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ECONOMIC STATISTICAL DESIGN OF X̄ CONTROL CHARTS FOR SYSTEMS WITH WEIBULL IN-CONTROL TIMES
;nd. EngngVol. 32, No. 3, pp. 575-586,1997 (C 1997ElsevierScienceLtd. All riahts reserved Printed in &eat Britain PII: S03s50-8352(96)00314-2 0360-8352/97 $17.01)+ 0,00 C
Pergamon
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ECONOMIC STATISTICAL DESIGN OF ~ CONTROL CHARTS FOR SYSTEMS WITH WEIBULL IN-CONTROL TIMES GUOQIANG ZHANG* and VICTOR BERARDI Department of Administrative Sciences,Graduate School of Management, Kent State University,
Kent, OH 44242-0001,U.S.A. (Received25 November 1997) Abstract—Thispaper considers the economicstatistical design of ~ control charts with Weibull failure properties. A Weibull failure model is appropriate for many systems in electrical and mechanical industries. The economic statistical control chart design model is developed and illustrated by two examples.It is shownthat for somesituations,the increasein costs associatedwith the improvedstatistical performancemay not be significantcompared to the economicdesign.A sensitivityanalysisis performed to examine the cost effect of changingthe controlled properties: Type I error rate, power, average time to signal (ATS); the expectedprocess shift to be detected; and the Weibull scale and shape parameters. Results show that cost is relativelyinsensitiveto improvementin Type I and Type 11error rates, but is highlysensitiveto changesin smaller upper bounds on ATS. Mis-specificationof the underlyingWeibull failure distribution parameters also may have significant economic consequences. o 1997 Elsevier ScienceLtd
1. INTRODUCTION
Control charts are the primary quality improvementtools used to establishand maintain statistical control of manufacturing processes.The effectiveuse of control charts is largely dependent upon their design, that is, selectionof the decision variables such as sample size, sampling period, and control limit based on some subjectiveand/or objectivecriteria. The design can affect the cost, statisticalproperties, and ultimatelyuser confidence.Cost considerationsare important for obvious reasons. Statisticalcriteria such as the magnitude of the false alarm rate and power as well as the length of time needed to detect undesired shiftscan have significantpractical consequenceson the effectiveimplementation and continued use of control charts. A large number of designmethods have been proposed in the literature. Although various design approaches exist for different types of control charts, they can be classified into four general categories: heuristic; economic; statistical; and economic statistical. Control charts were first designedin 1924by Shewhart [1]as a means to differentiatebetween the normal, expectedrandom causes and the specialor assignablecauses of the processvariability. In a Shewhart type control chart, the control limit coefficientL is generally taken to be equal to 3 based on normal distribution theory (3-sigmarule) and the sample size n is usually 4 or 5. No generalguidelinesexistfor the selectionof samplingintervalh [2-4].However,h = 1 (hour between samples)is used quite commonly in a mass production process based on industrial experience[5]. These heuristically designed Shewhart charts are the most popular in practice simply because they are easy to understand and implementwith a minimal amount of operator training required. Although the costs associatedwith Shewhart charts are implicitlyconsideredby setting the sample size and sampling frequency, the resulting charts are not guaranteed to be economicallyoptimal. In addition, not all important statistical properties are considered in Shewhart charts. Poor performance may result with regard to management’sdesire to find process shifts promptly and correctly. The lack of formal systematiccriteria in heuristicdesign of control charts led many researchers and practitioners to search for more structured methods. Statisticallydesigned control charts [6] *Correspondingauthor. Tel.: ( + 1)(330)672-2772Ext. 332. Fax: CAIE 32/3—D
575
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(330)672-2443.E-mail:gzhang@,kentvm. kent.~u.
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Guoqiang Zhang and Victor Berardi
are one of them. In a statistical design, the Type I error probability and power are usually prespecifiedat the desired levels.Thus, the sample size and control limitscan be determined. The average run lengths (ARLs) or average time to signal (ATS) can be used to find the sampling frequency. Saniga [7] applied the method to a joint design of ~ and R charts. An alternativeway of designingcontrol charts systematicallyis the economicdesign.The concept of an economic design was first introduced by Girshick and Rubin [8]. Although the optimal control rules in their model are too complex to have practical value, their work provided the basis for most cost-based models in control chart designs. Duncan [9]developed a complete economic design model of the ~ control chart. The decision variables, n, h, L, are selected such that the expected net income per unit of time is maximizedor the expectedcost per unit time is minimized. A considerableamount of research has been done in the economicdesignof various control charts after Duncan’s paper. In 1987,Lorenzen and Vance [10]proposed a unifiedapproach to economic control chart design which had major influenceon the subsequent research. For a more detailed reviewof the literature in this area, seeVance [11],Montgomery [12–14],Svoboda [15],and Ho and Case [16]. Both statistical control chart designs and economic designs have strengths and weaknesses. Statisticaldesignsyield charts with high power and low Type I error rate but may cost more than economic designs.On the other hand, economicdesignsfocus only on costs and ignore statistical properties. The economicstatisticaldesignwas firstproposed by Saniga [17]to combinethe benefits of both pure statistical and economic designs while minimizingtheir weaknesses. Economic statistical designs are determined via a non-linear constrained optimization. The objective is to minimize the expected total cost per unit time, as in economic design, subject to constraints on the Type I error rate, power, and ATS. Alternative and additional constraints can be specifieddependingon designer’sneeds. Economicstatisticaldesignsare the constrained version of economic designs.If the constraints alone are used in determining design parameters, without considering the cost objective, they become statistical designs. Economic statistical designs are generallymore costlythan economicdesignsdue to the added constraints. However,the tight limits on control chart statistical properties can lead to low process variability which enhances output quality. The principle of economic statistical design is fully consistent with the objective of statisticalquality control of simultaneouslyreducingcosts and maintaininghigh quality. Therefore, whenever possible, the economic statistical model should be considered as a viable alternative in the design of control charts. Little research has been published in the area of economic statistical designs. Saniga [17] considered an application of economic statistical principle to the joint design of an ~ chart and an R chart. Montgomery et al. [18]proposed an economicstatisticaldesignapproach to an EWMA control chart. Economic and economicstatisticaldesignsof control charts require a distribution of the process failure mechanism to be specified.In order to attain simplificationin model development, most designs in the literature assume a Poisson process, that is, the process fails immediatelyafter the first outside shock. This assumption is not always appropriate in practice, especiallyfor processes in which machine wear occurs over time. Processeswith increasinghazard functions are common in mechanicaland electricalindustries.Baker [19]reported that the optimal economiccontrol chart is relativelysensitiveto the assumption of processfailuremechanism.Hu [20]presented an abstract in which he discussedan economic design of an ~ control chart based on a non-Poisson process shift. Banerjeeand Rahim [21]proposed an economicdesignof an Xcontrol chart under a Weibull failure mechanism. They employed a variable sampling interval approach rather than the traditional constant sampling interval method for systems with increasing hazard rate. In this paper we consider an economic-statisticaldesign model for ~ control chart under the Weibull failure mechanism. The motivation for this study is that the ~ chart is a widely used statistical process control tool and the Weibull failure mechanism can be generally used for any systemwith an increasing,constant, or even decreasingfailure rate. The economicstatisticaldesign is the most comprehensiveand versatile approach to control chart design in that it incorporates both cost and statisticalconsiderationsof using control charts. The results of economic statistical design are compared to those of a pure economicmodel. An extensivesensitivitystudy of model parameters on the optimal design is performed and discussion of the results is provided.
Economicstatistical control chart design
577
2. THE MODEL
A general formulation for the economic statistical design model of an ~ control chart is: Min s.t.
F(rz,h,L)
(1)
a < MU,p > p}, ATS < ATSU
(2)
where F(.) is the expected total cost per unit of time. It is a function of decisionvariables:sample size(n), samplinginterval (h), and control limitcoefficient(L). Alpha (a), p, and ATS are the Type I error probability, power at the expected shift level 6, and the average time to signal when the process is out of control respectively,while au,pi, and ATSUare the corresponding bounds. Under the assumption of a Weibullfailure mechanism,that is, the in-control time of the process follows a Weibull distribution, we have f(t) = 2vt(’-’) exp{– W},
t >0, 1>0,
v >1
(3)
where~(t) is the density function of Weibull distribution with 1 being the scale parameter and v, the shape parameter. Note that when v = 1,~(t) becomes an exponential distribution. Banerjeeand Rahim [21]derived a cost function in their pure economicdesignunder the Weibull shock model. They also assumed that: (1) production stops during searches and repairs; (2) the length of sampling interval hjvaries with time becauseof the increasinghazard rate of the Weibull model and is chosen so that, givenno shift until the start of the interval, the probability of a shift occurring in an interval is the same for all intervals. This leads to a definition of tlj as #zj= ~“–
(j – 1)’/’]/2,,j = 1, 2, 3
(4)
Note that when v = 1 (Poisson shock model), hj = h] for all j which corresponds to uniform sampling intervals. Based on Banerjee and Rahim [21],the expected cost can be derived as ‘(n’ ‘“ ‘)=
E(C) E(T)
(5)
where E(T) and E(C) are given by E(T) = Z, + aZO(l– r)/r + h,rA(l – r) + (1 –p)hlr[rzt(l – r) –p~(l –p)]/(P – r)
(6)
E(c) = (a + bn)[(l –p)/p + l/r] + MY(1– r)/r + IDO– D1](l/2)i/’~(l+ l/v) + IMilr(l – r)ft(l – r) + (1 – p)hlD1r[rA(l– r) – PA(1 – p)]/(p – r) + D1h1r2A(l– r) + W (7) where A(x) = f (1 + u)’/kxo, r = 1 – exp( – M!) ,1=o In the cost function (5), three sets of parameters need to be specified.The definition of these parameters is given below. Time parameters Z. = the expected search time associated with a false alarm. Z, = the expected search time and repair time if a failure is detected, Cost parameters DO= the expected cost per hour due to producing a nonconforming item when the process is in control. D1= the expected cost per hour due to producing a nonconforming item when the process is out of control. W = the expected cost of locating an assignable cause and repairing the process (which includes the cost of down time).
578
Guoqiang Zhang and Victor Berardi Table 1. Economiccontrol chart designresults output SampleSize(n) First samplinginterval (h,) Control limit (L) Alpha Power Expectedcost
Rahim program
GRG2 formulation
20 units 5.48 h 1.4539 0.1460 0.7831 .S324.76h - ‘
20 units 5.48 h 1,4449 0.1418 0.7867 $324.77h - ‘
Y = the expected cost of false alarms (includingthe cost of searching and the cost of down time if production ceases during the search). a = the fixed cost per sample. b = the cost per unit sampled. Weibulldistributionparameters 1 = the scale parameter. v = the shape parameter, For a pure economicdesign, the decisionvariablesn, hi, and L are chosen so that the expected cost per unit time in (5) is minimized.In our optimization, constraints are added to ensure that statistical properties such as the false alarm rate, the power to detect a shift, and the average time to signal a shift are maintained at appropriate levels.Control on power requires that the shift size to be detected, d, be specified. Since the formulation (1) and (2) is a constrained nonlinear optimization problem, we elect to use a general nonlinear optimization software, GRG2, which is widely available for use on mainframes, workstations, PCs, and even spreadsheets[22]. Rahim [23]developeda computer program to solvetheir economicdesignmodel. To verify our GRG2 optimization, we use the exampleprovided by Rahim [23].The parameters are specifiedas follows: 20= 0.25 h, Z, = 1.00h, DO= $50.00, D, = $950.00, W = $1100.00, Y = $500.00, a = $20.00,b = !$4.22,A = 0.002,and v = 3.0.The shift sizeto be detected is set at ~ = ().5standard deviations.The program is first run with no constraints, which is equivalentto an economicdesign. Results using Rahim’s[23]program and GRG2 are compared in Table 1. From Table 1 we can see that GRG2 provides a consistent solution. The results also show the major limitation of the economiccontrol chart design. The Type I error rate is between IAO/o and 150/.which may be too high for many situations and would cause large number of false alarms. Similarly,the power is lessthan 79% which, whilenot exceptionallylow, may not be high enough for some applications. It should be noted that these numbers are reflectiveof this specificexample and for other casesthe statisticalperformancecould be much worse.With economiccontrol charts, a designeris simplynot capable of determininga prioriwhat the control chart statisticalproperties will actually be. In addition, he cannot determine how sensitivethe cost is to the improvement of these properties. Adding constraints on a, power and/or ATS into the pure economic model (l), we have the economic statistical model. Suppose the upper bound of ~ is set to be 0.05 and the lower bound of p at the expectedshift levelis to be 0.90.The ATS is left unconstrainedjust for simplicity.Using Rahim’s[23]data again yields results of economic statistical design from the GRG2 subroutines presented in Table 2. Table 2 shows that to achievethe desired statisticalproperties we need to pay a price for it. The economic statistical design more than doubles the sample size to 42 units, increasing the variable samplingcost by $92.84.The increasein overallexpectedcost, however, is surprisinglysmall. The increase is only 3.70/&from $324.77h- 1to $336.75h- ‘—which,in many situations, may be a relativelysmall price to pay in order to achievethe improved statisticalperformance of the control Table 2. Comparisonof economicand economicstatisticalcontrol chart design Economicstatistical Economic output 42 units 20 units Samplesize (n) 5.48 h 5.92 h First samplinginterval(h,) 1.960 I,445 Control limit (L) 0.05 0.1418 Alpha 0.90 0.7867 Power $336.75h““‘ $324.77h-‘ Expectedcost
Economicstatistical control chart design
579
charts. In this example,falsealarm rate (a) isreduced by nearly two-thirdswhilepower is improved by almost 15V0.The improvement of the st~tisticalperformance leads to less frequent sampling (the sampling interval is increased by about 10% from 5.48 h to 5.92 h) and a wider control limit (from 1.445to 1.960processstandard deviations).Theseeffectscombineto reduce the costs of false alarms, search and elimination, and product nonconformance. If the cost parameters are changed as follows: DO= $50.00, D, = $400.00, W = $800.00, Y = $300.00, a = $2.00, b = $5.00 and the rest of parameters and the bounds on statistical properties used in the economicstatisticaldesignremain the same as in the last example,we obtain the following results. The overall cost increases from $210.68for the pure economic design to $230.82 for the economic statistical design. Without statistical constraints, the economic design yields a very high false alarm rate, 39.5%, a power of only 0.718, and an average time to signal of 9.14 h. These examplesshow that economicdesignsdo guarantee the lowestcost yet they typicallyhave poor statistical performance, The price to pay for the improvementsof statisticalproperties could be small for some situations and warrants a more detailed investigation. In the next section, an extensivesensitivityanalysisis performed. The major purpose is to find how constraints on statisticalperformance measures such as u, p, and ATS affect the cost. Results of sensitivityanalysis will be valuable to users of control charts, providing guidelinesfor making trade-off decisions between cost and statistical properties. 3. SENSITIVITYANALYSIS
In this section,we investigatethe effectof varyingthe bounds of alpha (a), power (p), the average time to signal (ATS), and expected shift size (6) on the minimum expected cost and the three decision variables. The consequences of mis-specificationof the Weibull parameters are also studied. The model parameters are from Rahim [23]. The vaIues and ranges for the sensitivityanalyses are chosen as follows. The upper bound on U, when fixed, equals 0.05 while the lower bound on p is fixed to 0.90. The ATS upper bound is set to 7.0 as this is the economicdesign’sactual value. The investigatedvalues for each sensitivity variable is chosen to range from being relativelycost insensitiveto highly sensitive.Upper bound of IXranges from 0.005 to 0.30, lower bound of p ranges from 0.50 to 0.975, and upper bound of ATS varies from 2.0 to 10.0. The shift sizes to be detected ranges from 0.05 to 2.5. Figs 1–3contain the effect on expected cost per hour due to changing the bounds of a, p, and ATS respectively.Figure 4 gives the ii sensitivityresults. The ct.andp, sensitivitydata in Figs 1 and 2 indicate similarcost sensitivityover the investigated ranges. As the bounds are tightened,each showsthe expectedcost increasesto $350-$370h- The sample size,samplinginterval, and control limit interactions account for these changes.The effects of the bounds on a and p on decision variables n, h,, and L are also shown in the figures. The patterns of the effectsin these figures are consistent with what we can expect. For example, the control limit, L, is determined by u but is not related to power. As the sample size increases,the sampling frequency will generally decrease. The sensitivitydata in Fig. 3 show increasingcost as the ATS upper bound decreases, but on a much greater scale than for a and p. When ATS. is below 5.0, the cost increases rapidly to a high of $2246h-‘ at upper bound ATS of 2.0.This suggeststhat appropriate selectionof the bound on ATS may be important in terms of the economic consequence.The effect of upper bound of ATS on samplingsizeis quite interesting.Above 5.0 of ATS bound, the samplesizeis not sensitive to the bounds of ATS. From 2.0 to 5.0, the sample size increasesfirst and then decreaseswith a maximum value of 45 at ATS. of 3.0. This phenomenon may result from the interaction effect of the bounds on u, p, and ATS as suggestedin the data table in Fig. 3. However, the effectof ATS. on sample size is not significantin this example. Note that the control limit L is not affected by ATS upper bounds. The d analysis in Fig. 4 indicates a relative cost insensitivityfor shift values of 0.4 and above with an extreme sensitivityfor smaller values. Sample size is quite sensitiveto the shift levelwith a highest value of 4203 at d = 0.05 and a lowest value of 3 at 6 = 2.50. Note that control limit is also affected by the shift level.
580
Guoqiang Zhang and Victor Berardi
U.R
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Figures 1–3also illustrate that in some instances the economic statisticaldesign corresponds to a statisticaldesign.A statisticaldesignresultswhen the Type I error rate and power are constrained at their bounds. This occurs in the a sensitivityanalysis for a upper bounds of 0.15 and below. Above this u bound, p remains at its bound while the actual u relaxes from the strict equality constraint. For lower bounds of p at 0.85 and above or for ATS upper bounds of 6.0 and above, the designs are also pure statistical designs. Next the effectsof changing the Weibulldistribution parameters 2 and v are investigated.Often it is difficult for a designer to obtain estimates for J and v as the data may not be available. Therefore, the impact of mis-specifyingthese parameters is of interest. Sensitivity studies by McWilliams[24]and Banerjeeand Rahim [21]showed that economiccontrol chart designsare not sensitive to mis-specificationof Weibull distribution parameters. McWilliams considered two examplesfor fivecases and concludedthat the economiceffectin mis-specifyingthe Weibullshape parameter is trivial. While the McWilliams’conclusionis valid for economiccontrol chart design,
Economicstatistical control chart design
581
we find that this is not true when constraints on statistical properties are added to the economic design model. Tables 3 and 4 contain the Banerjee and Rahim [21]economic design results for the effect of changingWeibullscaleand shape parameters, 2 and v, on the optimal design,augmented to include the actual u, p, and ATS values realized at a shift level of 0.50. Notice that the actual a and p levelsof the design remain relativelystable throughout the ranges whilethe ATS value variesfrom less than 3.0 h to nearly 15h. Therefore, while the statistical error rates have changed very little, the responsiveness of the charts is severely impacted. This raises the question of what effect changing 2 and v will have on expected cost if the upper bound on ATS is constrained and the statistical error rates improved. As with previousanalyses,the originaleconomicdesignATS value of 7.0 is used as a responsivenessbenchmark here. The values of u and p are constrained at the 0.05 and 0.90 levels respectivelyto ensure improved statistical properties. Tables 5 and 6 give the effectsof changing J and v on the optimal economic statistical design. Our results show that cost patterns with the economic statistical design do not match those
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582
Guoqiang Zhang and Victor Berardi
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obtained through the economic design. Without regard to statistical properties and responsiveness, the economic design analysis leads one to conclude that for a constant mean time to failure, the effect of changing 2 or V,on the optimal design is fairly small. The cost pattern in Table 3 shows a 13.9°/0 increase as ). is raised 500°/0.The economic statistical design results in Table 5 give a different picture, however. Table 5 shows that adding the constraints has a major impact 011costs. The costs have a decreasing pattern as ). increases. Furthermore, the cost change over the range of ;. decreases by LIZ.070, suggesting that the accurate specification of 1. is a more important consideration for the economic statistical design compared to the economic design. [n addition, the direction of the cost pattern is contrary to that of the economic design and it can not be correctly anticip:lted from the economic design analysis alone. Note also that designs for ;. = 0.00 I ~ind ;. = 0.05 correspond to statistical designs. Table 6 contains the results from changing the Weibull slmpe parameter ~’on the expected cost
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(c)
Fig. 4. Delta sensitivityanalysis results
of the economic statisticaldesign.The economicdesignresults in Table 4 showsthat as v increases from 1.0 to 4.0, the expected cost steadily decreases by about 16?4.,while actual ATS increases fivefold.Economic statistical design analysis for this example, however, shows that the expected cost decreases slightly as v moves from 1.0 to 2.0 and increases rapidly at larger values. The maximum expected cost represents nearly a 2000/.increase over the minimum.
i O.0001
0.0002 0.0003 0.001 0.05
u 3.248 3.000 2.855 2.427 1.086
Table 3. The effectof changing), on the optimal economicdesign L Actualalpha Actualpower Actual ATS h, Expectedcost ($) n 13.226 0.116 0.796 10.53 1.57 228.22 23 12.566 0.119 0.799 10.04 1.56 231.30 23 12.165 0.119 0.799 9.72 1.56 233.19 23 10,650 1.57 0.119 0.813 8.66 239.16 24 3.420 0.116 0.836 2.86 1,56 260,03 26
Guoqiang Zhang and Victor Berardi
584
Table4, Theeffectof changingu on the outimal u I.0 2.0 3.0 4.0
i 0.0001 0.0002 0.0003 0.001 0.05
i 0.06549 0.00337 0.0002 0.0000124
v 3.248 3,000 2.855 2.427 1.086
Expectedcost ($) 261,17 24S,66 231.30 219.90
n 26 24 23 22
h, 2.36 7.23 10,04 11.64
eeonomic
L 1.56 1.57 1.56 I.55
design Actual alpha Actual power Actual ATS 0.839 0.119 2.813 0.810 0.116 8,921 0.799 12,566 0.119 0.787 14,794 0.121
Table 5. The effectof changing1 on the optimal economicstatisticaldesign Expectedcost ($) n h, L Actual alpha Actual power Actual ATS 463.94 47 6.50 1.96 0.05 0.925 7.000 399.73 45 6.43 1.96 0.916 7.000 0.05 44 368.79 6.39 1.96 0.05 0.908 7.000 301,27 42 6.30 1.96 0.05 0.900 7.000 267.1I 42 3.31 1.96 0.05 0.900 3.679
Table6. The effectof changingu on the optimal economicstatisticaldesign u 1.0 2.0 3.0 4.0
i 0.06549 0.00337 0.0002 0.0000124
Expectedcost ($) 268,37 265.80 399.73 768.22
n 42 42 45 50
h, 2.77 6.30 6.43 6.60
L 1,96 1.96 1.96 1.96
Actual alpha Actual power ActualATS 0.05 0.900 3.072 0.05 0.900 7.000 0.05 0.916 7.000 0.05 0.942 7.000
Sinceit is v that determineswhether the in-control time distribution is Weibull or exponential, the penalty cost for mis-specificationof the underlying failure mechanism can be quite high. From Tables 3–6, we also find that when ~ is high and v is low, the increase in the overall cost associated with the improved statistical performance is not significantcompared to the economic design. For example, when 2 = 0.05 and v = 1.086,the expected cost is $260.03(Table 3) for the economicdesignand $267.11(Table 5)for the economicstatisticaldesign.However,when 1 is quite low and v is quite high (see the situations for v = 4 and A= 0.0000124in Tables 4 and 6 and for 2 = 0.0001 and v = 3.248 in Tables 3 and 5), the economic statistical design can be much more costly than the economic design. 4. DISCUSSION
Economic statistical control chart design under a Weibull failure mechanism is a general and flexiblemodel that can be utilized for many industrial production processes.It givesdesignersthe capability to assess the impact of their decisions on several critical aspects that are either not available or not apparent via other design methods. Using economic statistical design with an appropriate sensitivity analysis, designers can readily observe the impact on cost, sample size, samplinginterval, and control limitsdue to the constraints on the statisticalerror rates and control chart responsiveness,and the specificationof the failure distribution parameters. Considering statistical properties in the quality control chart design has significantpractical implications.The Type I error rate or false alarm rate is the probability of concludingthe process mean has shifted due to assignablecause when in fact it has not. A high number of false alarms will quickly undermine operator confidencein the use of the charts and incur unnecessary search costs. In control chart terms, power is the probability of correctly identifyinga shift in the process when one exists. Power provides a performance measure of a control chart’s capability to detect undesirableshiftsthat may occur in the production process.A low power leadsnot only to excessive cost due to the inability of the chart to identify when a significantprocess shift has occurred, increasing quality losses,but to product variability and ultimately customer dissatisfaction.ATS refers to the promptness in which a significantprocess shift is in fact identified. It is a measure of the control chart design’sresponsivenessin detecting process shifts and is especiallyimportant when producing defectiveproducts has high penalty costs. There is no general rule governing the selection of bounds for the economic statistical design constraints. They should be chosen based on the specificproblem situation, the relevant cost information, as wellas the economicand statisticalconsequences.The sensitivityanalysisis useful to the designer in making these decisions.
Economic statistical control chart design
585
Our study has shown that in some situations, the statistical performance of control charts can be improved significantlywith only a slight increasein cost by using an economicstatisticaldesign instead of an economic design. The cost increase is further shown to be relatively insensitiveto the improvementin the Type I error rate and power throughout the investigatedrange. This implies that it maybe quite inexpensivefor practitionersto implementcontrol charts with lower falsealarm rates and with a higherprobability for detectinga processshiftwhen one actually exists.The bound on ATS should not be set too low sincethe expectedcost is highly sensitiveto small ATS bounds. A reasonable bound on ATS may be found from the actual ATS in the economicdesign.For larger shift levels in this example, it is inexpensiveto achieve significant improvements in statistical properties. Being capable of detecting extremely small process shifts in a responsive fashion, however, is an expensiveproposition to achieve with X charts. Additionally, because the specification of the Weibull shape and scale parameters has a significantimpact on the cost and responsivenessin the economicstatisticaldesign,the distribution parameters need to be estimated adequately to reflect the process characteristics. The economic design alone obscures this important result and actually may be misleadingin the direction of its cost change. Furthermore, our results illustrate that in many cases, the economicstatistical design corresponds to a statisticalcontrol chart design.The implication is that control chart designsthat are cost effectiveare often possiblewithout degradingthe statisticalproperties of the control charts 5. SUMMARY AND CONCLUSION
In this paper, the four control chart design methods, their advantages, and disadvantageswere discussed.It was shown that economic statistical control chart designs are the best for achieving desiredcontrol chart statisticalproperties whilesimultaneouslyminimizingthe associatedcost. The failure model considered was Weibull instead of the widelyused exponential because the Weibull model is more appropriate and general in many practical situations. The developmentof the Weibulleconomiccontrol chart designmodelwas reviewedand extended to an economic statistical design model. We showed that the power and Type I error rates could be improved significantlyover the economic design with a small increase in cost for certain situations. An extensivesensitivityanalysis was performed on the economic statistical model to investigatethe relativeeffectsof the bounds for statisticaland performance measures, such as Type I error rate, power, ATS, shift sizeto be detected, and Weibullshape and scale parameters on the minimum expected cost. The cost is relatively insensitiveto improvement in Type I and Type II error rates and to higher ATS bounds, but is highly sensitiveto the lower bounds on average time to signal (ATS). Mis-specificationof the underlying Weibull failure distribution parameters may have significanteconomicconsequences.The information from the sensitivityanalysis is valuable to control chart designers in making the trade-off decisions between the expected cost and the desired statistical properties. Acknowledgements—The authors wish to thank Dr O. F. Offodileand Dr M. Y. Hu for their valuable suggestionsand encouragement.
REFERENCES 1. Shewhart, WA., EconomicConlrolof Quality of ManufacturedProduct. Van Nostrand, New York, 1931. 2. Juran, J.M., Gryna, Jr. F.M. and Bingham, Jr. R.S., Quality Control Handbook.McGraw-Hill, New York, 1974. 3. Cowden, D.J., Statistical Methods in Quality Control.Prentice-Hall, EnglewoodCliffs, NJ, 1957. 4. Burr, I.W., EngineeringStatistics and Quality Control. McGraw-Hill, New York, 1953. 5. Feigenbaum, A.V., Total Quality Control.McGraw-Hill, New York, 1961. 6. Woodall, W. H., The statistical design of quality control charts. The Statistician, 1985,34, 155-160. 7. Saniga, EM., Isodynes for X and R control charts. Frontiersin Statistical Quality Control2, cd., H. J. Lertz, G. B. Wetherill and P.-T. Wilrich. Physica, Vienna, 1984,pp. 268-273. 8. Girshick, M. A. and Rubin, H., A Bayes’approach to a quality control model.Annalsof A4athematicalStarisiics, 1952, 23, 11+125. 9. Duncan, A. J., The economic design of X-bar-charts used to maintain current control of a process. Journalof the American Statistical Association, 1956,51, 228-242. 10. Lorenzen, T. J. and Vance, L. C., The economicdesign of control charts: a unifiedapproach. Technometrics,1987, 28, 3-lo.
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11. Vance,L. C., A bibliographyof statistical quality control chart techniques, 1970-1980.Journalof Qua/i?yTechnology, 1983,15, 59+2. 12. Montgomery, D. C., The economic design of control charts: a review and literature survey. Journal oj’ Quali/j Technology,1980,12, 75-87. 13. Montgomery, D. C., Introductionto Statistical Quality Control,2nd edn. Wiley, New York, 1991. 14. Montgomery, D. C., The use of statistical process control and design of experiments in product and process improvement.HE Transactions,1992,24, +17. 15. Svoboda,L., Economicdesignof control charts: a reviewand literature survey(1979–1989).StatisticalProcessCon(rol in Manufacturing,ed. J. B. Keats and D. C. Montgomery. Marcel Dekker, New York, 1991. 16. Ho, C. and Case, K. E., Economic design of control charts: a literature review for 1981–1991.Journal oj”Quality Technology,1994,26, 39–53. 17. Saniga, E. M., Economicstatistical control-chart designswith an application to ~ and R charts. Technornerrics,1989, 31, 313-320. 18. Montgomery,D. C., Torng, J. C.-C., Cochran, J. K. and Lawence,F. P., Statistical constrained economicdesign of the EWMA control chart. Journalof Quality Technology,1995,2?, 25&256. 19. Baker, K. R., Two process models iq the economicdesign of an X chart. AHE Transactions,1971,3, 257-263. 20. Hu, P. W., Economic design of an X control chart under non-Poisson process shift. Abs&acl, TIMS/ORSA Joint National Meeting, San Francisco,87, 1984. 21. Banerjee,P. K. and Rahim, M. A., Economicdesignof ~ control charts under Weibullshock models. Technometrics, 1988,30, 407-$14. 22. Lasdon, L. S. and Waren, A. D., G-RG2Users Guide.January 1986. 23. Rahim, M. A., Economicdesignon Xcontrol charts assumingWeibuliin-controltimes.Journalof Quality Technology, 1993,25, 296-305. 24. McWilIiams,T. P., Economiccontrol chart designs and the in-control time distribution: a sensitivitystudy. Journal of QUalWTechnology, 1989,21, 103–110.
Pergamon
@
ECONOMIC STATISTICAL DESIGN OF ~ CONTROL CHARTS FOR SYSTEMS WITH WEIBULL IN-CONTROL TIMES GUOQIANG ZHANG* and VICTOR BERARDI Department of Administrative Sciences,Graduate School of Management, Kent State University,
Kent, OH 44242-0001,U.S.A. (Received25 November 1997) Abstract—Thispaper considers the economicstatistical design of ~ control charts with Weibull failure properties. A Weibull failure model is appropriate for many systems in electrical and mechanical industries. The economic statistical control chart design model is developed and illustrated by two examples.It is shownthat for somesituations,the increasein costs associatedwith the improvedstatistical performancemay not be significantcompared to the economicdesign.A sensitivityanalysisis performed to examine the cost effect of changingthe controlled properties: Type I error rate, power, average time to signal (ATS); the expectedprocess shift to be detected; and the Weibull scale and shape parameters. Results show that cost is relativelyinsensitiveto improvementin Type I and Type 11error rates, but is highlysensitiveto changesin smaller upper bounds on ATS. Mis-specificationof the underlyingWeibull failure distribution parameters also may have significant economic consequences. o 1997 Elsevier ScienceLtd
1. INTRODUCTION
Control charts are the primary quality improvementtools used to establishand maintain statistical control of manufacturing processes.The effectiveuse of control charts is largely dependent upon their design, that is, selectionof the decision variables such as sample size, sampling period, and control limit based on some subjectiveand/or objectivecriteria. The design can affect the cost, statisticalproperties, and ultimatelyuser confidence.Cost considerationsare important for obvious reasons. Statisticalcriteria such as the magnitude of the false alarm rate and power as well as the length of time needed to detect undesired shiftscan have significantpractical consequenceson the effectiveimplementation and continued use of control charts. A large number of designmethods have been proposed in the literature. Although various design approaches exist for different types of control charts, they can be classified into four general categories: heuristic; economic; statistical; and economic statistical. Control charts were first designedin 1924by Shewhart [1]as a means to differentiatebetween the normal, expectedrandom causes and the specialor assignablecauses of the processvariability. In a Shewhart type control chart, the control limit coefficientL is generally taken to be equal to 3 based on normal distribution theory (3-sigmarule) and the sample size n is usually 4 or 5. No generalguidelinesexistfor the selectionof samplingintervalh [2-4].However,h = 1 (hour between samples)is used quite commonly in a mass production process based on industrial experience[5]. These heuristically designed Shewhart charts are the most popular in practice simply because they are easy to understand and implementwith a minimal amount of operator training required. Although the costs associatedwith Shewhart charts are implicitlyconsideredby setting the sample size and sampling frequency, the resulting charts are not guaranteed to be economicallyoptimal. In addition, not all important statistical properties are considered in Shewhart charts. Poor performance may result with regard to management’sdesire to find process shifts promptly and correctly. The lack of formal systematiccriteria in heuristicdesign of control charts led many researchers and practitioners to search for more structured methods. Statisticallydesigned control charts [6] *Correspondingauthor. Tel.: ( + 1)(330)672-2772Ext. 332. Fax: CAIE 32/3—D
575
( + 1)
(330)672-2443.E-mail:gzhang@,kentvm. kent.~u.
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Guoqiang Zhang and Victor Berardi
are one of them. In a statistical design, the Type I error probability and power are usually prespecifiedat the desired levels.Thus, the sample size and control limitscan be determined. The average run lengths (ARLs) or average time to signal (ATS) can be used to find the sampling frequency. Saniga [7] applied the method to a joint design of ~ and R charts. An alternativeway of designingcontrol charts systematicallyis the economicdesign.The concept of an economic design was first introduced by Girshick and Rubin [8]. Although the optimal control rules in their model are too complex to have practical value, their work provided the basis for most cost-based models in control chart designs. Duncan [9]developed a complete economic design model of the ~ control chart. The decision variables, n, h, L, are selected such that the expected net income per unit of time is maximizedor the expectedcost per unit time is minimized. A considerableamount of research has been done in the economicdesignof various control charts after Duncan’s paper. In 1987,Lorenzen and Vance [10]proposed a unifiedapproach to economic control chart design which had major influenceon the subsequent research. For a more detailed reviewof the literature in this area, seeVance [11],Montgomery [12–14],Svoboda [15],and Ho and Case [16]. Both statistical control chart designs and economic designs have strengths and weaknesses. Statisticaldesignsyield charts with high power and low Type I error rate but may cost more than economic designs.On the other hand, economicdesignsfocus only on costs and ignore statistical properties. The economicstatisticaldesignwas firstproposed by Saniga [17]to combinethe benefits of both pure statistical and economic designs while minimizingtheir weaknesses. Economic statistical designs are determined via a non-linear constrained optimization. The objective is to minimize the expected total cost per unit time, as in economic design, subject to constraints on the Type I error rate, power, and ATS. Alternative and additional constraints can be specifieddependingon designer’sneeds. Economicstatisticaldesignsare the constrained version of economic designs.If the constraints alone are used in determining design parameters, without considering the cost objective, they become statistical designs. Economic statistical designs are generallymore costlythan economicdesignsdue to the added constraints. However,the tight limits on control chart statistical properties can lead to low process variability which enhances output quality. The principle of economic statistical design is fully consistent with the objective of statisticalquality control of simultaneouslyreducingcosts and maintaininghigh quality. Therefore, whenever possible, the economic statistical model should be considered as a viable alternative in the design of control charts. Little research has been published in the area of economic statistical designs. Saniga [17] considered an application of economic statistical principle to the joint design of an ~ chart and an R chart. Montgomery et al. [18]proposed an economicstatisticaldesignapproach to an EWMA control chart. Economic and economicstatisticaldesignsof control charts require a distribution of the process failure mechanism to be specified.In order to attain simplificationin model development, most designs in the literature assume a Poisson process, that is, the process fails immediatelyafter the first outside shock. This assumption is not always appropriate in practice, especiallyfor processes in which machine wear occurs over time. Processeswith increasinghazard functions are common in mechanicaland electricalindustries.Baker [19]reported that the optimal economiccontrol chart is relativelysensitiveto the assumption of processfailuremechanism.Hu [20]presented an abstract in which he discussedan economic design of an ~ control chart based on a non-Poisson process shift. Banerjeeand Rahim [21]proposed an economicdesignof an Xcontrol chart under a Weibull failure mechanism. They employed a variable sampling interval approach rather than the traditional constant sampling interval method for systems with increasing hazard rate. In this paper we consider an economic-statisticaldesign model for ~ control chart under the Weibull failure mechanism. The motivation for this study is that the ~ chart is a widely used statistical process control tool and the Weibull failure mechanism can be generally used for any systemwith an increasing,constant, or even decreasingfailure rate. The economicstatisticaldesign is the most comprehensiveand versatile approach to control chart design in that it incorporates both cost and statisticalconsiderationsof using control charts. The results of economic statistical design are compared to those of a pure economicmodel. An extensivesensitivitystudy of model parameters on the optimal design is performed and discussion of the results is provided.
Economicstatistical control chart design
577
2. THE MODEL
A general formulation for the economic statistical design model of an ~ control chart is: Min s.t.
F(rz,h,L)
(1)
a < MU,p > p}, ATS < ATSU
(2)
where F(.) is the expected total cost per unit of time. It is a function of decisionvariables:sample size(n), samplinginterval (h), and control limitcoefficient(L). Alpha (a), p, and ATS are the Type I error probability, power at the expected shift level 6, and the average time to signal when the process is out of control respectively,while au,pi, and ATSUare the corresponding bounds. Under the assumption of a Weibullfailure mechanism,that is, the in-control time of the process follows a Weibull distribution, we have f(t) = 2vt(’-’) exp{– W},
t >0, 1>0,
v >1
(3)
where~(t) is the density function of Weibull distribution with 1 being the scale parameter and v, the shape parameter. Note that when v = 1,~(t) becomes an exponential distribution. Banerjeeand Rahim [21]derived a cost function in their pure economicdesignunder the Weibull shock model. They also assumed that: (1) production stops during searches and repairs; (2) the length of sampling interval hjvaries with time becauseof the increasinghazard rate of the Weibull model and is chosen so that, givenno shift until the start of the interval, the probability of a shift occurring in an interval is the same for all intervals. This leads to a definition of tlj as #zj= ~“–
(j – 1)’/’]/2,,j = 1, 2, 3
(4)
Note that when v = 1 (Poisson shock model), hj = h] for all j which corresponds to uniform sampling intervals. Based on Banerjee and Rahim [21],the expected cost can be derived as ‘(n’ ‘“ ‘)=
E(C) E(T)
(5)
where E(T) and E(C) are given by E(T) = Z, + aZO(l– r)/r + h,rA(l – r) + (1 –p)hlr[rzt(l – r) –p~(l –p)]/(P – r)
(6)
E(c) = (a + bn)[(l –p)/p + l/r] + MY(1– r)/r + IDO– D1](l/2)i/’~(l+ l/v) + IMilr(l – r)ft(l – r) + (1 – p)hlD1r[rA(l– r) – PA(1 – p)]/(p – r) + D1h1r2A(l– r) + W (7) where A(x) = f (1 + u)’/kxo, r = 1 – exp( – M!) ,1=o In the cost function (5), three sets of parameters need to be specified.The definition of these parameters is given below. Time parameters Z. = the expected search time associated with a false alarm. Z, = the expected search time and repair time if a failure is detected, Cost parameters DO= the expected cost per hour due to producing a nonconforming item when the process is in control. D1= the expected cost per hour due to producing a nonconforming item when the process is out of control. W = the expected cost of locating an assignable cause and repairing the process (which includes the cost of down time).
578
Guoqiang Zhang and Victor Berardi Table 1. Economiccontrol chart designresults output SampleSize(n) First samplinginterval (h,) Control limit (L) Alpha Power Expectedcost
Rahim program
GRG2 formulation
20 units 5.48 h 1.4539 0.1460 0.7831 .S324.76h - ‘
20 units 5.48 h 1,4449 0.1418 0.7867 $324.77h - ‘
Y = the expected cost of false alarms (includingthe cost of searching and the cost of down time if production ceases during the search). a = the fixed cost per sample. b = the cost per unit sampled. Weibulldistributionparameters 1 = the scale parameter. v = the shape parameter, For a pure economicdesign, the decisionvariablesn, hi, and L are chosen so that the expected cost per unit time in (5) is minimized.In our optimization, constraints are added to ensure that statistical properties such as the false alarm rate, the power to detect a shift, and the average time to signal a shift are maintained at appropriate levels.Control on power requires that the shift size to be detected, d, be specified. Since the formulation (1) and (2) is a constrained nonlinear optimization problem, we elect to use a general nonlinear optimization software, GRG2, which is widely available for use on mainframes, workstations, PCs, and even spreadsheets[22]. Rahim [23]developeda computer program to solvetheir economicdesignmodel. To verify our GRG2 optimization, we use the exampleprovided by Rahim [23].The parameters are specifiedas follows: 20= 0.25 h, Z, = 1.00h, DO= $50.00, D, = $950.00, W = $1100.00, Y = $500.00, a = $20.00,b = !$4.22,A = 0.002,and v = 3.0.The shift sizeto be detected is set at ~ = ().5standard deviations.The program is first run with no constraints, which is equivalentto an economicdesign. Results using Rahim’s[23]program and GRG2 are compared in Table 1. From Table 1 we can see that GRG2 provides a consistent solution. The results also show the major limitation of the economiccontrol chart design. The Type I error rate is between IAO/o and 150/.which may be too high for many situations and would cause large number of false alarms. Similarly,the power is lessthan 79% which, whilenot exceptionallylow, may not be high enough for some applications. It should be noted that these numbers are reflectiveof this specificexample and for other casesthe statisticalperformancecould be much worse.With economiccontrol charts, a designeris simplynot capable of determininga prioriwhat the control chart statisticalproperties will actually be. In addition, he cannot determine how sensitivethe cost is to the improvement of these properties. Adding constraints on a, power and/or ATS into the pure economic model (l), we have the economic statistical model. Suppose the upper bound of ~ is set to be 0.05 and the lower bound of p at the expectedshift levelis to be 0.90.The ATS is left unconstrainedjust for simplicity.Using Rahim’s[23]data again yields results of economic statistical design from the GRG2 subroutines presented in Table 2. Table 2 shows that to achievethe desired statisticalproperties we need to pay a price for it. The economic statistical design more than doubles the sample size to 42 units, increasing the variable samplingcost by $92.84.The increasein overallexpectedcost, however, is surprisinglysmall. The increase is only 3.70/&from $324.77h- 1to $336.75h- ‘—which,in many situations, may be a relativelysmall price to pay in order to achievethe improved statisticalperformance of the control Table 2. Comparisonof economicand economicstatisticalcontrol chart design Economicstatistical Economic output 42 units 20 units Samplesize (n) 5.48 h 5.92 h First samplinginterval(h,) 1.960 I,445 Control limit (L) 0.05 0.1418 Alpha 0.90 0.7867 Power $336.75h““‘ $324.77h-‘ Expectedcost
Economicstatistical control chart design
579
charts. In this example,falsealarm rate (a) isreduced by nearly two-thirdswhilepower is improved by almost 15V0.The improvement of the st~tisticalperformance leads to less frequent sampling (the sampling interval is increased by about 10% from 5.48 h to 5.92 h) and a wider control limit (from 1.445to 1.960processstandard deviations).Theseeffectscombineto reduce the costs of false alarms, search and elimination, and product nonconformance. If the cost parameters are changed as follows: DO= $50.00, D, = $400.00, W = $800.00, Y = $300.00, a = $2.00, b = $5.00 and the rest of parameters and the bounds on statistical properties used in the economicstatisticaldesignremain the same as in the last example,we obtain the following results. The overall cost increases from $210.68for the pure economic design to $230.82 for the economic statistical design. Without statistical constraints, the economic design yields a very high false alarm rate, 39.5%, a power of only 0.718, and an average time to signal of 9.14 h. These examplesshow that economicdesignsdo guarantee the lowestcost yet they typicallyhave poor statistical performance, The price to pay for the improvementsof statisticalproperties could be small for some situations and warrants a more detailed investigation. In the next section, an extensivesensitivityanalysisis performed. The major purpose is to find how constraints on statisticalperformance measures such as u, p, and ATS affect the cost. Results of sensitivityanalysis will be valuable to users of control charts, providing guidelinesfor making trade-off decisions between cost and statistical properties. 3. SENSITIVITYANALYSIS
In this section,we investigatethe effectof varyingthe bounds of alpha (a), power (p), the average time to signal (ATS), and expected shift size (6) on the minimum expected cost and the three decision variables. The consequences of mis-specificationof the Weibull parameters are also studied. The model parameters are from Rahim [23]. The vaIues and ranges for the sensitivityanalyses are chosen as follows. The upper bound on U, when fixed, equals 0.05 while the lower bound on p is fixed to 0.90. The ATS upper bound is set to 7.0 as this is the economicdesign’sactual value. The investigatedvalues for each sensitivity variable is chosen to range from being relativelycost insensitiveto highly sensitive.Upper bound of IXranges from 0.005 to 0.30, lower bound of p ranges from 0.50 to 0.975, and upper bound of ATS varies from 2.0 to 10.0. The shift sizes to be detected ranges from 0.05 to 2.5. Figs 1–3contain the effect on expected cost per hour due to changing the bounds of a, p, and ATS respectively.Figure 4 gives the ii sensitivityresults. The ct.andp, sensitivitydata in Figs 1 and 2 indicate similarcost sensitivityover the investigated ranges. As the bounds are tightened,each showsthe expectedcost increasesto $350-$370h- The sample size,samplinginterval, and control limit interactions account for these changes.The effects of the bounds on a and p on decision variables n, h,, and L are also shown in the figures. The patterns of the effectsin these figures are consistent with what we can expect. For example, the control limit, L, is determined by u but is not related to power. As the sample size increases,the sampling frequency will generally decrease. The sensitivitydata in Fig. 3 show increasingcost as the ATS upper bound decreases, but on a much greater scale than for a and p. When ATS. is below 5.0, the cost increases rapidly to a high of $2246h-‘ at upper bound ATS of 2.0.This suggeststhat appropriate selectionof the bound on ATS may be important in terms of the economic consequence.The effect of upper bound of ATS on samplingsizeis quite interesting.Above 5.0 of ATS bound, the samplesizeis not sensitive to the bounds of ATS. From 2.0 to 5.0, the sample size increasesfirst and then decreaseswith a maximum value of 45 at ATS. of 3.0. This phenomenon may result from the interaction effect of the bounds on u, p, and ATS as suggestedin the data table in Fig. 3. However, the effectof ATS. on sample size is not significantin this example. Note that the control limit L is not affected by ATS upper bounds. The d analysis in Fig. 4 indicates a relative cost insensitivityfor shift values of 0.4 and above with an extreme sensitivityfor smaller values. Sample size is quite sensitiveto the shift levelwith a highest value of 4203 at d = 0.05 and a lowest value of 3 at 6 = 2.50. Note that control limit is also affected by the shift level.
580
Guoqiang Zhang and Victor Berardi
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(c)
Fig. 1. Alpha sensitivityanalysis results
Figures 1–3also illustrate that in some instances the economic statisticaldesign corresponds to a statisticaldesign.A statisticaldesignresultswhen the Type I error rate and power are constrained at their bounds. This occurs in the a sensitivityanalysis for a upper bounds of 0.15 and below. Above this u bound, p remains at its bound while the actual u relaxes from the strict equality constraint. For lower bounds of p at 0.85 and above or for ATS upper bounds of 6.0 and above, the designs are also pure statistical designs. Next the effectsof changing the Weibulldistribution parameters 2 and v are investigated.Often it is difficult for a designer to obtain estimates for J and v as the data may not be available. Therefore, the impact of mis-specifyingthese parameters is of interest. Sensitivity studies by McWilliams[24]and Banerjeeand Rahim [21]showed that economiccontrol chart designsare not sensitive to mis-specificationof Weibull distribution parameters. McWilliams considered two examplesfor fivecases and concludedthat the economiceffectin mis-specifyingthe Weibullshape parameter is trivial. While the McWilliams’conclusionis valid for economiccontrol chart design,
Economicstatistical control chart design
581
we find that this is not true when constraints on statistical properties are added to the economic design model. Tables 3 and 4 contain the Banerjee and Rahim [21]economic design results for the effect of changingWeibullscaleand shape parameters, 2 and v, on the optimal design,augmented to include the actual u, p, and ATS values realized at a shift level of 0.50. Notice that the actual a and p levelsof the design remain relativelystable throughout the ranges whilethe ATS value variesfrom less than 3.0 h to nearly 15h. Therefore, while the statistical error rates have changed very little, the responsiveness of the charts is severely impacted. This raises the question of what effect changing 2 and v will have on expected cost if the upper bound on ATS is constrained and the statistical error rates improved. As with previousanalyses,the originaleconomicdesignATS value of 7.0 is used as a responsivenessbenchmark here. The values of u and p are constrained at the 0.05 and 0.90 levels respectivelyto ensure improved statistical properties. Tables 5 and 6 give the effectsof changing J and v on the optimal economic statistical design. Our results show that cost patterns with the economic statistical design do not match those
~1 L..
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582
Guoqiang Zhang and Victor Berardi
U.B.
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obtained through the economic design. Without regard to statistical properties and responsiveness, the economic design analysis leads one to conclude that for a constant mean time to failure, the effect of changing 2 or V,on the optimal design is fairly small. The cost pattern in Table 3 shows a 13.9°/0 increase as ). is raised 500°/0.The economic statistical design results in Table 5 give a different picture, however. Table 5 shows that adding the constraints has a major impact 011costs. The costs have a decreasing pattern as ). increases. Furthermore, the cost change over the range of ;. decreases by LIZ.070, suggesting that the accurate specification of 1. is a more important consideration for the economic statistical design compared to the economic design. [n addition, the direction of the cost pattern is contrary to that of the economic design and it can not be correctly anticip:lted from the economic design analysis alone. Note also that designs for ;. = 0.00 I ~ind ;. = 0.05 correspond to statistical designs. Table 6 contains the results from changing the Weibull slmpe parameter ~’on the expected cost
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of the economic statisticaldesign.The economicdesignresults in Table 4 showsthat as v increases from 1.0 to 4.0, the expected cost steadily decreases by about 16?4.,while actual ATS increases fivefold.Economic statistical design analysis for this example, however, shows that the expected cost decreases slightly as v moves from 1.0 to 2.0 and increases rapidly at larger values. The maximum expected cost represents nearly a 2000/.increase over the minimum.
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Table 3. The effectof changing), on the optimal economicdesign L Actualalpha Actualpower Actual ATS h, Expectedcost ($) n 13.226 0.116 0.796 10.53 1.57 228.22 23 12.566 0.119 0.799 10.04 1.56 231.30 23 12.165 0.119 0.799 9.72 1.56 233.19 23 10,650 1.57 0.119 0.813 8.66 239.16 24 3.420 0.116 0.836 2.86 1,56 260,03 26
Guoqiang Zhang and Victor Berardi
584
Table4, Theeffectof changingu on the outimal u I.0 2.0 3.0 4.0
i 0.0001 0.0002 0.0003 0.001 0.05
i 0.06549 0.00337 0.0002 0.0000124
v 3.248 3,000 2.855 2.427 1.086
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n 26 24 23 22
h, 2.36 7.23 10,04 11.64
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L 1.56 1.57 1.56 I.55
design Actual alpha Actual power Actual ATS 0.839 0.119 2.813 0.810 0.116 8,921 0.799 12,566 0.119 0.787 14,794 0.121
Table 5. The effectof changing1 on the optimal economicstatisticaldesign Expectedcost ($) n h, L Actual alpha Actual power Actual ATS 463.94 47 6.50 1.96 0.05 0.925 7.000 399.73 45 6.43 1.96 0.916 7.000 0.05 44 368.79 6.39 1.96 0.05 0.908 7.000 301,27 42 6.30 1.96 0.05 0.900 7.000 267.1I 42 3.31 1.96 0.05 0.900 3.679
Table6. The effectof changingu on the optimal economicstatisticaldesign u 1.0 2.0 3.0 4.0
i 0.06549 0.00337 0.0002 0.0000124
Expectedcost ($) 268,37 265.80 399.73 768.22
n 42 42 45 50
h, 2.77 6.30 6.43 6.60
L 1,96 1.96 1.96 1.96
Actual alpha Actual power ActualATS 0.05 0.900 3.072 0.05 0.900 7.000 0.05 0.916 7.000 0.05 0.942 7.000
Sinceit is v that determineswhether the in-control time distribution is Weibull or exponential, the penalty cost for mis-specificationof the underlying failure mechanism can be quite high. From Tables 3–6, we also find that when ~ is high and v is low, the increase in the overall cost associated with the improved statistical performance is not significantcompared to the economic design. For example, when 2 = 0.05 and v = 1.086,the expected cost is $260.03(Table 3) for the economicdesignand $267.11(Table 5)for the economicstatisticaldesign.However,when 1 is quite low and v is quite high (see the situations for v = 4 and A= 0.0000124in Tables 4 and 6 and for 2 = 0.0001 and v = 3.248 in Tables 3 and 5), the economic statistical design can be much more costly than the economic design. 4. DISCUSSION
Economic statistical control chart design under a Weibull failure mechanism is a general and flexiblemodel that can be utilized for many industrial production processes.It givesdesignersthe capability to assess the impact of their decisions on several critical aspects that are either not available or not apparent via other design methods. Using economic statistical design with an appropriate sensitivity analysis, designers can readily observe the impact on cost, sample size, samplinginterval, and control limitsdue to the constraints on the statisticalerror rates and control chart responsiveness,and the specificationof the failure distribution parameters. Considering statistical properties in the quality control chart design has significantpractical implications.The Type I error rate or false alarm rate is the probability of concludingthe process mean has shifted due to assignablecause when in fact it has not. A high number of false alarms will quickly undermine operator confidencein the use of the charts and incur unnecessary search costs. In control chart terms, power is the probability of correctly identifyinga shift in the process when one exists. Power provides a performance measure of a control chart’s capability to detect undesirableshiftsthat may occur in the production process.A low power leadsnot only to excessive cost due to the inability of the chart to identify when a significantprocess shift has occurred, increasing quality losses,but to product variability and ultimately customer dissatisfaction.ATS refers to the promptness in which a significantprocess shift is in fact identified. It is a measure of the control chart design’sresponsivenessin detecting process shifts and is especiallyimportant when producing defectiveproducts has high penalty costs. There is no general rule governing the selection of bounds for the economic statistical design constraints. They should be chosen based on the specificproblem situation, the relevant cost information, as wellas the economicand statisticalconsequences.The sensitivityanalysisis useful to the designer in making these decisions.
Economic statistical control chart design
585
Our study has shown that in some situations, the statistical performance of control charts can be improved significantlywith only a slight increasein cost by using an economicstatisticaldesign instead of an economic design. The cost increase is further shown to be relatively insensitiveto the improvementin the Type I error rate and power throughout the investigatedrange. This implies that it maybe quite inexpensivefor practitionersto implementcontrol charts with lower falsealarm rates and with a higherprobability for detectinga processshiftwhen one actually exists.The bound on ATS should not be set too low sincethe expectedcost is highly sensitiveto small ATS bounds. A reasonable bound on ATS may be found from the actual ATS in the economicdesign.For larger shift levels in this example, it is inexpensiveto achieve significant improvements in statistical properties. Being capable of detecting extremely small process shifts in a responsive fashion, however, is an expensiveproposition to achieve with X charts. Additionally, because the specification of the Weibull shape and scale parameters has a significantimpact on the cost and responsivenessin the economicstatisticaldesign,the distribution parameters need to be estimated adequately to reflect the process characteristics. The economic design alone obscures this important result and actually may be misleadingin the direction of its cost change. Furthermore, our results illustrate that in many cases, the economicstatistical design corresponds to a statisticalcontrol chart design.The implication is that control chart designsthat are cost effectiveare often possiblewithout degradingthe statisticalproperties of the control charts 5. SUMMARY AND CONCLUSION
In this paper, the four control chart design methods, their advantages, and disadvantageswere discussed.It was shown that economic statistical control chart designs are the best for achieving desiredcontrol chart statisticalproperties whilesimultaneouslyminimizingthe associatedcost. The failure model considered was Weibull instead of the widelyused exponential because the Weibull model is more appropriate and general in many practical situations. The developmentof the Weibulleconomiccontrol chart designmodelwas reviewedand extended to an economic statistical design model. We showed that the power and Type I error rates could be improved significantlyover the economic design with a small increase in cost for certain situations. An extensivesensitivityanalysis was performed on the economic statistical model to investigatethe relativeeffectsof the bounds for statisticaland performance measures, such as Type I error rate, power, ATS, shift sizeto be detected, and Weibullshape and scale parameters on the minimum expected cost. The cost is relatively insensitiveto improvement in Type I and Type II error rates and to higher ATS bounds, but is highly sensitiveto the lower bounds on average time to signal (ATS). Mis-specificationof the underlying Weibull failure distribution parameters may have significanteconomicconsequences.The information from the sensitivityanalysis is valuable to control chart designers in making the trade-off decisions between the expected cost and the desired statistical properties. Acknowledgements—The authors wish to thank Dr O. F. Offodileand Dr M. Y. Hu for their valuable suggestionsand encouragement.
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