Optimal tool replacement for processes with low fraction defective

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Production, Manufacturing and Logistics

Optimal tool replacement for processes

with low fraction defective

W.L. Pearn

*

_{, Ya-Chen Hsu}

Department of Industrial Engineering and Management, National Chiao Tung University, 1001 TA Hsueh Road, Taiwan, ROC Received 22 March 2005; accepted 12 May 2006

Available online 28 August 2006

Abstract

Tool wear is a frequent and natural part in many machining processes and is a systematic assignable cause. The fraction of defectives would rise as the tool deteriorates. When the fraction defective reaches a certain level, the tool must be replaced. To minimize the defective parts and the overall tool costs, the optimal tool replacement time needs to be deter-mined. Process capability indices (PCIs) have been eﬀectively used in the manufacturing industry to measure the fraction of defectives. Conventional methods of capability measurement become inaccurate since the process data is contaminated by the assignable cause variation. In order to determine the optimal tool replacement time to maintain maximum product quality, conventional capability calculation must be modiﬁed. Considering process capability changes dynamically, an

esti-mator of Cpmkis investigated. We obtain an exact form of the sampling distribution in the presence of a systematic

assign-able cause. This study provides an eﬀective management policy for optimal tool replacement under low fraction of defectives. To illustrate the application of this procedure, a case study involving the tool wear problem is presented. Ó 2006 Elsevier B.V. All rights reserved.

Keywords: Quality management; Critical value; Process capability index; Replacement time; Tool wear

1. Introduction

In automated machines, tools occupy a prominent place in producing quality goods. The tool will wear gradually as the manufacturing process proceeds. For instance, the machining operation shapes a production part using, cutting, drilling, or grinding operations, and so on. While such wear is unavoidable, tools must be controlled to maintain product quality and eﬃcient tool utilization. One important issue for tool wear control is the tool replacement policy. The tool should be replaced when product quality becomes worse. Process capability indices have been widely used in the manufacturing industry for measuring process quality, partic-ularly, for processes with low fraction of defectives. In practice, a minimal capability requirement would be preset by the customers/engineers in order to maintain a low fraction of defectives. When the capability index fails to reach the prescribed minimum value, one could conclude that the process is incapable of reaching the

* _{Corresponding author.}

E-mail address:[email protected](W.L. Pearn).

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desired production quality and the tool must be reset. In this study, we investigate an eﬀective management policy based on process capability calculation for optimal tool replacement time with low fraction of defec-tives to meet manufacturing requirement.

In the manufacturing industry, process capability indices have been widely used to provide numerical mea-sures on process reproduction capability, which are convenient and powerful tools for quality assurance and guidance for process improvement. Those indices are easy to understand and straightforward to apply in many industries such as automotive, semiconductor and IC assembly manufacturing industries. Among them, Cpand Cpk(seeKane, 1986) are the most extensively-used two in the manufacturing industry. Those indices

have been defined explicitly as the following: Cp¼ USL LSL 6r ; Cpk¼ min USL l 3r ; l LSL 3r ; Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2_{þ ðl T Þ}2 q ; Cpmk¼ min USL l 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2_{þ ðl T Þ}2 q ; l LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2_{þ ðl T Þ}2 q 8 > < > : 9 > = > ;;

where T is the target value, l is the process mean and r is the standard deviation of the characteristic, USL and LSL are the upper and lower specification limits, respectively. On the topic of PCIs, several authors have presented the use and examined their associated properties with different degrees of completeness. Examples are Kushler and Hurley (1992), Rodriguez (1992), Kotz and Johnson (1993), Va¨nnman and Kotz (1995), Bothe (1997), Spiring (1997), Kotz and Lovelace (1998), Palmer and Tsui (1999), Pearn and Shu (2003), Va¨nn-man and Hubele (2003), and references therein.Kotz and Johnson (2002)provided a compact survey for the development of PCIs with interpretations and comments on some 170 publications appeared during 1992– 2000. Spiring et al. (2003) consolidated the research findings in the field of process capability analysis for the period 1990–2002.

To understand and correctly interpret process capability indices, the process under investigation must be free from any special or assignable cause (i.e., in-control). Unfortunately, such condition is hardly met in many industrial applications. For example, when the assignable cause is in the form of tool wear, the output values inherently will show a certain increasing or decreasing trend. The causes such as tool wear are respon-sible for inducing autocorrelation and are not physically removable from the process. As a result, processes with uncontrollable trend are quite common in practice, and process capability analysis becomes a diﬃcult task for practitioners. Quality researchers see this fact, and several approaches have been suggested to deal with problems of assignable cause. Some approaches attempt to remove the variability associated with the sys-tematic assignable cause. For instance, Montgomery (1985)proposed ﬁtting the AR(1) time series model to the auto-correlated data.Yang and Hancock (1990)recommended that in computing the Cpindex, the

unbi-ased estimator of r can be obtained as r/(1 q)1/2, where q is deﬁned as the average correction factor. Time series modeling trend data had been also suggested by Alwan and Roberts (1988), who recommend using residuals in monitoring the process. Other approaches make the general assumption of linear degradation in the tool. For example, Long and De Coste (1988) investigated the procedure to remove the linearity by regressing on the means of the subgroups and then determined the process capability.Quesenberry (1988)also suggested that tool wear can be modeled over an interval of tool life by a regression model and assumes that the tool wear rate is known or a good estimate of it is available, and that the process mean can be adjusted after each batch without an error.

Most of the previous works reviewed above, however, did not consider a dynamic process capability over a cycle. By considering the process capability dynamic within a cycle, as well as from cycle to cycle, we could circumvent some of the problems encountered.Spiring (1991)has devised a modiﬁcation of Cpmindex for this

dynamic process under the inﬂuence of systematic assignable causes. Pearn et al. (1992) proposed an index called Cpmk, which combines the merits of the three basic indices Cp, Cpk, and Cpm. In this paper, we consider

capability index Cpmkfor the dynamic process under the inﬂuence of systematic assignable cause. This study is

divided into six sections beginning with introduction. Section2contains the concept of process capability mea-sure when the process involves tool wear problem. In Section3, a modiﬁed estimator of Cpmkis proposed and

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its explicit form of the sampling distribution is derived. Section4provides a method for managing a process exhibiting assignable cause. Practitioners can use the proposed approach to determine whether their process meets the preset capability requirement, and make reliable decisions on when to stop the process for tool replacement. In Section5, an application example involving tool wear is presented. Section 6 concludes the paper by a brief summary and discussion.

2. Measuring process capability with tool wear

Before assessing process capability, it is necessary to ensure that the process is under statistical control and the observations are statistically independent. However, it is not always the case.Porter and Oakland (1991)

pointed out that the two specific conditions which make the process capability assessment to be difficult are: (1) ensuring stability of the mean and of the standard deviation; and (2) an absence of any special causes. In practice, processes with uncontrollable but acceptable trend are common. This is also referred to as a constant or consistent process drift, and other examples include accumulation of contaminants and temperature change drift must be quantified and removed before the remaining variability can be analyzed for statistical control (Kotz and Lovelace, 1998). The tool wear problems are responsible for inducing correlation and are not phys-ically removable from the process. The issues of correlation among the samples and its effect on control chart limits have been examined by many authors (seeVasilopoulos and Stamboulis, 1978, Burr, 1979). Although various authors have looked at the issue of correlation from the point of control charts, process capability aspects have seldom been considered.

Fig. 1illustrates an example of tool wear problem with four cycles, which displays information regarding process speciﬁcations (i.e., USL, LSL and T), the starting, stopping, tool replacement times (i.e., t0, t1, t2, t3,

t4), and the process output. InFig. 1, the solid line illustrates the general systematic tool wear process with

non-linear cycles over time/production. The change times may represent chronological time but are more likely to represent production qualities. The traditional measurement of process capability index Cpmk is

aﬀected by tool wear slope (see the dashed line inFig. 1). The causes such as tool wear are responsible for inducing autocorrelation and are not physically removable from the process. Ignoring the unknown trend pat-terns, the presence of assignable cause variations will make the result of any capability index meaningless.

In order to calculate process capability accurately, the eﬀects such as tool wear with systematic assignable causes must be considered. When systematic assignable causes are present and tolerated, the overall variation of the process (r2) is composed of the variation due to random causesðr2

rÞ and the variation due to assignable

causesðr2

aÞ. That is, r2¼ r2rþ r2a. The traditional PCI measures fails to acknowledge that portions of the

over-all variation, (in the presence of tool wear), will be due to assignable causes. Hence any estimates of the process capability will confound the true capability with these two sources. In order to get a true measure of process capability, any variation due to an assignable cause must be removed from the measure of process capability.

Wallgren (1996)has also studied the properties and implications of the index Cpm when the consecutive

measurements represent observations of dependent variables stemming from a Markov process in discrete

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time. This occurs, for example, when consecutive measurements from a process are serially corrected. The author had also developed an augmentation of the index Cpm, denoted Cpmrbased on the ﬁrst-order

autore-gressive model (AR(1)).Spring (1989, 1991)viewed this as a dynamic process which is in a constant change as a process, tool, age, etc. In this dynamic model, the capability of the process may vary, possibly in a predict-able manner. Spiring has devised a modiﬁcation of Cpmindex for this dynamic process under the inﬂuence of

systematic assignable causes. In this scenario the goal is to maintain some minimum level of capability at all times. To become pro-active in the area of tool wear, steps should be taken to eliminate variation due to an assignable cause.

3. Statistical properties of the estimated Cpmk

In this section, we first introduce a modification of Cpmkindex for the dynamic process under the influence

of systematic assignable causes. Accordingly, an explicit form of the cumulative distribution function of the dynamic estimator of Cpmkis obtained, which can be expressed in terms of a mixture of the chi-square

distri-bution and the non-central chi-square distridistri-bution. We then obtain the rth moment, and the mean and the variance as well as the bias and the mean square error (MSE) of the estimated Cpmk for dynamic process.

3.1. Estimation of Cpmk with tool wear

Using process capability index can monitor the changing ability of the process. Considering process capa-bility changes dynamically, the goal is to maintain some minimum level of capacapa-bility. We proposed a modi-ﬁcation of Cpmkindex for dynamic processes at time period t under the inﬂuence of systematic assignable cause

as Cpmk¼ min USL lt 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 rtþ ðlt T Þ 2 q ; lt LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 rtþ ðlt T Þ 2 q 8 > < > : 9 > = > ;; ð1Þ

where ltrepresents the process mean and r2rtis the variation (due to random causes only) of the process at time

period t. Utilizing the identity min{a ,b} = (a + b)/2 ja bj/2, the index Cpmkdeﬁned in Eq.(1)can be

alter-natively rewritten as Cpmk¼ d jlt Mj 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 rtþ ðlt T Þ 2 q ; ð2Þ

where d = (USL LSL)/2 is half of the length of the speciﬁcation interval, M = (LSL + USL)/2 is the mid-point between the lower and the upper speciﬁcation limits. Finding the value of Cpmkor a suitable estimate at

various times t over each cycle in the lifetime of the tool is required for monitoring a process’s capability. In its simplest and most common form, tool wear data tend to have an upward or a downward slope over time. Assuming the eﬀect of the tool deterioration is linear over the sampling window only, then the tool wear data can be modeled by a regression model over the sampling window of tool life. Once control has been acti-vated, the estimates of Cpmkare available without involving contribution of the assignable causes. Hence, the

proposed estimator of process capability can be obtained by replacing ltand rrtby the estimators Xtn and

[(n 2)MSEt/(n 1)]1/2, respectively. Then we have

b Cpmk¼ min USL Xtn; Xtn LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ r2 rtþ ðXtn T Þ 2 q ¼ d Xtn M 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn2ÞMSEt ðn1Þ þ ðXtn T Þ 2 q ; ð3Þ where Xtn ¼ Pn i¼1Xti=n;MSEt¼ Pn i¼1 Xti bXti 2

=ðn 2Þ, n denotes the subgroup sample size, and Xti

repre-sents the ith value of the quality characteristic in the sampling period t. The variation r2

rt is removed by

con-sidering of the sequentially selected points (i.e., t = 0, 1, 2, . . . , n) instead of the sample variance. The MSEtis the

mean square error associated with the regression equation bXti ¼ ^atþ ^btti, where tiis the sequence number of

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^

bt is large, it indicates that the tool wear trend is signiﬁcant, and the tool will soon be producing too many

defectives. In such situation, the practitioner would choose to replace the tool even if the current process had adequate capability. Alternatively, the practitioner could choose to sample the data more frequently to monitor the tool wear more closely. On the other hand, small value of ^bindicates the tool wear situation is not that serious. In general, our method provides a reference for the practitioner to follow for decision making. Assuming the sampling scheme to be sequential and using ordinary least square (OLS) estimates of ^atand

^

bt, the computational formula for MSEtcan be derived alternatively as follows:

MSEt¼ Xn i¼1 ðXti bXtiÞ 2 ðn 2Þ = ¼ X n i¼1 X2_t i ^at Xn i¼1 Xti ^bt Xn i¼1 iXti # " , ðn 2Þ; where ^at¼2ð2nþ1Þ_ðn1Þ Xtn 6Pn i¼1iXti nðn1Þ , and ^bt¼ 12Pn i¼1Xti

nðn2_1Þ _ðn1Þ6 Xtn. Then we have that

MSEt¼ Xn i¼1 X2_t i ^at Xn i¼1 Xti ^bt Xn i¼1 iXti # " , ðn 2Þ ¼ Pn i¼1X 2 ti ðn 1Þ 2nð2n þ 1Þ ðn 1Þ2 X 2 tn 12 Pn_i¼1iXti 2 nðn2_{1Þðn 1Þ}þ 12Xtn Pn i¼1ðiXtiÞ ðn 1Þ2 # " , ðn 2Þ: Therefore b Cpmk¼ d Xtn M 3 Pn i¼1X 2 ti ðn1Þ 2nð2nþ1Þ ðn1Þ2 X2tn 12Pn i¼1iXti ð Þ2 nðn2_1Þðn1Þ þ 12XtnP n i¼1ðiXtiÞ ðn1Þ2 þ nðXtnT Þ2 ðn1Þ 1=2; ð4Þ where M = T.

3.2. Sampling distribution of the estimated Cpmk

From Eq.(3), estimator bCpmk for the dynamic process can be rewritten as follows:

b Cpmk¼ d Xtn M 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðn2Þ_ðn1ÞMSEtþ ðXtn T Þ 2 q b ffiffiffi n p H 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin n1Kþ H 2 q ; ð5Þ where b = d/rrt, K¼ ðn 2ÞMSEt=r2rt; H ¼ ffiffiffi n p ðXtn MÞ=rrt , and Xtn¼ Pn

i¼1Xti=n. Under the assumption of

normality, K is distributed as v2

n2;a chi-square distribution with n 2 degrees of freedom, H 2

is distributed as v02

1;k a non-central chi-square distribution with one degree of freedom and non-centrality parameter

k¼ nðlt T Þ 2

=r2

rt. And H is distributed as a folded-normal distribution, Nðn

ffiffiffi n p

;1Þ with probability density function fHðhÞ ¼ /ðh þ n ffiffiffin

p

Þ þ /ðh npffiffiffinÞ for h P 0, where /(Æ) is the probability density function of the standard normal distribution and n = (lt T)/rrt.

For x > 0, the cumulative distribution function of bCpmk can be derived as

F bCpmkðxÞ ¼ P bCpmk 6_x ¼ P ðb ffiffiffi n p H Þ 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin n1Kþ H 2 q 6_x 0 B @ 1 C A ¼ 1 P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n 1Kþ H 2 r <ðb ffiffiffi n p H Þ 3x ¼ 1 Z 1 0 P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_n n 1Kþ H 2 r <ðb ffiffiffi n p H Þ 3x H ¼ h fHðhÞdh ¼ 1 Z 1 0 P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n 1Kþ h 2 r <ðb ffiffiffi n p hÞ 3x fHðhÞdh ¼ 1 Z 1 0 P K <ðn 1Þðb ffiffiffi n p hÞ2 9nx2 ðn 1Þh2 n ! fHðhÞdh:

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Since K is distributed as v2 n2, we have P K <ðn 1Þðb ffiffiffi n p hÞ2 9nx2 ðn 1Þh2 n ! ¼ 0 for h > bpffiffiffin=ð1 þ 3xÞ: Therefore, F bCpmkðxÞ ¼ 1 Z bpffiffin=ð1þ3xÞ 0 P K <ðn 1Þðb ffiffiffi n p hÞ2 9nx2 ðn 1Þh2 n ! fHðhÞdh ¼ 1 Z bpffiffin=ð1þ3xÞ 0 G ðn 1Þðb ffiffiffi n p hÞ2 9nx2 ðn 1Þh2 n ! fHðhÞdh for x > 0; ð6Þ

where G(Æ) is the cumulative distribution function of v2

n2. Substituting fH(h) into Eq.(6) leads to the result:

F bCpmkðxÞ ¼ 1 Z bpffiffin=ð1þ3xÞ 0 G ðn 1Þðb ffiffiffi n p tÞ2 9nx2 ðn 1Þt2 n ! /ðt þ npffiffiffinÞ þ /ðt npffiffiffinÞ dt for x > 0: ð7Þ The proposed sampling scheme is similar to the schemes used in monitoring a process for control charting procedures. The general format is to gather k subgroups of size n from each cycle (e.g., the period form t0

to t1in Fig. 1) over the lifetime of the tool. The value of k will be unique to each process and, in fact, may

change from cycle to cycle within a process. On the other hand, sample size of less than ﬁve (i.e., n < 5) are cautioned against, while larger samples (e.g., n > 30) may also pose a problem. The optimal sample size for assessing process capability in the presence of systematic assignable cause will vary for each process considered (Spiring, 1991).

3.3. The rth moment of the estimated Cpmk

Under the assumption of normality and for the general case with T = M,Pearn et al. (1992)derived the rth moment of bCpmk. Using the similar technique to derive the rth moment of the modiﬁed bCpmk, we obtained as

follows: Eð bCr_pmkÞ ¼e k=2 3r Xr i¼0 ð1Þi r i _d r ffiffiffi n 2 r ri X 1 j¼0 ðk 2Þ j j! Cðiþ1 2 þ jÞ Cð1 2þ jÞ Cðn1rþi 2 þ jÞ Cðnþi 2 þ jÞ " # : ð8Þ

Taking r = 1 and 2, the expected value and variance of bCpmk can be expressed as the following:

Eð bCpmkÞ ¼ ek=2 3 X1 j¼0 ðk 2Þ j j! d r ffiffiffi n 2 r Cð n 2 1 þ jÞ Cðn 2þ jÞ j!Cð n1 2 þ jÞ Cð1 2þ jÞCðnþ12 þ jÞ " # ; and

Varð bCpmkÞ ¼ Eð bC2pmkÞ ½Eð bCpmkÞ 2 ; where Eð bC2_pmkÞ ¼e k=2 9 X1 j¼0 k 2 j j! dpffiffiffin r 2 1 nþ 2j 3 dpffiffiffiffiffi2n r j! Cð1 2þ jÞ 2 nþ 2j 2þ 1þ 2j nþ 2j 1 " # ;

where CðuÞ ¼R₀1tu1etdt is a gamma function. Therefore, the bias and the MSE of bCpmk are: Biasð bCpmkÞ ¼

Eð bCpmkÞ Cpmk;MSEð bCpmkÞ ¼ Varð bCpmkÞ þ ½Biasð bCpmkÞ 2

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4. Testing procedure for process capability

Under the assumption of normality, the cumulative distribution function of bCpmk for dynamic process can

be expressed in terms of a mixture of the chi-square distribution and the non-central chi-square distribution. Therefore, to test whether a given process is capable, we can consider the following statistical testing hypotheses:

H0: Cpmk6C ðprocess is not capableÞ;

H1: Cpmk> C ðprocess is capableÞ:

Using the index bCpmk, the engineers can access the process performance and monitor the manufacturing

pro-cesses on routine basis. A testing procedure similar to those used in monitoring a process with control chart can be used to monitor the process and determine whether the process should stop and reset the tool to avoid producing non-conforming products. Deﬁning the decision making rule /*_{(x) as the following:}

/ðxÞ ¼ 1 if bCpmk > ca; 0 otherwise; (

where a(ca) = a is the type I error, the chance of incorrectly concluding an incapable process (Cpmk6C) as

capable (Cpmk> C), thus, the test /*(x) rejects the null hypothesis H0(Cpmk6C) if bCpmk > ca. Based on the

cumulative distribution function of bCpmk expressed in Eq.(7), given values of capability requirement C, the

a-risk, the sample size n and the parameter n, hence the critical value cacan be obtained by solving the

equa-tion Pð bCpmkP cajCpmk¼ CÞ ¼ a using available numerical integration methods. That is,

Z bpffiffin=ð1þ3CÞ 0 G ðn 1Þðb ffiffiffi n p tÞ2 9nC2 ðn 1Þt2 n ! /ðt þ npffiffiffinÞ þ /ðt npffiffiffinÞ dt¼ a: ð9Þ

Note that, for ﬁxed values of C, n and a, Eq.(9)is an even function of n. Thus, we obtain the same critical value cafor both n = n0and n =n0. Since the process parameters l and r are unknown, the distribution

characteristic parameter n = (l T)/r is also unknown, which has to be estimated in real applications, nat-urally by substituting l and r by the sample mean and the sample variance. To eliminate the need for estimat-ing the parameter n, we examine the behavior of the critical values caagainst the parameter 0 6 n 6 3. Further,

we perform extensive calculations to obtain the critical values cafor 0 6 n 6 3, n = 5, 10, 20, 30 and Cpmkform

0 to 2 with risk a = 0.05. Note that the parameter values we investigated, 0 6 n 6 3, cover a suﬃciently wide range of applications with process capability analysis.Figs. 2(a)–2(d)display the surface plots of the critical value ca versus the parameter 0 6 n 6 3, 0 6 Cpmk62 with type I error a = 0.05 for sample size

n = 5, 10, 20, 30, respectively. The results indicate that (i) the critical value cais increasing in n, and is

decreas-ing in n, (ii) the critical value caobtains its maximum at n = 0.5 in all cases with accuracy up to 103. Hence,

for practical purpose we may solve Eq.(9)with n = 0.5 to obtain the required critical values cafor given Cpmk,

n, and a, without having to further estimate the parameter n. Thus, the risk a can be ensured, and the decisions made based on such approach are indeed more reliable.

Therefore, for users’ convenience in applying our proposed procedure, we tabulate the critical values of Cpmkfor various values of a = 0.01, 0.025 and 0.05 with n = 5(1)30 inTable 1for commonly recommended

minimum capability requirement C = 1.00, 1.33, 1.50, 1.67 and 2.00. For example, if C = 1.00 is the minimum capability requirement, then for a = 0.05, with sample size n = 15 we can ﬁnd ca= 1.60 fromTable 1. That is,

as the estimated process capability drops below the critical value of Cpmk, the practitioner should stop the

pro-cess and reset the tool because there is an evidence to consider that the propro-cess is nearing the end of its ability to produce agreeable product. Otherwise, if the values of Cpmkgreater than the critical value, then the process

is considered capable and is allowed to continue. In the following, we calculate the power of the test as pðCpmkÞ ¼ 1 b ¼ P ð bCpmk> CjCpmkÞ ¼ Z bpffiffin=ð1þ3CÞ 0 G ðn 1Þðb ffiffiffi n p tÞ2 9nC2 ðn 1Þt2 n ! /ðt þ npffiffiffinÞ þ /ðt npffiffiffinÞ dt: ð10Þ

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In Figs. 3(a)–3(d), we plotted the power curves, p(Cpmk) versus ca value, for the quality conditions with

C = 1.00, 1.33, a-risk = 0.01, 0.05 and n = 10(5)30. It can be seen from Figs. 3(a)–3(d), the power is quite good.

5. Capability testing with applications 5.1. Capability requirement

In the general case, a manufacturing process is said to be inadequate if Cpmk< 1.00; it indicates that the

process is not adequate with respective to the manufacturing tolerances, the process variation needs to be reduced (often using design of experiments). The fraction of defectives for such process exceeds 2700 ppm (parts per million). A manufacturing process is said to be marginally capable if 1.00 6 Cpmk< 1.33; it indicates

that caution needs to be taken regarding the process consistency and some process control is required (usually

Fig. 2b. Surface plot of cawith 0 6 n 6 3 and 0 6 Cpmk62 for n = 10 and a = 0.05.

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using R or S control charts). The fraction of defectives for such process is within 66–2700 ppm. A manufac-turing process is said to be satisfactory if 1.33 6 Cpmk< 1.67; it indicates that process consistency is

satisfac-tory, material substitution may be allowed, and no stringent precision control is required. The fraction of defectives for such process is within 0.54–66 ppm. A manufacturing process is said to be excellent if 1.67 6 Cpmk< 2.00; it indicates that process precision exceeds satisfactory. The fraction of defectives for such

process is within 0.002–0.54 ppm. Finally, a manufacturing process is said to be super if CpmkP2.00. The

fraction of defectives for such process is less than 0.002 ppm. 5.2. An example

To illustrate the practicality of our proposed approach to actual data, we consider the following real case taken from a metal crown company engaged mainly in making aluminum lids components, which are pro-duced on a press. Each press contains 22 dies and the differences exist die-to-die and press-to-press. Slight fluc-tuations are observed with lot-to-lot changes in steel. Since the press of the interfaces may affect wear rates of

Fig. 2c. Surface plot of cawith 0 6 n 6 3 and 0 6 Cpmk62 for n = 20 and a = 0.05.

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tool, the process exhibits tool wear. Each day, lids from each station are sampled and lid height is measured. Data collected over one-week period. Each sample contains a single lid from each station. We investigated a particular type of the lid product with the upper and lower speciﬁcation limits of the key characteristic, lid height, are set to USL = 68.4 mm, LSL = 64.65 mm, respectively and the target value is set to T = 66.525 mm. The lid height is measured and recorded when the product comes out of the process. The collected

Table 1

Critical values cafor Cpmk= 1.00, 1.33, 1.50, 1.67, 2.00, n = 5(1)30, and a = 0.01, 0.025, 0.05

Cpmk 1.00 1.33 1.50 1.67 2.00 n a 0.01 0.025 0.05 0.01 0.025 0.05 0.01 0.025 0.05 0.01 0.025 0.05 0.01 0.025 0.05 5 4.02 3.12 2.55 5.21 4.06 3.33 5.83 4.54 3.73 6.44 5.02 4.13 7.64 4.02 3.12 6 3.27 2.65 2.24 4.24 3.45 2.92 4.74 3.86 3.27 5.25 4.27 3.62 6.22 3.27 2.65 7 2.85 2.37 2.05 3.70 3.09 2.67 4.13 3.46 2.99 4.57 3.82 3.31 5.42 2.85 2.37 8 2.58 2.19 1.92 3.35 2.85 2.50 3.74 3.19 2.80 4.14 3.53 3.11 4.91 2.58 2.19 9 2.39 2.06 1.83 3.10 2.68 2.38 3.47 3.00 2.67 3.84 3.32 2.96 4.55 2.39 2.06 10 2.24 1.96 1.75 2.92 2.55 2.29 3.26 2.86 2.56 3.61 3.16 2.84 4.29 2.24 1.96 11 2.14 1.88 1.70 2.78 2.45 2.21 3.11 2.75 2.48 3.44 3.04 2.75 4.08 2.14 1.88 12 2.05 1.82 1.65 2.66 2.37 2.15 2.98 2.66 2.41 3.30 2.94 2.67 3.92 2.05 1.82 13 1.98 1.77 1.61 2.57 2.30 2.10 2.88 2.58 2.36 3.19 2.86 2.61 3.78 1.98 1.77 14 1.92 1.72 1.58 2.50 2.25 2.06 2.79 2.52 2.31 3.09 2.79 2.56 3.67 1.92 1.72 15 1.87 1.69 1.55 2.43 2.20 2.02 2.72 2.46 2.27 3.01 2.73 2.52 3.58 1.87 1.69 16 1.82 1.65 1.52 2.37 2.16 1.99 2.66 2.42 2.23 2.94 2.68 2.48 3.49 1.82 1.65 17 1.78 1.62 1.50 2.32 2.12 1.96 2.60 2.38 2.20 2.88 2.63 2.44 3.42 1.78 1.62 18 1.75 1.60 1.48 2.28 2.09 1.94 2.55 2.34 2.17 2.83 2.59 2.41 3.36 1.75 1.60 19 1.72 1.58 1.46 2.24 2.06 1.92 2.51 2.31 2.15 2.78 2.55 2.38 3.30 1.72 1.58 20 1.69 1.56 1.45 2.21 2.03 1.90 2.47 2.28 2.13 2.74 2.52 2.36 3.25 1.69 1.56 21 1.67 1.54 1.43 2.17 2.01 1.88 2.44 2.25 2.11 2.70 2.49 2.33 3.20 1.67 1.54 22 1.65 1.52 1.42 2.15 1.99 1.86 2.40 2.23 2.09 2.66 2.47 2.31 3.16 1.65 1.52 23 1.63 1.50 1.41 2.12 1.97 1.84 2.37 2.20 2.07 2.63 2.44 2.29 3.13 1.63 1.50 24 1.61 1.49 1.40 2.10 1.95 1.83 2.35 2.18 2.05 2.60 2.42 2.28 3.09 1.61 1.49 25 1.59 1.48 1.39 2.07 1.93 1.82 2.32 2.16 2.04 2.57 2.40 2.26 3.06 1.59 1.48 26 1.57 1.46 1.38 2.05 1.91 1.80 2.30 2.15 2.02 2.55 2.38 2.24 3.03 1.57 1.46 27 1.56 1.45 1.37 2.03 1.90 1.79 2.28 2.13 2.01 2.52 2.36 2.23 3.00 1.56 1.45 28 1.54 1.44 1.36 2.02 1.88 1.78 2.26 2.11 2.00 2.50 2.34 2.22 2.98 1.54 1.44 29 1.53 1.43 1.35 2.00 1.87 1.77 2.24 2.10 1.99 2.48 2.33 2.20 2.95 1.53 1.43 30 1.52 1.42 1.34 1.98 1.86 1.76 2.22 2.09 1.98 2.46 2.31 2.19 2.93 1.52 1.42 Cpmk Power value 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5

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Cpmk Power value 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5

Fig. 3b. Power curves for C = 1.00 and n = 10(5)30 (from bottom to top in the plot) with a = 0.05.

Cpmk Power value 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5

Fig. 3c. Power curves for C = 1.33 and n = 10(5)30 (from bottom to top in the plot) with a = 0.01.

Cpmk Power value 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0

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data consist of 105 observations arranged in seven subgroups of 15 each. The plot of the individual values in the series is depicted in Fig. 4. The increasing trend of the individual values due to tool wear appears to be linear in nature. Also, the values of the lid height of each component is inﬂuenced by the amount of tool wear at that instant, which is likely to be dependent on the condition of the tool when previous component was processed. Now, the goal is to maintain some minimum level of capability at all times and to monitor/manage this processes under the inﬂuence of tool wear problem. When the measure of process capability comes closer to the minimum acceptable level, the processing should be stopped and the tool should be replaced.

Suppose for this particular process under consideration to be capable, the process index Cpmkmust reach at

least a certain level C, say, 1. Thus, applying the proposed capability measure for dynamic, the practitioners can monitor the process by calculating the measure of Cpmk. The proposed testing procedure for a process

involving tool wear is similar to those used in monitoring a process with control chart. In this case we can obtain the critical value of bCpmk is 1.55 by checking Table 1under the given values of risk a = 0.05, sample

size n = 15 and minimum capability requirement C = 1.00. While the estimated process capability drops below the critical value of bCpmk, the practitioner should stop the process and reset the tool because there is

an evidence to consider that the process is nearing the end of its ability to produce agreeable product. As regards the values of bCpmkgreater than 1.55 the process is considered capable and is allowed to continue. Based

on the data listed inTable 2, the calculated bCpmk for dynamic process at each time period are summarized in

Table 3.Fig. 5plots the measure of process capability bCpmk for dynamic process at each time period over a

0 20 40 60 80 100 Observations LSL 66 67 USL mm

Fig. 4. Plot of the original data.

Table 2

The collected 7 subgroups each of 15 observations (unit: mm) i Time period t t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 1 66.100 66.335 66.470 66.542 66.670 66.722 66.872 2 66.261 66.295 66.387 66.551 66.665 66.722 66.931 3 66.147 66.335 66.456 66.501 66.684 66.715 66.860 4 66.214 66.361 66.402 66.504 66.644 66.777 66.836 5 66.133 66.314 66.468 66.568 66.689 66.724 66.922 6 66.223 66.335 66.430 66.546 66.715 66.770 66.943 7 66.216 66.428 66.480 66.470 66.695 66.803 66.907 8 66.288 66.337 66.428 66.572 66.732 66.770 66.900 9 66.159 66.397 66.413 66.618 66.665 66.753 66.929 10 66.252 66.337 66.499 66.625 66.606 66.789 66.919 11 66.288 66.418 66.387 66.599 66.717 66.758 66.862 12 66.242 66.416 66.504 66.656 66.675 66.805 66.922 13 66.297 66.423 66.432 66.596 66.727 66.774 66.836 14 66.304 66.361 66.516 66.594 66.708 66.800 66.929 15 66.221 66.435 66.546 66.665 66.739 66.781 66.950

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single cycle of the process. It is observed that the estimated bCpmk reaches maximum at time period t = 4 and

then drops below the line of critical values 1.55 at time period t = 7. Therefore, based on these results obtained we would suggest that the process should be stopped and the tool should be replaced at time period t = 7 to avoid produce unacceptable components.

6. Conclusions

In most manufacturing industries, a tool replacement policy is essential in order to minimize the fraction defective and the manufacturing cost. Therefore, an eﬀective tool management policy is essential for the man-ufacturing industry in order to meet customer’s requirements. Capability indices are eﬀective methods for quantifying process performance and for conveying critical information regarding the suitability of a manu-facturing process to meet the required quality standards. The index Cpmkcombines the merits of the two

indi-ces Cpk, and Cpmto provide numerical measures on process performance. In this paper, we have applied the

process capability index Cpmk to determine the optimal tool replacement time under tool wear condition.

Under the assumption of normality, the sampling distribution of the estimated Cpmk is a mixture of the

chi-square and the non-central chi-square distributions. We implemented the derived results to develop an eﬀective procedure to assess process capability at each time period over a process cycle, and to calculate the critical values for various sample sizes. Practitioners can use the proposed procedure to determine whether their process meets the preset capability requirement, and to make reliable decisions in determining the opti-mal time for tool replacements.

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Fig. 5. Capability plot for dynamic process at each time period. Table 3

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b

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