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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 717184, 13 pages
http://dx.doi.org/10.1155/2012/717184
Research Article

Computational Procedure of Performance Assessment of Lifetime Index of Products for the Weibull Distribution with the Progressive First-Failure-Censored Sampling Plan

1Department of Information Management, Shih Chien University, Kaohsiung Campus, Kaohsiung 84550, Taiwan
2Department of International Business, Chang Jung Christian University, Tainan 71101, Taiwan
3Department of Applied Mathematics, National Chiayi University, Chiayi City 60004, Taiwan

Received 31 January 2012; Accepted 6 March 2012

Academic Editor: Vu Phat

Copyright © 2012 Ching-Wen Hong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Process capability analysis has been widely applied in the field of quality control to monitor the performance of industrial processes. In practice, lifetime performance index 𝐶𝐿 is a popular means to assess the performance and potential of their processes, where L is the lower specification limit. This study will apply the large-sample theory to construct a maximum likelihood estimator (MLE) of 𝐶𝐿 with the progressive first-failure-censored sampling plan under the Weibull distribution. The MLE of 𝐶𝐿 is then utilized to develop a new hypothesis testing procedure in the condition of known L.

1. Introduction

Effectively managing and measuring the business operational process is widely seen as a means of ensuring business survival through reduced time to market, increased quality, and reduced costs. Process capability analysis is an effective means of measuring process performance and potential capability. In the manufacturing industry, process capability indices are utilized to assess whether product quality meets the required level. For instance, Montgomery [1] (or Kane [2]) proposed the process capability index 𝐶𝐿 (or 𝐶𝑃𝐿) for evaluating the lifetime performance of electronic components, where 𝐿 is the lower specification limit, since the lifetime of electronic components exhibits the larger-the-better quality characteristic of time orientation. Tong et al. [3] constructed a uniformly minimum variance unbiased estimator (UMVUE) of 𝐶𝐿 under an exponential distribution. Moreover, the UMVUE of 𝐶𝐿 is then utilized to develop the hypothesis testing procedure. The purchasers can then employ the testing procedure to determine whether the lifetime of electronic components adheres to the required level. Manufacturers can also utilize this procedure to enhance process capability. Hong et al. [4] also constructed a maximum likelihood estimator (MLE) of 𝐶𝐿 with the type II right censored sample under a pareto distribution. Moreover, the MLE estimator of 𝐶𝐿 is then utilized to develop a hypothesis testing procedure. The managers can then employ the testing procedure to assess the business performance. Lee et al. [5, 6] also constructed an MLE of 𝐶𝐿 under the Burr XII distribution with progressively type II right censored sample and the Gompertz distribution with the first-failure-censored sample, respectively. Moreover, the MLE of 𝐶𝐿 is then utilized to develop a hypothesis testing procedure. The managers can then employ the testing procedure to assess the quality performance of product.

In this study, process capability analysis is also utilized to assess product quality. The lifetime performance index 𝐶𝐿 is also utilized to measure product quality with the Weibull distribution based on the progressive first-failure-censored sampling plan. The Weibull distribution is useful in a great variety of applications, particularly as a model for product life. It has also been used as the distribution of strength of certain materials. It is named after Weibull [7], who popularized its use among engineers. One reason for its popularity is that it has a great variety of shapes. This makes it extremely flexible in fitting data, and it empirically fits many kinds of data (see Nelson [8]). The Weibull distribution includes the exponential and the Rayleigh distributions as special cases. The exponential and the Rayleigh distributions have been recognized as a useful model for the analysis of lifetime data. The Weibull distribution family has played an important role in the analysis of lifetime data. The probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) of the Weibull distribution are as follows, respectively, 𝑓𝑋𝛽(𝑥)=𝛼𝛽𝑥𝛽1𝑥exp𝛼𝛽𝐹,𝑥>0,𝛼>0,𝛽>0,(1.1)𝑋𝑥(𝑥)=1exp𝛼𝛽,𝑥>0,𝛼>0,𝛽>0.(1.2) The parameter 𝛽 is called the shape parameter, and the parameter 𝛼 is called the scale parameter. For the special case 𝛽=1, the Weibull distribution is the simple exponential distribution. For the special case 𝛽=2, the Weibull distribution is the Rayleigh distribution. In addition, for 3𝛽4, the shape of the Weibull distribution is close to that of the normal distribution (see Nelson [8]).

In life testing experiments, the experimenter may not always be in a position to observe the life times of all the products (or items) put on test. This may be because of time limitation and/or other restrictions (such as money, material resources, mechanical or experimental difficulties) on data collection. Therefore, censored samples may arise in practice. In this study, we consider the case of progressive first-failure-censored sampling plan. The progressive first-failure-censored sampling plan is the combination of first-failure-censored sampling plan and progressively type II right censored sampling plan. Owing to, sometimes the lifetime of a product is quite long. Thus, a right type II censored sample plan for such a product can be too long. Johnson [9] proposed the first-failure-censored sampling plan in which the experimenter can decide to group the test units into several sets (each set is an assembly of test units), and then run all the test units simultaneously until the first failure in each group. Such plans are usually feasible when test facilities are scarce but test material is relatively cheap. Balasooriya [10] examined the failure-censored sampling plan for the 2-parameter exponential distribution based on testing r random samples, each of size n, one after the other. That procedure is based on exact results, and only the first failure time of each sample is needed. The Balasooriya sampling plan is compared with traditional sampling plans using a sample of size 𝑟𝑛 (see Wu et al. [11]). The first-failure-censored sampling plan has an advantage in terms of shorter test time and a saving of resources. Note that a first-failure-censoring scheme is terminated when the first failure in each set is observed. If an experimenter desires to remove some sets of test units before observing the first failures in these sets, the above-described scheme will not be of use to the experimenter. The first-failure-censored sampling plan does not allow for sets to be removed from the test at the points other than the final termination point. However, this allowance will be desirable when some sets of the surviving units in the experiment that are removed early on can be used for some other tests. As in the case of accidental breakage of experimental units or loss of contact with individuals under study, the loss of test units at points other than the termination point may also be unavoidable (see Wu and Kuş [12]). Therefore, we also consider the case of the progressively type II right censoring in this study. Progressive type II right censoring is a useful scheme in which a specific fraction of individuals at risk may be removed from the experiment at each of several ordered failure times (see Fernández [13]). The experimenter can remove units from a life test at various stages during the experiments, possibly resulting in a saving of costs and time (see Sen [14]). Therefore, the progressive first-failure-censored sampling plan has an advantage in terms of shorter test time, a saving of resources, and in which a specific fraction of individuals at risk may be removed from the experiment at each of several ordered failure times. The progressive first-failure-censored sampling plan is illustrated as follows.

Suppose that 𝑚 is the number of failures observed before termination and n independent groups with k items within each group are put in a life test. 𝑅1 groups and the group in which the first failure is observed are randomly removed from the test as soon as the first failure (say 𝑋1) has occurred, 𝑅2 groups and the group in which the second failure is observed are randomly removed from the test as soon as the first failure (say 𝑋2) has occurred, and finally 𝑅𝑚(𝑚𝑛) groups and the group in which the mth failure is observed are randomly removed from the test as soon as the mth failure (say 𝑋𝑚) has occurred. Then 𝑋1𝑋2𝑋𝑚 are called the progressive first-failure-censored order statistics with censoring scheme 𝑅=(𝑅1,𝑅2,,𝑅𝑚). It is clear that 𝑛=𝑚+𝑅1+𝑅2++𝑅𝑚. The familiar complete, type II right censored, first-failure-censored, and progressively type II right censored samples are special cases of the progressive first-failure-censored sampling plan. Note that if 𝑅1=𝑅2==𝑅𝑚=0, then the progressive first-failure-censored sampling plan reduces to the first-failure-censored sampling plan. If 𝑘=1, then the progressive first-failure-censored sampling plan reduces to the progressively type II right censored sampling plan. If 𝑘=1 and 𝑅1=𝑅2==𝑅𝑚=0, then 𝑛=𝑚 and the progressive first-failure-censored sampling plan reduces to the complete sampling plan. If 𝑘=1, 𝑅1=𝑅2==𝑅𝑚1=0, and 𝑅𝑚=𝑛𝑚, then the progressive first-failure-censored sampling plan reduces to type II right censored sampling plan (see Wu and Kuş [12]).

Hong et al. [4], and Lee et al. [5, 6] proposed the data transformation method to construct a MLE of 𝐶𝐿. In this study, the large sample in place of the data transformation method. Under the assumption of Weibull distribution, the main aim of this paper will apply the large-sample theory to construct an MLE of 𝐶𝐿 with the progressive first-failure-censored sampling plan. The MLE of 𝐶𝐿 is then utilized to develop a new hypothesis testing procedure in the condition of known 𝐿. The new testing procedure can be employed by managers to assess whether the lifetime of products adheres to the required level in the condition of known 𝐿.

The rest of this paper is organized as follows. Section 2 introduces some properties of the lifetime performance index for lifetime of product with the Weibull distribution. Section 3 discusses the relationship between the lifetime performance index and conforming rate. Section 4 then presents the MLE of the lifetime performance index and its statistical properties with Weibull distribution based on the progressive first-failure-censored sampling plan. Section 5 will apply the large-sample theory to develop a new hypothesis testing procedure for the lifetime performance index. One numerical example and concluding remarks are made in Sections 6 and 7, respectively.

2. The Lifetime Performance Index

Suppose that the lifetime (in years) of products may be modeled by a Weibull distribution. Let 𝑋 denote the lifetime of such a product and 𝑋 has the Weibull distribution with the p.d.f. as given in (1.1). Clearly, a longer lifetime implies a better product quality. Hence, the lifetime is a larger-the-better-type quality characteristic. The lifetime is generally required to exceed 𝐿 unit times to both be economically profitable and satisfy customers. Montgomery [1] developed a capability index 𝐶𝐿 for properly measuring the larger-the-better quality characteristic. 𝐶𝐿 is defined as follows: 𝐶𝐿=𝜇𝐿𝜎,(2.1) where the process mean is 𝜇, the process standard deviation is 𝜎, and 𝐿 is the lower specification limit.

To assess the lifetime performance of products, 𝐶𝐿 can be defined as the lifetime performance index. Under 𝑋 has the Weibull distribution and there are several important properties, as follows.(i)The lifetime performance index 𝐶𝐿 can be rewritten as 𝐶𝐿=𝜇𝐿𝜎=𝛼Γ(1/𝛽+1)𝐿𝛼2Γ(2/𝛽+1)𝛼2Γ2,𝐶(1/𝛽+1)𝐿<Γ(1/𝛽+1)Γ(2/𝛽+1)Γ2,(1/𝛽+1)(2.2) where the process mean 𝜇=𝐸(𝑋)=𝛼Γ(1/𝛽+1), the process standard deviation 𝜎=Var(𝑋)=𝛼2Γ(2/𝛽+1)𝛼2Γ2(1/𝛽+1), 𝐿 is the lower specification limit, and the gamma function Γ(𝜆)=0𝑦𝜆1𝑒𝑦𝑑𝑦 for 𝜆>0.(ii)The failure rate function 𝑟𝑋(𝑥) is defined by 𝑟𝑋(𝑓𝑥)=𝑋(𝑥)1𝐹𝑋=(𝑥)𝛽/𝛼𝛽𝑥𝛽1exp(𝑥/𝛼)𝛽exp(𝑥/𝛼)𝛽=𝛽𝛼𝛽𝑥𝛽1,𝑥>0,𝛼>0,𝛽>0.(2.3) When the mean 𝛼Γ(1/𝛽+1)(>𝐿), then the lifetime performance index 𝐶𝐿>0. From (2.2) and (2.3), we can see that, for example, as given 𝛽>0, the larger 𝛼 (i.e., the larger the mean 𝛼Γ(1/𝛽+1)), then the smaller the failure rate and the lager the lifetime performance index 𝐶𝐿. Therefore, the lifetime performance index 𝐶𝐿 reasonably and accurately represents the lifetime performance of new product.

3. The Conforming Rate

If the lifetime of a product 𝑋 exceeds the lower specification limit 𝐿, then the product is defined as a conforming product. The ratio of conforming products is known as the conforming rate and can be defined as 𝑃𝑟=𝑃(𝑋>𝐿)=𝐿𝛽𝛼𝛽𝑥𝛽1𝑥exp𝛼𝛽Γ1𝑑𝑥=exp𝛽+1𝐶𝐿Γ2𝛽+1Γ21𝛽+1𝛽,(3.1) where 𝐶𝐿<Γ(1/𝛽+1)/Γ(2/𝛽+1)Γ2(1/𝛽+1) and 𝛽>0.

Obviously, a strictly increasing relationship exists between conforming rate 𝑃𝑟 and the lifetime performance index 𝐶𝐿 with given 𝛽. Since a one-to-one mathematical relationship exists between the conforming rate 𝑃𝑟 and the lifetime performance index 𝐶𝐿, therefore, utilizing the one-to-one relationship between 𝑃𝑟 and 𝐶𝐿, lifetime performance index can be a flexible and effective tool, not only for evaluating product quality, but also for estimating the conforming rate 𝑃𝑟. For given 𝛽 and 𝐶𝐿, the conforming rate 𝑃𝑟 can calculated by (3.1).

4. Maximum Likelihood Estimator of Lifetime Performance Index

In lifetime testing experiments of products, the experimenter may not always be in a position to observe the lifetimes of all the items on test due to time limitation and/or other restrictions (such as money, material resources, mechanical or experimental difficulties) on data collection. In this study, we consider the case of the progressive first-failure-censored sampling plan. The progressive first-failure-censored sampling plan has an advantage in terms of shorter test time, a saving of resources, and in which a specific fraction of individuals at risk may be removed from the experiment at each of several ordered failure times.

Let 𝑋 denote the lifetime of such a product, and 𝑋 has a Weibull distribution with the p.d.f. 𝑓𝑋(𝑥) as (1.1) and c.d.f. 𝐹𝑋(𝑥) as (1.2). 𝑋1,𝑋2,,𝑋𝑚 are the progressively first-failure-censored order statistics from the Weibull distribution with censoring scheme 𝑅=(𝑅1,𝑅2,,𝑅𝑚). Since the joint p.d.f. of 𝑋1,𝑋2,,𝑋𝑚 is given as follows: 𝑓𝑋1,𝑋2,,𝑋𝑚=𝑐𝑘𝑚𝑚𝑗=1𝑓𝑋𝑗𝑋1𝐹𝑗𝑘(𝑅𝑗+1)1=𝑐𝑘𝑚𝑚𝑗=1𝛽𝛼𝛽𝑋𝑗𝛽1𝑋exp𝑗𝛼𝛽𝑘(𝑅𝑗+1),0<𝑋1𝑋2𝑋𝑚<,(4.1) where 𝑐=𝑛(𝑛𝑅11)(𝑛𝑅1𝑅22)(𝑛𝑅1𝑅2𝑅𝑚1𝑚+1), so, the likelihood function is 𝐿(𝛼,𝛽)=𝑐𝑘𝑚𝑚𝑗=1𝛽𝛼𝛽𝑋𝑗𝛽1𝑋exp𝑗𝛼𝛽𝑘(𝑅𝑗+1).(4.2) The log-likelihood function is ln𝐿(𝛼,𝛽)=ln𝑐𝑘𝑚+𝑚ln𝛽𝑚𝛽ln𝛼+(𝛽1)𝑚𝑗=1ln𝑋𝑗𝑘𝑚𝑗=1𝑅𝑗𝑋+1𝑗𝛼𝛽.(4.3) Assuming that 𝛼 and 𝛽 are both unknown, the differentiation of (4.3) with respect to 𝛼 and 𝛽 yields 𝜕ln𝐿(𝛼,𝛽)𝜕𝛼=𝑚𝛽𝛼+𝑘𝑚𝑗=1𝑅𝑗𝑋+1𝛽𝑗𝛽𝛼𝛽1,𝜕ln𝐿(𝛼,𝛽)=𝑚𝜕𝛽𝛽𝑚ln𝛼+𝑚𝑗=1ln𝑋𝑗𝑘𝑚𝑗=1𝑅𝑗𝑋+1𝑗𝛼𝛽𝑋ln𝑗𝛼.(4.4) The maximum likelihood estimator (MLE) 𝛼 of 𝛼 and the MLE ̂𝛽 of 𝛽 can be derived by solving the equations 𝑚̂𝛽𝛼+𝑘𝑚𝑗=1𝑅𝑗𝑋̂𝛽+1𝑗̂𝛽𝛼̂𝛽1𝑚=0,(4.5)̂𝛽𝑚ln𝛼+𝑚𝑗=1ln𝑋𝑗𝑘𝑚𝑗=1𝑅𝑗𝑋+1𝑗̂𝛽𝑋𝛼ln𝑗𝛼=0.(4.6) By (4.5), the MLE 𝛼 of 𝛼 is given by 𝑘𝛼=𝑚𝑗=1𝑅𝑗𝑋̂𝛽+1𝑗𝑚1/̂𝛽.(4.7) The substitution of (4.7) into (4.6) yields the equation 1̂𝛽+1𝑚𝑚𝑗=1ln𝑋𝑗𝑚𝑗=1𝑅𝑗𝑋̂𝛽+1𝑗ln𝑋𝑗𝑚𝑗=1𝑅𝑗𝑋̂𝛽+1𝑗=0.(4.8) By (4.8), the MLE ̂𝛽 of 𝛽 can be found by Newton’s method.

By using the invariance of MLE (see Zehna [15]), the MLE of 𝐶𝐿 can be written as 𝐶𝐿=̂𝛼Γ1/𝛽+1𝐿𝛼2Γ̂2/𝛽+1𝛼2Γ2̂,𝐶1/𝛽+1𝐿<Γ̂1/𝛽+1Γ̂2/𝛽+1Γ2̂,1/𝛽+1(4.9) where the MLEs 𝛼 and ̂𝛽 can be found by Newton’s method with (4.7) and (4.8).

The asymptotic normal distribution for the 𝐶𝐿 can be obtained in large-sample theory. From the log-likelihood function in (4.3), we have 𝜕2ln𝐿(𝛼,𝛽)𝜕𝛼2=𝑚𝛽𝛼2𝑘𝛽(𝛽+1)𝛼2𝑚𝑗=1𝑅𝑗𝑋+1𝑗𝛼𝛽,𝜕2ln𝐿(𝛼,𝛽)=𝜕𝛼𝜕𝛽𝑚𝛼+𝑘𝛼𝑚𝑗=1𝑅𝑗𝑋+1𝑗𝛼𝛽𝑋𝛽ln𝑗𝛼,𝜕+12ln𝐿(𝛼,𝛽)𝜕𝛽2=𝑚𝛽2𝑘𝑚𝑗=1𝑅𝑗𝑋+1𝑗𝛼𝛽𝑋ln𝑗𝛼2.(4.10) And the Fisher information matrix is given by 𝜕𝐼(𝛼,𝛽)=𝐸2ln𝐿(𝛼,𝛽)𝜕𝛼2𝜕𝐸2ln𝐿(𝛼,𝛽)𝜕𝜕𝛼𝜕𝛽𝐸2ln𝐿(𝛼,𝛽)𝜕𝜕𝛼𝜕𝛽𝐸2ln𝐿(𝛼,𝛽)𝜕𝛽2.(4.11) Under some mild regularity conditions (see Theorem 5.2.2 of Sen and Singer [16]), ̂(𝛼,𝛽) is asymptotically bivariately normal distribution with mean (𝛼,𝛽) and covariance matrix 𝐼1(𝛼,𝛽), that is, ̂(𝛼,𝛽)𝐷𝑁((𝛼,𝛽),𝐼1(𝛼,𝛽)).

Let 𝐶𝐿=𝛼Γ(1/𝛽+1)𝐿𝛼2Γ(2/𝛽+1)𝛼2Γ2(1/𝛽+1)(𝛼,𝛽).(4.12) By using the delta method (see Casella and Berger [17, page 245, Theorem 5.5.28]), we have 𝐶𝐿𝐶𝐿𝐷𝑁0,𝜕(𝛼,𝛽),𝜕𝛼𝜕(𝛼,𝛽)𝐼𝜕𝛽1(𝛼,𝛽)𝜕(𝛼,𝛽),𝜕𝛼𝜕(𝛼,𝛽)𝜕𝛽𝑇.(4.13) And by using Theorem 5.6.1 of Sen and Singer [16] (or Lawless [18, page 549]), 𝐶𝐿𝐶𝐿2̂𝛽Var𝛼,𝐷𝜒21,(4.14) where ̂𝛽=Var𝛼,𝜕(𝛼,𝛽),𝜕𝛼𝜕(𝛼,𝛽)𝜕𝛽𝛼=𝛼𝛽=̂𝛽𝐼1̂𝛽𝛼,𝜕(𝛼,𝛽),𝜕𝛼𝜕(𝛼,𝛽)𝜕𝛽𝑇𝛼=𝛼𝛽=̂𝛽,𝜕(𝛼,𝛽)=𝐿𝜕𝛼𝛼2Γ(2/𝛽+1)Γ2,(1/𝛽+1)𝜕(𝛼,𝛽)=𝜕𝛽𝛽2Γ(1/𝛽+1)Ψ(1/𝛽+1)Γ(2/𝛽+1)Γ2𝛽(1/𝛽+1)2[]𝛼Γ(1/𝛽+1)𝐿Γ(2/𝛽+1)Ψ(2/𝛽+1)+Γ2(1/𝛽+1)Ψ(1/𝛽+1)𝛼Γ(2/𝛽+1)Γ2(1/𝛽+1)3/2,(4.15) the digamma function Ψ(𝑥)=Γ(𝑥)/Γ(𝑥),𝑥>0, and the observed information matrix 𝐼̂𝛽=𝜕𝛼,2ln𝐿(𝛼,𝛽)𝜕𝛼2𝜕2ln𝐿(𝛼,𝛽)𝜕𝜕𝛼𝜕𝛽2ln𝐿(𝛼,𝛽)𝜕𝜕𝛼𝜕𝛽2ln𝐿(𝛼,𝛽)𝜕𝛽2𝛼=𝛼𝛽=̂𝛽.(4.16)

5. Testing Procedure for the Lifetime Performance Index

Construct a statistical testing procedure to assess whether the lifetime performance index adheres to the required level. Assuming that the required index value of lifetime performance is larger than 𝑐, where 𝑐 denotes the target value, the null hypothesis 𝐻0𝐶𝐿𝑐 and the alternative hypothesis 𝐻1𝐶𝐿>𝑐 are constructed.

Firstly, by using 𝐶𝐿, the MLE of 𝐶𝐿 as the test statistic, the rejection region can be expressed as {𝐶𝐿>𝐶0}. Given the specified significance level 𝛼, the critical value 𝐶0 can be calculated as follows: Sup𝐶𝐿𝑐𝑃𝐶𝐿>𝐶0𝛼,𝐶𝑃𝐿𝐶𝐿̂𝛽>𝐶Var𝛼,0𝐶𝐿̂𝛽Var𝛼,𝐶𝐿𝑐𝛼,𝐶𝑃𝐿𝐶𝐿̂𝛽>𝐶Var𝛼,0𝐶𝐿̂𝛽Var𝛼,𝐶𝐿=𝑐=𝛼,𝐶𝑃𝐿𝑐̂𝛽>𝐶Var𝛼,0𝑐̂𝛽Var𝛼,=𝛼,𝐶𝑃𝐿𝑐̂𝛽Var𝛼,2>𝐶0𝑐̂𝛽Var𝛼,2=𝛼,𝐶𝑃𝐿𝑐̂𝛽Var𝛼,2𝐶0𝑐̂𝛽Var𝛼,2=1𝛼,(5.1) where ̂𝛽=Var𝛼,𝜕(𝛼,𝛽),𝜕𝛼𝜕(𝛼,𝛽)𝜕𝛽𝛼=𝛼𝛽=̂𝛽𝐼1̂𝛽𝛼,𝜕(𝛼,𝛽),𝜕𝛼𝜕(𝛼,𝛽)𝜕𝛽𝑇𝛼=𝛼𝛽=̂𝛽(5.2) and under 𝐶𝐿𝐶=𝑐,𝐿𝑐̂𝛽Var𝛼,2𝐷𝜒2(1).(5.3) From (5.1), by utilizing CHIINV(1𝛼) function which represents the lower 100(1𝛼)th percentile of 𝜒2(1), then 𝐶0𝑐̂𝛽Var𝛼,2=CHIINV1𝛼(5.4) is obtained. Thus, the following critical value can be derived: 𝐶0=𝑐+̂𝛽Var𝛼,CHIINV(1𝛼),(5.5) where 𝑐 and 𝛼 denote the target value and the specified significance level, respectively.

The managers can then employ the one-sided hypothesis testing to determine whether the lifetime performance index adheres to the required level. The proposed testing procedure about 𝐶𝐿 can be organized as follows.

Step 1. Determine the lower lifetime limit 𝐿 for products and performance index value 𝑐; then the testing null hypothesis 𝐻0𝐶𝐿𝑐 and the alternative hypothesis 𝐻1𝐶𝐿>𝑐 are constructed.

Step 2. Specify a significance level 𝛼.

Step 3. Calculate the value of test statistic 𝐶𝐿=̂𝛼Γ1/𝛽+1𝐿𝛼2Γ̂2/𝛽+1𝛼2Γ2̂,𝐶1/𝛽+1𝐿<Γ̂1/𝛽+1Γ̂2/𝛽+1Γ2̂,1/𝛽+1(5.6) where the MLEs 𝛼 and ̂𝛽 can be found by Newton’s method with (4.7) and (4.8).

Step 4. Obtain the critical value 𝐶0=𝑐+̂𝛽Var𝛼,CHIINV(1𝛼),(5.7) where ̂𝛽=Var𝛼,𝜕(𝛼,𝛽),𝜕𝛼𝜕(𝛼,𝛽)𝜕𝛽𝛼=𝛼𝛽=̂𝛽𝐼1̂𝛽𝛼,𝜕(𝛼,𝛽),𝜕𝛼𝜕(𝛼,𝛽)𝜕𝛽𝑇𝛼=𝛼𝛽=̂𝛽(5.8) and 𝑐 and 𝛼 denote the target value and the specified significance level.

Step 5. The decision rule of statistical test is provided as follows.
If 𝐶𝐿>𝐶0, it is concluded that the lifetime performance index of product meets the required level.
Based on the proposed testing procedure, the lifetime performance of products is easy to assess. One numerical example of the proposed testing procedure is given in Section 6, and the numerical examples illustrate the use of the testing procedure.

6. Numerical Examples

In this section, we propose the new hypothesis testing procedure to one simulated large-sample data set. Example 6.1 considered is a simulated large-sample data with 𝑘=5, 𝑛=50, and 𝑚=30 from a Weibull distribution.

Example 6.1. The following data are the progressive first-failure-censored sample of a computer generated from a Weibull distribution with p.d.f. as given in (1.1) and 𝛼=40, 𝛽=1, 𝐿=4, 𝐶𝐿=0.9. The simulated progressive first-failure-censored sample and the simulated progressive first-failure-censored scheme are given as follows:
{𝑥𝑖,𝑖=1,,30}={0.10971, 0.11117, 0.78476, 1.27366, 1.30471, 1.78242, 1.85144, 1.88851, 2.70589, 2.93703, 3.53395, 3.65632, 3.76333, 4.10132, 4.50531, 4.94733, 5.06265, 7.04528, 7.52044, 8.08150, 9.07310, 9.27218, 10.6786, 11.7043, 12.4732, 13.1637, 13.8520, 13.9263, 14.7226, 19.5564}, 𝑅=(0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 3, 0, 0, 5, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 1), 𝑘=5, 𝑛=50, and 𝑚=30.
Then, we also state the proposed testing procedure about 𝐶𝐿 as follows.

Step 1. The lower lifetime limit 𝐿 is assumed to be 4 by the simulation condition 𝐿=4, that is, if the lifetime of a product exceeds 4, then the product is defined as a conforming product. To deal with the product managers’ concerns regarding operational performance, the conforming rate 𝑃𝑟 of operational performances is required to exceed 80 percent. By the simulation condition 𝛽=1 and (3.1), the 𝐶𝐿 value of the operational performances is required to exceed 0.78. Thus, the performance index value is set at 𝑐=0.78. The testing hypothesis 𝐻0𝐶𝐿0.78 versus 𝐻1𝐶𝐿>0.78 is constructed.

Step 2. Specify a significance level 𝛼=0.05.

Step 3. Calculate the value of test statistic 𝐶𝐿=̂𝛼Γ1/𝛽+1𝐿𝛼2Γ̂2/𝛽+1𝛼2Γ2̂=1/𝛽+140.3104Γ(1/1.17825+1)440.31042Γ(2/1.17825+1)40.31042Γ2(1/1.17825+1)=1.30537,(6.1) where the MLEs 𝛼=40.3104 and ̂𝛽=1.17825 can be found by Newton’s method with (4.7) and (4.8).

Step 4. Obtain the critical value 𝐶0=0.78+Var((40.3104,1.17825))CHIINV(10.05)=0.78+0.013501×3.841=1.00774,(6.2) according to 𝐼01(40.3104,1.17825)=0.0256311.295101.2951098.94971=,115.2061.507861.507860.029842𝜕(𝛼,𝛽),𝜕𝛼𝜕(𝛼,𝛽)𝜕𝛽𝛼=40.3104̂𝛽=1.17825=[],[]0.003057811,0.81798Var((40.3104,1.17825))=0.003057811,0.81798115.2061.507861.507860.0298420.0030578110.81798=0.013501,(6.3) the target value 𝑐=0.78, and the significance level 𝛼=0.05.

Step 5. Because 𝐶𝐿=1.30537>𝐶0=1.00774, so we do reject to the null hypothesis 𝐻0𝐶𝐿0.78. Thus, we can conclude that the lifetime performance index of product does meet the required level.

7. Conclusions

Process capability analysis has been widely applied in the field of quality control to monitor the performance of industrial processes. In practice, lifetime performance index 𝐶𝐿 is a popular means to assess the performance and potential of their processes, where L is the lower specification limit. Moreover, in life testing experiments, the experimenter may not always be in a position to observe the life times of all the businesses (or items) put on test. This may be because of time limitation and/or other restrictions (such as money, material resources, mechanical or experimental difficulties) on data collection. Therefore, censored samples may arise in practice. The progressive first-failure-censored sampling plan has an advantage in terms of shorter test time, a saving of resources, and in which a specific fraction of individuals at risk may be removed from the experiment at each of several ordered failure times. The familiar complete, type II right censored, first-failure-censored, and progressively type II right censored samples are special cases of the progressive first-failure-censored sampling plan. The Weibull distribution has been recognized as a useful model for the analysis of lifetime data. So, we consider the case of the progressive first-failure-censored sampling plan, and our study applied the large-sample theory to construct an MLE of 𝐶𝐿 under the Weibull distribution. Moreover, the MLE of 𝐶𝐿 is utilized to develop a new testing procedure for the performance index of products. The new hypothesis testing procedure is a quality performance assessment system in Enterprise Resource Planning (ERP). The managers can then employ the new testing procedure to determine whether the lifetime performance of products adheres to the required level. The managers can also utilize this procedure to enhance product process capability.

Acknowledgments

The authors wish to thank the referee for valuable suggestions which led to the improvement of this paper. This paper was partially supported by the National Science Council, Taiwan (Plan no. NSC 100-2221-E-158-004).

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