Fuzzy process capability indices with asymmetric tolerances

Abstract

Process performance can be analyzed by using process capability indices (PCIs), which are summary statistics to depict the process location and dispersion successfully. Traditional PCIs are generally used for a process which has a symmetric tolerance when the target value (T) locates on the midpoint of the specification interval (m). When this is not the case (T ≠ m), there are serious disadvantages in the casual use and interpretation of traditional PCIs. To overcome these problems, PCIs with asymmetric tolerances have been developed and applied successfully. Although PCIs are very usable statistics, they have some limitations which prevent a deep and flexible analysis because of the crisp definitions for specification limits (SLs), mean, and variance. In this paper, the fuzzy set theory is used to add more information and flexibility to PCIs with asymmetric tolerances. For this aim, fuzzy process mean, $\tilde{\mu }$ and fuzzy variance, ${\tilde{\sigma }}^2$, which are obtained by using the fuzzy extension principle, are used together with fuzzy specification limits (SLs) and target value (T) to produce fuzzy PCIs with asymmetric tolerances. The fuzzy formulations of the indices ${\stackrel{\sim }{C}}_{\mathit{pk}}^{″}\text{,}{\stackrel{\sim }{C}}_{\mathit{pm}}^{\ast }\text{,}{\stackrel{\sim }{C}}_{\mathit{pmk}}^{″}$, which are the most used PCIs with asymmetric tolerances, are developed. Then a real case application from an automotive company is given. The results show that fuzzy estimations of PCIs with asymmetric tolerances include more information and flexibility to evaluate the process performance when it is compared with the crisp case.

Introduction

Process capability indices (PCIs), which provide numerical measures on whether a process meets the customer requirements or not, have been popularly applied for evaluating process performance. They are summary statistics which measure the actual or potential performance of the process characteristics relative to the target and SLs. Several PCIs such as C_p, C_pk, C_pm, and C_pmk are used to estimate the capability of a process (Kotz & Johnson, 2002). These PCIs have been proposed to provide unitless measures of process potential and performance for processes with symmetric tolerances. C_p is defined as the ratio of specification width over the process spread. The specification width represents customer and/or product requirements. The process variations are represented by the specification width. If the process variation is very large, the C_p value is small and it represents a low process capability. It can be obtained by using the following formula (Montgomery, 2005):

$$C_p=\frac{\text{Allowable Process Spread}}{\text{Actual Process Spread}}=\frac{\mathit{USL}-\mathit{LSL}}{6\sigma }$$

where σ is the standard deviation of the process, USL and LSL represent the upper and lower specification limits, respectively.

C_p indicates how well the process fits between upper and lower specification limits. It never considers any process shift and simply measures the spread of the specifications relative to the six-sigma spread in the process. If the process average is not centered near the midpoint of specifications limits (m), the C_p index gives misleading results. Therefore Kane (1986) introduced C_pk which is used to provide an indication of the variability associated with a process. It shows how a process confirms to its specification. The index is usually used to relate the natural tolerance (3σ) to the specification limits and describes how well the process fits within the specification limits, taking into account the location of the process mean. C_pk is calculated as follows (Montgomery, 2005):

$$C_{\mathit{pk}}=\min \{C_{\mathit{pl}}\text{,}{C}_{\mathit{pu}}\}=\frac{\min \{\mathit{USL}-\mu \text{,}\mu -\mathit{LSL}\}}{3\sigma }$$

Since the designs of C_p and C_pk are independent of T, they can fail to account for process loss incurred by the departure from the target. A well-known pioneer in the quality control, Taguchi, pays special attention on the loss in product’s worth when one of product’s characteristics deviates from the customers’ ideal value T (target value). To take this factor into account, Hsiang and Taguchi (1985) introduced the index C_pm. Chan, Cheng, and Spiring (1988) developed the index C_pm, which provides indicators of both process variability and deviation of process mean from target value, and also provides a quadratic loss interpretation, taking into account the process departure. As a result the index C_pm incorporates with the variation of production items with respect to T and SLs, and emphasizes on measuring the ability of the process to cluster around the target. The index C_pm is defined as follow (Wu, Pearn, & Kotz, 2009):

$$C_{\mathit{pm}}=\frac{\mathit{USL}-\mathit{LSL}}{6\sqrt{{\sigma }^2+(\mu -T{)}^2}}=\frac{d}{3\sqrt{{\sigma }^2+(\mu -T{)}^2}}$$

where $d=\frac{\mathit{USL}-\mathit{LSL}}{2}$.

Pearn, Kotz, and Johnson (1992) proposed the process capability index C_pmk, which combines the features of the three earlier indices C_p, C_pk, and C_pm. The index C_pmk alerts the user whenever the process variance increases and/or the process mean deviates from its T. The index C_pmk is defined as follow (Wu et al., 2009):

$$C_{\mathit{pmk}}=\min \left\{\frac{\mathit{USL}-\mu }{3\sqrt{{\sigma }^2+(\mu -T{)}^2}}\text{,}\frac{\mu -\mathit{LSL}}{3\sqrt{{\sigma }^2+(\mu -T{)}^2}}\right\}$$

If the process mean departs from the target value, the reduced value of C_pmk is more significant than three indices C_p, C_pk and C_pm. Hence, the index C_pmk responds to the departure of the process mean from the T faster than the other three basic indices C_p, C_pk and C_pm, while it remains sensitive to the changes of process variation (Chang, 2009).

Fuzzy logic is a branch of mathematics that allows a computer to model the real world in the same way that people do. It provides a simple way to reason with vague, ambiguous, and imprecise input or knowledge. This has provided more information and more sensitiveness on PCIs. In this paper, to the best of our knowledge, PCIs with asymmetric tolerances are developed under fuzzy environment for the first time. The rest of this paper is organized as follows: PCIs with asymmetric tolerances are introduced in Section 2. The fuzzy estimations of process mean and process variance are analyzed and the formulations of fuzzy PCIs with asymmetric tolerances are derived in Section 3. Section 4 includes a real case application from an automotive company for the fuzzy PCIs with asymmetric tolerances. Concluding remarks and future research directions are discussed in Section 5.