(h,n,k)=gn+ 1Ihv[(b*(e""-1)-2c*aX1-IlY

(e""+a-Il)-a*n]

(12)

(13)

(14)

It can be shown that an optimal solution exists if 2a* < b*. If there is a positive benefit from adjusting a process that is in state II, then maximizing 1T'*(h , n, k) is equivalent to maximizing the linear transformation [1T'* (h,n,k) - gIl]v/Ab* where

[7t*(h,n,k)-gn]v/Ab* = 1IAh[«~-1)-2c*a/b*) (1-{3)/( ~ + a - {3)-a*n/b*] (15)

Since the objective function is very flat around the optimal solution, we can replace the integer variable n by the continuous variable y , where y = sVn. If we also make the substitutions x = )Jz and a = a*(b* 51) and set z = k, then from the right-hand side of (14) we obtain the standardized objective function

1T'(x,y,z)=(l/x)[«e'-l) -2c* a/b* (l-Il)l

(e'+a -Il)-ay] (16)

We call the positive real numbers (x* ,y* ,z*) maximiz­ing 1T'(x,y,z) an optimal standardized adjustment policy. Another simplification is obtained by setting the actual cost of adjustment, c*, to zero. In practice, the process

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of adjustment is often quite simple and therefore inex­pensive. Perhaps more importantly, c* is small relative to the benefit of adjustment to b*, and therefore the term 2c*a/b* may be neglected. Proceeding like this, we obtain the approximate standardized objective func­tion

7r(x,y,z)=(lIx)[e'-l) (l-Il)/(e'+a -Il)-ay] (17)

with

a = 2<1>( - z),

and

Il =<1> (z - y) (18)

Maximizing .rr(x,y,z) is equivalent to maximizing :!'"(x,,r,z)Awith c* = O. We call the positive real numbers x*, y*, z* an approximate optimal standardized adjust­ment policy. An approximation of the optimal adjust­ment policy is obtained analogously by setting:

A

h* = x*/A

A A

n* = nearest positive integer to (y*/5'j,

k* = z* (19)

Note that the approximate standardized objective function is written in terms of only one input parame­ter, a. Thus each solution for a particular "a" represents a number of standard cases making it possible to inves­tigate the properties of the optimal solution in general, and, most importantly, to develop a simple graphical solution algorithm.

To solve the economic adjustment problem, the quality engineer must specify the magnitude of the expected shift 5 expressed in multiples of the process standard deviation, the average length of time before a shift occurs 1IA, the sampling cost per unit a* , and the benefit per adjustment b* . The solution is found by the following algorithm.

Adjustment Algorithm:

Step 1: Calculate the standardized sampling cost a:

a = a*/(b* 52)

Step 2: Take from Figure 2 the approximate standardized

optimal design parameters A A A x* ,y* ,z*

liE Transactions, November 1994

I

Step 3: The approximate optimal design parameters are

given by:

sampling interval: h* = x* fA A

sample size: n* = nearest positive integer to (y*/lil,

control limit: k* = z*

Some numerical results

Tables 1 and 2 serve to illustrate the accuracy of the approximate optimal standardized adjustment policies (c* = 0) when compared with the exact ones. Although there are differences in the standardized design parameters, these differences in terms of the objective function 11" or 11"* are negligible.

Table 2 contains some numerical results obtained 'from a computer representing approximate optimal

Table 1. Exact Optimal Standardized Adiustment Policies

2C'" I:f'

a 0.01 0.001 0.001 0.0 10-7 )(' 0.00235 0.00214 0.00207 0.00206

V' 5.59 5.23 5.13 5.J~ Z" 4.27 3.96 3.88 3.87

10-' )(' 0.00664 0.00612 0 .00602 0.00600

Y" 5.05 4.80 4.76 4.75 Z" 3.80 3.60 3.56 3.56

10-' )(' 0.01840 0 .01740 0.01725 0.01723 V' 4.51 4.36 4.35 ~.34 Z" 3.34 3 .23 3.22 3.22

10-' )(' 0.05045 0.04876 0.04856 0.04854

Y" 3.94 3.87 3.86 3 .86 Z" 2.88 2.83 2.83 2.82

10-3 )(' 0.13649 0.13360 0.13328 0.13328 yo 3.27 3:23 3.23 3.~~ Z" 2.36 2.33 2.33 2.33

10-2 )(' 0.35533 0.34816 0.34724 0.34710

V' 2.24 2.22 2.21 2.21 Z" 1.59 1.57 1.57 1.57

November 1994, lIE Transactions

Standardized Sampling Interval (~

. i

, i .l t

c:: . -~. ~C· ~t· ...... ~ .. -~:::t;: . .".t_;t:; :f .:t:.:it= 1:ls;_i IT- ~'F f-,-- ' .- ' /.J.. ~L I ,- . -f-'-+-t-"-+--+-:-. -H"H' C I , . Lh; --r

. ,

E=btr·: :-=f::t:++=t:tR=t=+.:..-~ ~ -+ +-\T-~+-~r-r-~~hH-4~-+' - f . i ~~ ~·t -f· +--+--+-+-t+t--+-tl-++i .. : --:- t-

I . . _..\-:-+

H-,. ~. . I .;.._., . t~ ' . -,. .. t ~ - r--+---7-.. j • - : ' r

t-:-rt : .. i--. _ T-I-H.--t · . ! ' t 'T ' ,. - ' j; ': '..l .l J

J..zl I!WlI 1000UOO pue (".{) az!s aldweg peZ!p.lepulnS

v

0.- u; 0 0._ U 01 .s

0'- Ci E 01

(J) ..... '" CD --- N

--- 'E 01

'" .. ~ c: S (J)

.-

. .-.. -. .-. .-

. .-0._ N

~ ::J Cl u:

standardized economic adjustment designs for a = 10-8, 10-', . .. ,10-2

• A graphical solution to the prob­lem can also be obtained using Figure 2.

Note from Table 2 that the larger values of "a" may not be very representative of typical values of "a" in practice and are provided for illustration. For example, the implication of a = 10-2 is that the cost of sampling 100 units is equal to the benefit of adjusting a process that has been affected by an assignable cause resulting in a shift of 8. We would expect this benefit to be typi­cally much larger.

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The statistical properties of the designs with small­er values of "a" are quite good in that both the Type I and Type II error probabilities are small. This result is plausible in an adjustment model because there are explicit penalties invoked for underadjusting or over­adjusting a process. On the same note, it is quite inter­esting to compare the general nature of the statistical properties of an economic adjustment design to an eco­nomic desi~, which does not explicitly consider the possibility of process overadjustment. The Type II error probabilities for Duncan's [5] economic designs for x charts are generally small. Indeed, simplified eco­nomic designs for x charts proposed by Chiu and Wetherill [1] are based upon fixing the Type II error probability at a small level. With several charts used in combination such as the x and R chart, the Type II error probabilities are even smaller, as shown by Saniga [7]. The similarity of the result of a small Type II error probability between both types of models is not sur­prising because this probability relates to detecting an assignable cause of poor quality, a factor considered in both approaches.

On the other hand, the Type I error probabilities for the adjustment model are always small for typical con­figurations of input parameters. Equivalently, k is always large. Comparing this to Duncan's [5] 25 sam­ple problems shows that k for the adjustment model is substantially larger than that for an economic model. This result, too, is not surprising since economic models do not consider the possibility of process overadjustment.

Looking at the solution from another perspective, we see that the column in Table 2 representing E(AII) is relatively invariant with respect to "a." E(AII) repre­sents the average run length before an assignable cause is detected and the values near 1 reflect the fact that this assignable cause will be detected almost immediately. The rapid decrease in E(AV is plausible in that the more expensive sampling is, relative to the benefit of sam­pling, the less frequently sampling will be performed.

It is interesting to note the stability of E(AF) for a = 10-8 to 10-2

• While "a" increases in magnitude by the factor 1()6, the average number of incorrect adjustments increases only by the factor 6 over this range. This result occurs because the effect of an increase in sam­pling cost is a joint increase in sampling frequency and Type I error probability, which together stabilize the number of deadjustments per renewal cycle.

The last column of Table 2 contains a parameter T(h,1l,le) representing the ratio of the expected time operating out of control and the expected length of a renewal cycle. We refer to T(h,n,le) as the expected per­centage of time that the process is out of control, where

1i(h ,Ie) = £!A,.A.~-£(t.c! 100 ,n flA,.A.JIt

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Table 2 indicates that when "a" is large, say a ~ 10-3, the approximate optimal policy is such that the process may operate out of control an unacceptable percentage of time. For example, with a = 10-1, T(h,1l,k) = 30.46%. While our model balances the cost of adjusting the process with the benefits of this adjust­ment, one could be inclined to believe that such a pol­icy might not be in the best overall interest of the firm. For example, although operating in state II 305% of the time is the economically optimal policy, the firm's strategic goal might be one of producing a high-quali­ty product. In such a situation it would be better to require tighter control than is economically optimal.

One method of ensuring the control chart signals shifts correctly and quickly is for the sampling interval to be small and the power to be high. If the Type I error probability is also small, there is also a small chance for process deadjustment. Saniga [8] has proposed designs that meet the constraints of a small h and Type I and Type II error probabilities; these are called eco­nomical statistical designs or statistically constrained designs. In particular a design (h$7 ns, ks) is called a sta­tistically constrained design if

-rr*(hs,1l~s) = max -rr*(h,n,le)

Subject to, say:

a ~ 0.01,

~ ~ 0.10,

h ~ 0.05 (21)

The following example illustrates the use of a con­strained design in the case where a = 10-2

• Note that a = 10-2 is a rather extreme case, and therefore not completely covered by Figure 2. The reader could, however, obtain the approximate optimal design using Table 1 where the last row refers to the case a = 10-2

•

But in order to indicate once again the accuracy of the approximation (even in extreme cases), the exact eco­nomically adjustment design has been calculated by a numerical search program.

Let a* = 1.0, b* = 100, 8 = 1.0, A = 1.0 so that a = 10-2• The exact economically optimal adjustment design is h* = 0.3521, n* = 5, and k*= 1.57. That is, a sample of size 5 is taken every 0.3521 hours and the process is adjusted if the mean of the sample falls out­side the control limits. For this policy the profit is 7J(h*, n*, k*) = 55.47 and the error probabilities are a = 0.1164 and ~ = 0.2527. We find that the expected time in percent of operating out of control for this pol­icy is T(h*,n*,k*) = 30.34%.

lIE Transactions, November 1994

While this is the economically optimal adjustment policy, the example illnstrates the need to explore eco­nomic policies further before implementation. That is, it is doubtful if a production manager would implement a statistical process control policy that has such high error probabilities and allows the process to operate out of control 30.34% of the time.

Solving the same problem.with the constraints given in (21) above shows that h I = 0.5000, n; = 15 and 2~ = 2.58. The profit drops to 45.00 and the percent­age oftime out of control drops to 25%. While this pol­icy is more plausible because of the small error probabilities of a = 0.01 and Jj = 0.10, the percentage of time out of control may still be unacceptable. This number can be reduced by systematically reducing the upper bound on the constraint on h or by adding a con­straint to restrict T(h ,n ,k) to some acceptable value. We have solved the constrained problem using the latter method and we find that the optimal constrained policy is now hs = 0.162, ns = 15, and ks = 2.576. Here, we have set an upper bound on T (hs, n$) k s) of T( ) ~ 10%. Thus, at optimality we have T() = 10%, a = 0.01 and Jj = 0.10. The profit of this policy is 24.07, which is a substantial reduction when compared to the optimal profit of 55.47. Of course, it is difficult to determine which is the best policy to implement in general. On the one hand, the firm can choose the profit maximiz­ing policy at the expense of quality. On the other hand, the firm can choose to maximize profit while keeping some bound on quality, which is illustrated with the constrained problem. While these types of decisions are not addressed in this paper, we mention that the use of our methods will allow decision makers to choose the type of policy that is best suited to the policies of their particular firm.

Summary

We have developed a model of a process whose quality can be affected by the occurrence of an assign­able cause which causes a shift in the mean of a process or by an incorrect adjustment of the process when it is operating according to its capability. This is a common situation in practice and is discussed at length by Deming [4]; we provide actual data showing the effect on quality of incorrect adjustment.

While of substantial importance in practice, the problem has not been addressed explicitly in the 35-year history of economic models for control chart design dating back to Duncan [5], although Saniga [8] has allowed implicit control of incorrect operator adjustment by constraining the Type I error probability in his economic statistical model.

The model we have presented can be extended to all other situations where economic models are developed for the purpose of control chart design. It also has the advan­tage of requiring the estimation of only a small number of

November 1994, lIE Transactions

parameters when compared to other models and allows a design to be determined with only a calculator.

Acknowledgements

The authors wish to thank the referees for their sub­stantive comments, especially Saeed Maghsoodloo. Elart von Collani's research was supported by the DFG (German Research Foundation). Erwin M. Saniga's research was supported by the Center for Information Systems Management, Education and Research at the University of Delaware.

REFERENCES (I] Chiu, W. K., and Wetherill, G. B., "A Simplified Scheme

for the Economic Design of X Charts," Jou17lll1 of QUillily Technology , Vol. 6, 63-69 (1974).

[2] Collani, E. v., 'The Economic Design of Control C}ums, Teubner Verlag, Stuttgart (1989).

[3] Collani, E. v., "Economic Quality Control-A Survey on Some New Results." (In German), Operations Research­Spelarum, Vol. 12, 1-23 (1990).

[4] Deming, W. E., QUillily, Productivily and Competitive Position, M.I.T. Press (1982).

[5] Duncan, A. 1., "The Economic Design of X Charts Used to Maintain Current Control of a Process," Journal of the American SUltisticaI Association, Vol. 51, 228-242 (1956).

[6] Montgomery, D. c., "The Economic Design of Control Charts: A Review and Literature Survey," Journal of QUil/ily Technology, Vol. 12, 75-78 (1980).

[7] Saniga, E. M., "Joint Economically Optimal Design ofX/R Control Charts," MlVIQgemenr Science, Vol. 24, 420-431 (1971).

[8] Saniga, E . M , "Economic Statistical Control Chart Designs with an Application toX andR Charts" Tec:Juwmetric.s, Vol. 31, 313-320 (1989).

[9] Vance, L. C., "A Bibliography of Statistical Quality Control Chart Techniques, 1970-1980," JourfUll of QUillily Technology, Vol. 15, 59-62 (1983).

[10] Woodall, W . H., "Weaknesses of The Economic Design of Control Charts," TechnometrU:s, Vol. 28, 4()8.4()9 (1986).

Elart von Collani is a Professor of Stochastics at the University of Wiirzburg, Germany, from which he also received the degree of Dr. rer. nat. habil. His research interests are statistical quality control and maintenance theory. He has published two books and more than 80 articles and is managing editor of Economic QUillily Control.

Erwin M. Saniga is a Professor of Business Administration and head of the area of Operations Management at the University of Delaware. He received his Ph.D. from Pennsylvania State University. His research and application interests are in applied statistics and opera­tions management in general and statistical quality control in partic­ular. He bas published articles on these topics in a number of journals, including lIE Transactions.

Christoph Weigand received the degree of Dr. rer. naL from the University of Wiirzburg, Germany. Until 1991 he was a member of the Wiirzburg Research Group on Quality Control. Presently he is working for the Deutsche Bank AG.

Received July 1989; revised February and September 1992. Handled by the Department of Engineering Statistics and Applied Probability.

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