BUS606: Operations and Supply Chain Management

Process capability indices have been used in industries to measure the performance of the processes. A process is called capable if the product meets customer expectations. There are copious studies on process capability estimations. Kotz and Johnson reviewed the studies on process capability indices during the years 1992–2000. Yum and Kim provided a bibliography of approximately 530 journal papers and books published during the years 2000–2009. The process capability indices $C_{\mathrm{p}}$, $C_{\mathrm{pk}}$ and $C_{\mathrm{pm}}$ are the most commonly used indices. The process capability index $C_{\mathrm{p}}$ was proposed by Juran. This process capability index cannot evaluate the capability of the process properly when the process mean and target value are not equal. Kane introduced $C_{\mathrm{pk}}$ index to solve this problem. Hsiang and Taguchi proposed $C_{\mathrm{pm}}$ index. The $C_{\mathrm{pm}}$ process capability index considers the deviation of the process mean from its target value as well as the process variance. This index shows the ability of the process to manufacture products around the target value. Pearn et al. proposed the process capability index $C_{\mathrm{pkm}}$ that have a same link between $C_{\mathrm{p}}$ and $C_{\mathrm{pk}}$with $C_{\mathrm{pm}}$. Boyles proposed $S_{\mathrm{pk}}$ index based on the process yield of the process. Sharma and Rao employed define, measure, analysis, improvement and control (DMAIC) approach to reduce the variation of the engine connecting rod machining process and as a result to improve the capability of the process. Also, Sharma and Rao used DMAIC approach to improve the capability of the engine crankshaft manufacturing process.

The process capability indices are proposed under the assumption that each stage is independent from the other stages. While in the multistage processes, the process in each stage is affected by the processes in the previous stages. Therefore, the capability of the process in each stage is dependent on the capability of the processes in the previous stages and the process capability indices calculate the overall capability of the process in this stage. Most of the researches on multistage processes are related to control chart and multistage process capability analyses are not studied much. Zhang was the first who designed a control charts for multistage processes. The cause-selecting control charts are developed by Zhang for multistage processes. Zhang developed the cause-selecting control chart. Zhang proposed two kinds of process capability indices in each stage of multistage processes including total and specific process capability indices. Total process capability index calculates the process capability in each stage when the quality characteristic in the current stage is affected by the quality characteristics of precedent processes. Specific process capability index calculates the capability of the process in the current stage when the effects of precedent processes are omitted. The total and specific process capability indices are calculated by Eq. (1).

$\begin{aligned} C_{\mathrm{pt}} &=\frac{\mathrm{USL}-\mathrm{LSL}}{6 \sigma_{t}} \\ C_{\mathrm{ps}} &=\frac{\mathrm{USL}-\mathrm{LSL}}{6 \sigma_{s}} \end{aligned}$                     (1)

where USL and LSL are the upper and lower specification limits, respectively, $\sigma_{t}$ is the total standard deviation of the process in each stage and $\sigma_{s}$ is the specific standard deviation of process that is obtained from cause-selecting control chart. Wade and Woodall proposed using prediction limits to improve the statistical performance of cause-selecting control charts. Yang proposed a new approach to compute the cost model for a two-stage process. Then, he designed economic X chart and cause-selecting control chart to monitor a two-stage process. Yang and Yang proposed an approach to monitor two-stage process when data are autocorrelated. Yang and Yeh proposed a cause-selecting control chart for the two-stage process with attribute data. Ghahyazi et al. examined the effects of cascade property in multistage process in monitoring simple linear profile in Phase II. Davoodi et al. monitored multistage Poisson count processes using exponentially weighted moving average (EWMA) and C control charts and proposed a changed point estimator to find the real time of a change in the process parameters. Pirhooshyaran and Niaki proposed a double-max multivariate exponentially weighted moving average (MEWMA) control chart to monitor the parameters of the multistage processes when data are multivariate and autocorrelated. Goodarzi et al. developed two cause-selecting control charts to monitor the censored lognormal reliability data in a three-stage process. Linn et al. proposed a method to determine the priority of the process variation reduction in multistage processes to improve the overall capability index. Chen et al. proposed a process capability index CPMPCI for the complex product machining process (multistage process) based on Taguchi loss function.

In the multistage processes, the process capability indices are calculated when the process is in control. The conditions of the processes are determined using case-selecting control charts. As mentioned former, in a two-stage process, the quality characteristic in the first stage affects the quality characteristic in the second stage referred to as the cascade property. Therefore, using the traditional process capability index to measure the capability of the second stage may provide misleading results due to ignoring the cascade effect. In other words, if one is going to measure the capability of the second stage specifically, she cannot use the traditional process capability indices on the quality characteristic in the second stage and she must use a statistic independent from the quality characteristic in the first stage. This is the main point which is not considered in any studies so far.

In this paper, we propose a method to eliminate the effects of the first stage on the quality characteristic of the second stage and calculate the specific capability of the second stage in a two-stage process. In our method, the process capability indices are calculated for the residuals in the second stage and these indices represents the specific capability of that stage instead of the overall capability. The residuals in the second stage are independent from the quality characteristic in the first stage; therefore, the process capability indices for the residuals calculate the specific capability of the second stage. In addition, a method is proposed to estimate the specification limits of the residuals based on the specification limits of the quality characteristics in the first and second stages. The performance of the proposed method is evaluated for two-stage processes with different mean and standard deviation values of the quality characteristic in the first and second stages. The structure of the paper is as follows: The proposed method is outlined in the following section. A simulation study to evaluate the performance of the proposed method is presented in the next section which is followed by a section on a real case study and finally conclusion is presented in the last section.