Production quality and yield assurance for processes with multiple independent characteristics

Abstract

Process capability indices have been widely used in the manufacturing industry providing numerical measures on process potential and process performance. Capability measure for processes with single characteristic has been investigated extensively, but is comparatively neglected for processes with multiple characteristics. In real applications, a process often has multiple characteristics with each having different specifications. Singhal [Singhal, S.C., 1990. A new chart for analyzing multiprocess performance. Quality Engineering 2 (4), 397–390] proposed a multi-process performance analysis chart (MPPAC) for analyzing the performance of multi-process product. Using the same technique, several MPPACs have been developed for monitoring processes with multiple independent characteristics. Unfortunately, those MPPACs ignore sampling errors, and consequently the resulting capability measures and groupings are unreliable. In this paper, we propose a reliable approach to convert the estimated index values to the lower confidence bounds, then plot the corresponding lower confidence bounds on the MPPAC. The lower confidence bound not only gives us a clue minimum actual performance which is tightly related to the fraction of non-conforming units, but is also useful in making decisions for capability testing. A case study of a dual-fiber tip process is presented to demonstrate how the proposed approach can be applied to in-plant applications.

Introduction

During the last decade, numerous process capability indices (PCIs), including C_p, C_a, C_pu, C_pl, and C_pk, have been proposed in the manufacturing industry to provide numerical measures on process performance, which are effective tools for quality/reliability assurance (see Kane, 1986, Chan et al., 1988, Pearn et al., 1992, Pearn et al., 1998, Kotz and Lovelace, 1998, Kotz and Johnson, 2002 for more details). These indices are defined as

$$\begin{array}{cc}\hfill & C_p=\frac{\mathrm{USL}-\mathrm{LSL}}{6\sigma }\text{,}{C}_{\mathit{pu}}=\frac{\mathrm{USL}-\mu }{3\sigma }\text{,}\hfill \\ \hfill & C_{\mathit{pl}}=\frac{\mu -\mathrm{LSL}}{3\sigma }\text{,}{C}_a=1-\frac{|\mu -m|}{d}\text{,}\hfill \\ \hfill & C_{\mathit{pk}}=\min \left\{\frac{\mathrm{USL}-\mu }{3\sigma }\text{,}\frac{\mu -\mathrm{LSL}}{3\sigma }\right\}\text{,}\hfill \end{array}$$

where USL and LSL are the upper and the lower specification limits, respectively, μ is the process mean, σ is the process standard deviation, m = (USL + LSL)/2 is the mid-point of the specification interval and d = (USL − LSL)/2 is half the length of the specification interval. For normally distributed processes, C_p, C_a and C_pk indices are appropriate measures for processes with two-sided specifications. The index C_p measures the overall process variation relative to the specification tolerance, therefore only reflects process potential (or process precision). The index C_a measures the degrees of process centering, which alerts the user if the process mean deviates from its center. Therefore, the index C_a only reflects process accuracy. The index C_pk takes into account process variation as well as process centering, providing process performance in terms of yield (proportion of conformities). Given a fixed value of C_pk, the bounds on process yield p can be expressed as 2Φ(3C_pk) − 1 ⩽ p ⩽ Φ(3C_pk) (Boyles, 1991), where Φ(·) is the cumulative distribution function of the standard normal distribution. For instance, if C_pk = 1.00, then it guarantees that the yield will be no less than 99.73%, or equivalent to no more than 2700 parts per million (ppm) of non-conformities. On the other hand, the indices C_pu and C_pl have been designed particularly for processes with one-sided manufacturing specifications, which measure the-smaller-the-better and the-larger-the-better process capabilities, respectively. For normally distributed processes with one-sided specification limit, USL or LSL, the relationship between the one-sided capability indices and the process yield can be calculated as p_u = P(X < USL) = Φ(3C_pu) and p_l = P(X > LSL) = Φ(3C_pl).

In factory applications a product usually has multiple characteristics with each having different specifications, which need to be monitored and controlled hence is a difficult and time-consuming task for factory engineers. A multi-process performance analysis chart (MPPAC) proposed by Singhal (1990), which evaluates the performance of a multi-process product with symmetric bilateral specifications. Singhal (1991) further presented a MPPAC with several well-defined capability zones by using the process capability indices C_p and C_pk for grouping the processes in a multiple process environment into different performance categories on a single chart. Using the same technique, several modified control charts have been developed for monitoring processes with single or multiple independent characteristics. Pearn and Chen (1997) proposed a modification to the C_pk MPPAC combining the more-advanced process capability indices, C_pm or C_pmk, to identify the problems causing the processes failing to center around the target. By combining Singhal’s MPPAC with asymmetric process capability index C_pa, Chen et al. (2001) introduced a process capability analysis chart (PCAC) to evaluate process performance for an entire product composed of multiple characteristics with symmetric and asymmetric specifications. Pearn et al. (2002) introduced a MPPAC to the chip resistors applications based on the incapability index C_pp. Chen et al. (2003) also developed a control chart for processes with multiple characteristics based on the generalization of yield index S_pk proposed by Boyles (1994). We should note that the process mean μ and the process variance σ² are usually unknown in practice. In order to calculate the index value, sample data must be collected and a great degree of uncertainty may be introduced into capability assessments due to sampling errors. However, those existing research works on MPPAC are restricted to assuming the value of μ and σ² are known or obtaining quality information from one single sample of each process ignoring sampling errors. The information provided from the existing MPPAC, therefore, is unreliable and misleading resulting in incorrect decisions. In this paper, we propose a reliable approach to obtain the lower confidence bounds and apply it to the modified C_pk MPPAC. A real-world application to the dual-fiber tips manufacturing process is presented for illustration.