1. INTRODUCTION
The existence of an intensely competitive business environment and the general trend toward higher expectations of the quality of products obliges manufacturers to monitor the quality of their products in the most efficient and economical way possible. When the inspection is for the purpose of lot sentencing, in light of conformance to a standard, the method utilized is typically called acceptance sampling (Montgomery, 2013). Acceptance sampling is among the primary techniques of statistical process control (SPC). Judicious use of it can supplement and support other applications of SPC (Schilling and Neubauer, 2009). The use of acceptance sampling on its own provides a proven resource for the evaluation of products. An acceptance sampling scheme comprises the acceptance or rejection criteria and the sample size.
When the fraction of defects is very low, the sample size required becomes very large in order to effectively reveal the true quality of the lot. To reduce the sample size, yield index based variable sampling plans have been proposed by different authors including Pearn and Wu (2006, 2007), Wu and Pearn (2008) and Wu and Liu (2014). These plans are proven resources for lot sentencing.
However, the inspection efficiency is still relatively low. For example, if the quality level that the consumer desires is S_{pk} = 1.50 and the lowest quality level the consumer is willing to accept is S_{pk} = 1.33 , this requires 546 samples to achieve a producer risk of 0.01 and a consumer risk of 0.05. To improve the inspection efficiency, the accumulated quality history from previous lots might be included. The EWMA considers the past and current lot information. The weights decay geometrically; the older the observation, the smaller the weight. This EWMA measurement has been widely utilized in control charts, which ended up being effective at identifying small shifts (Čisar and Čisar, 2011;Hunter, 1986;Lucas and Saccucci, 1990;Montgomery, 2013). Aslam et al. (2013) introduced an acceptance sampling scheme using the EWMA yield index. Yen et al. (2014) provided a single variable plan using the EWMA process yield index S_{pk} and Aslam et al. (2015b) employed the EWMA method for the quality characteristic. Aslam et al. (2015a) and Aslam et al. (2016) proposed variable repetitive and mixed sampling plans based on the EWMA statistic, assuming that the quality characteristics follow the normal distribution. Recently, Afzal et al. (2017) developed a yield index based multiple dependent repetitive sampling plan for supplier selection. However, the above methods are based on the basic assumption that the measurement data is independent and they consider only a single quality characteristic, and therefore cannot be applied to either dependent or profile data.
Noorossana et al. (2011a) discussed standard SPC applications, where the concern is to monitor a product or process based on measurements of a single or a vector of quality characteristics at a given space or time. However, advances in technology have permitted the gathering of extensive measurements to reconstruct the whole functional relationship for the product or process performance. The functional relationship is typically referred to as a profile. Figure 1 illustrates linear profiles with four levels of the independent variable, each level having fixed values. On the monitoring of linear profiles, Woodall (2007) provided a summary of research ideas. With the assumption that the process data are uncorrelated, many investigations have been done on the monitoring of simple nonlinear/ linear profiles (Chuang et al., 2013;Ghahyazi et al., 2014;Li and Wang, 2010;Noorossana et al., 2010;Noorossana et al., 2011b). For simple nonlinear and linear profiles, a processyield index S_{pkA} and its normal approximation are proposed by Wang and Guo (2014) and Wang (2014), respectively. The authors assume that the process data is independent.
For linear profiles with the independence assumption, Wang (2016) provides an EWMA based sampling scheme. However, in continuous manufacturing processes data are often autocorrelated. Wang and Tamirat (2014) proposed a processyield index S_{pkA; AR}_{(1)} and the corresponding lower confidence bound to deal with the presence of autocorrelation between profiles. The authors found that process autocorrelations affected the standard error of the yield index. That is, the method proposed by Wang (2016) cannot be directly applied to correlated data. To the best of our knowledge, there is no detailed study on the acceptance sampling plans to deal with the firstorder autoregressive model between linear profiles.
In this manuscript, we propose a new variable single sampling scheme for economic appraisal of materials. In the presence of a firstorder autoregressive between linear profiles, we present a variable single sampling scheme using the EWMA method with a yield index. Taking into account the acceptable quality level (AQL) at the producer’s risk and the lot tolerance percent defective (LTPD) at the consumer’s risk, a nonlinear optimization method is proposed to determine the number of profiles required for inspection and the corresponding acceptance or rejection criteria. The flow of this paper is as follows: first, a process yield index S_{pkA; AR}_{(1)} is summarized, and then section 3 describes the proposed sampling scheme using the EWMA method. Section 4 compares the performance of the new plan and the single sampling plan proposed by Wang (2016). In section 5 a simulated example is used to demonstrate the applicability of the new scheme. Conclusions and suggestions for future studies are offered in section 6.
2. PROCESS YIELD INDEX FOR LINEAR PROFILES WITH TWOSIDED SPECIFICATIONS
In this section, we review the process yield index for a firstorder autoregressive between linear profiles. Let y_{ij} represent the quality characteristics of a process, which is the response variable. Thus, for linear profiles the firstorder autocorrelation is modeled as:(1)
where x_{i} is the i^{th} level of the independent variable, n is the number of levels for x_{i}, k means the number of profiles, ε_{ij} is correlated random error, α and β represent the intercept and slope of linear profiles respectively, ρ is the autocorrelation coefficient, and ${a}_{ij}\sim N(0,\hspace{0.17em}{\sigma}^{2})$.
For a process with the twosided specification, let USL_{i} and LSL_{i} denote the upper and lower specification limits of the response variable at the i^{th} level of x_{i} . At the i^{th} level of x_{i} , μ_{i} and ${\sigma}_{i}{}^{2}$ are the process mean and the variance. The relationship between manufacturing specifications and actual process performance can be derived from the method proposed by Boyles (1994). The index at the i^{th} level of x_{i} is defined as follows:
where ${C}_{d{r}_{i}}=\left({\mu}_{i}{m}_{i}/{d}_{i}\right),\hspace{0.17em}\hspace{0.17em}{C}_{dpi}={\sigma}_{i}/{d}_{i},\hspace{0.17em}{m}_{i}=USL{L}_{i}+LS{L}_{i}/2,\hspace{0.17em}{d}_{i}=US{L}_{i}LS{L}_{i}/2$, Φ is the cumulative distribution function of a standard normal distribution, andΦ−1 is the inverse function of Φ. For instance, Figure 1 considers a linear profile with four levels (n = 4), and 25 profiles (k = 25). The upper and lower specification limits, the location and scale parameters at each level are shown in Figure 1. The yield index at each level is determined using Equation (2); the corresponding process yield is given as: ${p}_{i}=2\Phi (3{S}_{pki})1$. The total process yield for linear profiles with n levels can be defined as:(3)
For a firstorder autoregressive between linear profiles, Wang and Tamirat (2014) derived the estimator in Equation (4). Table 1 lists various ${S}_{pkA;AR(1)}$ values, with the related process yield and nonconformity in parts per million (ppm). For practical applications, a process is classified as shown in Table 1.
where
$\begin{array}{l}{\widehat{S}}_{pki}=\frac{1}{3}{\text{\Phi}}^{1}\left[\frac{1}{2}\text{\Phi}\left(\frac{US{L}_{i}{\overline{y}}_{i}}{{S}_{i}}\right)+\frac{1}{2}\Phi \left(\frac{{\overline{y}}_{i}LS{L}_{i}}{{S}_{i}}\right)\right]\\ =\frac{1}{3}{\text{\Phi}}^{1}\left[\frac{1}{2}\text{\Phi}\left(\frac{1{\widehat{C}}_{d{r}_{i}}}{{\widehat{C}}_{d{p}_{i}}}\right)+\frac{1}{2}\text{\Phi}\left(\frac{1+{\widehat{C}}_{d{r}_{i}}}{{\widehat{C}}_{d{p}_{i}}}\right)\right]\end{array}$
${\widehat{C}}_{d{r}_{i}}=\left({\overline{y}}_{i}{m}_{i}/{d}_{i}\right),\hspace{0.17em}{\widehat{C}}_{dpi}={S}_{i}/{d}_{i},\hspace{0.17em}{\overline{y}}_{i}$ and ${S}_{i}{}^{2}$ are the sample mean and variance at the i^{th} level of the independent variable, which might be acquired from a steady process, and ${\widehat{S}}_{pkA;\hspace{0.17em}AR(1)}$ is the estimator of the processyield index ${S}_{pkA;AR(1)}$.
The exact distribution of ${\widehat{S}}_{pkA;\hspace{0.17em}AR(1)}$ is numerically difficult to find, as it includes a complex function of the sample mean and the sample variance. Wang and Tamirat (2014) applied the firstorder expansion of a onedimensional Taylor series and central limit theorem to derive the asymptotic normal distribution of index ${\widehat{S}}_{pkA;\hspace{0.17em}AR\left(1\right)}$ and this is given as follows:(5)
where $\begin{array}{l}{a}_{i}={d}_{i}/\sqrt{2}{\sigma}_{i}\left\{(1{C}_{d{r}_{i}})\varphi \left(\frac{1{C}_{d{r}_{i}}}{{C}_{d{p}_{i}}}\right)+(1+{C}_{d{r}_{i}})\varphi \left(\frac{1+{C}_{d{r}_{i}}}{{C}_{d{p}_{i}}}\right)\right\}\\ {b}_{i}=\varphi \left(\frac{1{C}_{d{r}_{i}}}{{C}_{d{p}_{i}}}\right)\varphi \left(\frac{1+{C}_{d{r}_{i}}}{{C}_{d{p}_{i}}}\right),f=1\frac{2}{k(k1)}{\displaystyle \sum _{i=1}^{k1}(ki)}{\rho}_{i},\\ g=1+\frac{2}{k}{\displaystyle \sum _{i=1}^{k1}(ki)}{\rho}_{i},\hspace{0.17em}F=k+2{\displaystyle \sum _{i=1}^{k1}(ki){\rho}^{2i}}\\ +\frac{1}{{k}^{2}}{\left[k+2{\displaystyle \sum _{i=1}^{k1}(ki){\rho}^{i}}\right]}^{2}\frac{2}{k}{\displaystyle \sum _{i=0}^{k1}{\displaystyle \sum _{j=0}^{ki}(kij){\rho}^{i}}}{\rho}^{j},\hspace{0.17em}{\rho}_{i}\end{array}$ is the i^{th} lag autocorrelation, and ϕ is the probability density function of a standard normal distribution.
3. PROPOSED SAMPLING PLAN FOR AUTOCORRELATION BETWEEN LINEAR PROFILES
In this study, two quality levels are considered for determining the plan parameters. The process yield level desired by the consumer (C_{AQL}) and the worst level of process yield the consumer can accept (C_{LTPD}). Thus, the parameters for the sampling plan are determined by the operating characteristics curve, which passes through (C_{AQL}, 1−α) and (C_{LTPD}, β) , where α and β represent the producer’s and consumer’s risk respectively.
In some situations, the relationship between a supplier and buyer is longterm; the quality history from preceding submitted lots is available. To take advantage of such information availability, we propose the variable sampling scheme using the EWMA yield index. The sampling procedure is described below:

Step 1: Select (α) and (β) and the process capability requirements (C_{AQL}, C_{LTPD}) at two risks respectively.

Step 2: Select a random number of k profiles at the current time t and collect the preceding acceptance lots with their yield index values. Then, compute the following EWMA sequence, say Z_{t} for t = 1, 2, 3, …, T.
where λ ranging from 0 to 1 is a smoothing constant, and the choice of λ value is based on minimizing the aggregate of the square errors, $SSE={\displaystyle {\sum}_{t=2}^{T}{\left({Z}_{t}{\widehat{S}}_{pkA;AR(1);t}\right)}^{2}}$, where ${Z}_{2}={\widehat{S}}_{pkA;AR(1);1}$ (Hunter, 1986). The optimal λ value is found using R Development Core Team (2017) and the DEoptim algorithm (Ardia et al., 2011). The code is available on request from the authors.

Step 3: Accept the lot from the supplier if Z_{t} ≥ c_{a}.

Step 4: Reject the lot if Z_{t} < c_{a} , where c_{a} is the critical value.
The OC function of our proposed sampling scheme is given as follows:
In Equation (6), the mean and variance of Z_{t} can be obtained
as $E({Z}_{t})={S}_{pkA;AR(1)}+{\displaystyle \sum _{i=1}^{n}\frac{{a}_{i}(1f)}{12n\varphi (3{S}_{pkA;AR(1)})}}$ and $Var({Z}_{t})=\left(\frac{\lambda}{2\lambda}\right){\displaystyle \sum _{i=1}^{n}\left[\frac{k{a}_{i}{}^{2}F}{{\left(k1\right)}^{2}}+{b}_{i}{}^{2}g\right]}/36{n}^{2}k\varphi {(3{S}_{pkA;AR(1)})}^{2}.$. Therefore, Equation (7) can be rewritten as Equation (8):
where Z is a standard normal random variable. The lot acceptance probability, say ${\pi}_{A}\left({Z}_{t}\right)$, is derived as follows:(9)
The parameters of the proposed scheme can be found using the nonlinear optimization technique given in Equation (10), where the number of profiles (k) and critical value (c_{a}) are decision variables. For a particular sampling plan, the producer is interested in finding the probability that a type I error can be committed. Using Equation (10), the producer may find a sampling scheme which ensures that the lot acceptance probability is greater than the desired confidence level, 1−α, the process yield level desired by the consumer (C_{AQL}). Concurrently the consumer desires that, based on sample information, the probability that a bad (quality) population will be accepted is smaller than β. That is, ${S}_{pkA;\hspace{0.17em}AR(1)}={C}_{AQL}$ for the producer and ${S}_{pkA;AR(1)}={C}_{LTPD}$ for the consumer.
Given ρ, λ, C_{AQL}, C_{LTPD}, α, and β as inputs, we evaluate the constraints (10b) and (10c), with the objective of minimizing the number of profiles required for inspection. A search procedure is applied to determine the plan parameters. To determine critical values and the number of profiles, a computer program is written in R Core Team (2017); the code is available from the authors. First, 1000 combinations of k and c_{a} are randomly generated, where k ranges from 2 to 3000 and c_{a} follows a uniform distribution from C_{AQL} to C_{LTPD}. The above procedure is repeated 1000 times. In Tables 2, 3, and 4, we tabulate the sampling plan parameters for various combinations of (C_{AQL}, C_{LTPD}) at α = 0.05 and β = 0.10. The sampling parameters are found under a given λ = 0.10, 0.20, 0.50, and 1.0, considering two different autocorrelation coefficients ρ = 0.5 and 0.75 and two levels of the independent variable n = 4 and n = 10, respectively. The number of profiles required for acceptance or rejection with a smoothing parameter λ < 1 is smaller than the traditional single sampling plan (λ = 1 ). The smaller the value of the smoothing constant, the lower the required number of profiles for inspection. In practice, relatively small values of λ generally work best when the EWMA is the most appropriate model.
For instance, when C_{AQL} = 1.5, C_{LTPD} = 1.2, and n = 4 , at given values of α = 0.05 , β = 0.10 , and ρ = 0.5 , the plan parameters (k and c) obtained with λ = 0.10, 0.20, and 0.50 are (7 and 1.4314), (22 and 1.3708), and (172 and 1.3366), respectively. In addition, with a givenρ = 0.75 , we found that the plan parameters (k and c_{a}) obtained are (25 and 1.4248), (97 and 1.3575), and (816 and 1.3350), respectively. Examining Tables 2 – 4, it is apparent that increasing the autocorrelation coefficient from 0.5 to 0.75 significantly increases the number of profiles required. Increasing the number of levels from 4 to 10 reduces the number of profiles required, to achieve the anticipated levels of protection for consumers and producers.
4. COMPARISON STUDY
This section compares the proposed sampling scheme and an existing single sampling plan. Figure 2 displays the discriminatory power of the sampling scheme under fixed values of n = 100, c_{a} = 1.33, and ρ = 0.5. Figure 2 shows that the new sampling scheme is superior to the variable single acceptance sampling plan (λ =1) in terms of power. That is, the steeper the slope, the superior the power. Also, the smaller the value of λ , the better the power. For example, if the quality level desired by the consumer is Spk =1.50 , the probability of acceptance at λ values of 0.25, 0.5, 0.75, and 1.0 becomes 1.0, 0.99, 0.96, and 0.91 respectively. That is, the associated producer’s risk turns out to be 0, 0.01, 0.04, and 0.09 respectively. Moreover, Figure 3 compares the efficiency of the new sampling scheme with a traditional single sampling plan (λ =1) at C_{AQL} = 1.50, α = 0.01, and β = 0.05 for several values of C_{LTPD}. The result indicates that the new plan requires a smaller number of profiles; that is, it is more efficient.
5. AN ILLUSTRATIVE EXAMPLE
To demonstrate the applicability of the new acceptance sampling scheme, we considered a case from (Natrella, 2010). The three control measurements on the line widths of three photomask reference standards were measured over six days with an optical imaging system that had been calibrated from similar measurements on ten artifacts. Measurements were taken at the lower, midpoint, and upper end of the calibration interval. The phase I analysis resulted in a control process with estimation of profile parameters as given in Equation (11) (Natrella, 2010).
The upper and lower specification limits are shown in Table 5. We assume that the contract is formulated by the supplier and the consumer and we assume that the values of C_{AQL} and C_{LTPD} are agreed to be 1.50 and 1.25 at two levels of risk,α = 0.025 and β = 0.05, respectively. The sampling plan must provide a probability of at least 1−α of accepting the lot if the lot proportion defective is at C_{AQL} = 1.5 and a probability of acceptance of no more than β if the lot proportion defective is at C_{LTPD} = 1.25. From the quality history of 10 previously accepted lots, the optimal value of λ is found to be 0.1273. In light of the quality levels in the agreement, and using Equations (10a) – (10c), the plan parameters became c = 1.2827 and k = 36. Thus, a data set of 36 profiles is measured for further inspection. The estimated value of ${\widehat{S}}_{pkA;AR(1)}$ obtained is 1.5116. Since Z_{t} =1.4637 > c_{a} =1.2827 , we can conclude that the current calibration system can be accepted.
6. CONCLUSION
We developed a new variable sampling scheme using the process yield index ${S}_{pkA;\hspace{0.17em}AR(1)}$ to manage the lot acceptance or rejection problem for the firstorder autoregressive between linear profiles. With a given λ =1 , the proposed plan is reduced to a traditional single sampling for linear profiles. Since the proposed method takes the quality history of the previous lot’s information into consideration, subsequently the number of profiles required for inspection is smaller than the yield index based single variable sampling plan. In addition, we tabulated the required number of profiles k and the critical acceptance value c_{a} for various combinations of two quality levels (C_{AQL}, C_{LTPD}) at α = 0.05 and β = 0.10 and with λ = 0.10, 0.20, 0.50, and 1.0 and ρ = 0.5 and 0.75 under n = 4 and n = 10. Finally, an example is provided to illustrate the practical application of the proposed method. Multivariate and nonlinear profiles can be included in future studies. The proposed method can also be extended to other types of autocorrelation structures, such as moving average and autoregressive integrated moving average models.