# Capability Index Definition

### Capability index (Cp) Definition

Process capability index (Cpk) is an arithmetic tool used for quantifying the proficiency of a process to produce products within the customer specification range. It values the producer’s ability to produce output within the acceptable customer range. Cpk is important in approximating the closeness between you and the target set, and the steadily one is to the mean performance. Cpk portrays the best case situation for the current development

### A Little More on What is Process Capability Index (Cpk)

Cpk is given by this complex equation:

[Minimum (mean – LSL, USL – mean)] / (0.5*NT).

LSL refers to Lower Specification Limit, and USL stands for Upper Specification Limit.

Normally Cpk is described as the ability of the process in achieving whether or not the mean is located at the middle of the measurement range.

Cp and Cpk, these two data share a lot. The lesser the standard deviation, the greater the Cp and Cpk. Under the appropriate situation, Cp and Cpk are the same in value.

Cpk is a regular index to illustrate the ability of one procedure. Cpk value is directly proportional to the degree of goodness of a procedure. The higher the Cpk value, the better the procedure. For instance, Machine 1 has a Cpk of 1.7 and machine 2 has a Cpk of 1.1. In regards to the Cpk, one can conclude that machine 1 is better than 2. We can also calculate the yield produced and the ineffectiveness of the machine because Cpk applies the use of the gradation range.

### Relationship to measures of process fallout

The plotting from capability indices like Cpk to quantification of process impacts is forthright. Process fallout measures the number of flows a process results into and id measured by DPMO or PPM. Process output is the supplement of process fallout, and it is estimated similar to the area under the probability density curve.

Cpk equal to or greater than 1.33 shows that the process is able and matches specification range. Any value less than 1.33 suggests fluctuation is wider in comparison to the description or the process mean if far from the provision made.

Example 1:

Upper specification limit (USL) =16

Lower Specification limit (LSL) = 4

Mean (μ) = 10 & Standard deviation (σ) = 2

Given the formula to calculate Cpk is

Cpk = min [USL−μ/3σ, μ−LSL/3σ]

= min [16-10/6, 10-4/6]

= min [1, 1]

= 1

Statistical clarification when curve stretches from +3 to -3, it is believed to occupy 99.73% and here the machine is making 99.73% fit parts.

Example 2:

Upper specification limit (USL) =18

Lower Specification limit (LSL) = 0

Standard deviation (σ) = 2

Cpk = min [USL−μ/3σ, μ−LSL/3σ]

= min [18-10/6, 10-0/6]

= min [1.33, 1.67]

= 1.33

Here at least, 99.99% of the outputs from the machine are good.

### Academic Research for Capability Index

• A multivariate process capability index over a rectangular solid tolerance zone, Chen, H. (1994). Statistica Sinica, 749-758. A multivariate process capability index (PCI) was suggested over an overall acceptable region including ellipsoidal and rectangular three-dimensional ones as unique cases. The calculating aspect of the suggested multivariate PCI is deeply examined emphasising on the bivariate regular process. Resampling procedures and Monte Carlo procedure are proposed to solve the difficulty in dealing with the distributional and inferential traits.
• Process capability indices, Kane, V. E. (1986). Journal of quality technology, 18(1), 41-52. The paper outlines the presentation and association of capability indices of Cp, CPU, CPL, k, and Cpk process factors. These Indices are viewed to make a supplementary system of measurement of process performance and can be applied with both unilateral and bilateral tolerances regardless of target values. There is a discussion of numerous applications of in=dices together with mathematical sampling considerations.
• A process capability index sensitive to skewness, Wright, P. A. (1995). Journal of statistical computation and simulation, 52(3), 195-203. The paper suggests a new process capability index Cs which expands the more important index to date. Cpmk, by considering both the possibility of process average not lying in the middle of specification range and includes a consequence if the average varies from its target and including the skewness punishment Authors proposed an estimator and its biases and variation are examined in the absence of skewness. Therefore, the its advocated use for supervising close normal processes where capability loss results to disproportionateness.
• Interval estimation of process capability index Cpk, Zhang, N. F., Stenback, G. A., & Wardrop, D. M. (1990). Communications in Statistics-Theory and Methods, 19(12), 4455-4470. The paper presents numerous sampling distribution factors of the estimator for Cpk, and the assumption of normal, isolated and similarly distributed data, in specific the anticipation of variance and skewness are derived. Asymmetric interval estimator for Cpk might be rational given that sampling distribution is weakly skewed. The authors also established systematic interval estimator and undertook a simulation experiment to explore its scope probabilities.
• Capability index using principal components analysis, Wang, F. K., & Chen, J. C. (1998). Quality Engineering, 1 1(1), 21-27.  The paper outlines that the process capability indices are measures of the ability of a process to produce output in line with some assigned specifications associated with some features of the objects produced. Indices are useful in the multivariate process and refer to the geometric average of some univariate indices for every significant part.
• Deciding the and evaluate process capability index Cp with fuzzy numbers, Tsai, C. C., & Chen, C. C. (2006). The International Journal of Advanced Manufacturing Technology, 30(3-4), 334-339. The wide applications of process capability index Cp in the manufacturing industry has been extended by this paper to uncertain surroundings with a procedure of carrying out the tests on Cp of fuzzy numbers. Two nonlinear functions are formulated to evaluate the alpha cut of index C˜p from numerous values of alpha. Besides classical tests, mathematical decision suggested in this paper illustrates a grade of acceptability of the null hypothesis and alternative hypothesis. Using crisp values, the traditional method cannot only boil down to the conventional formula for Cp determination but also lead to two decisions of either accepting or rejecting the null hypothesis.
• A new generalisation of process capability index Cpk, Pearn, W. L. (1998). Journal of Applied Statistics, 25(6), 801-810. Mathematical measurement of processability and performance was provided in the manufacturing industry from the use of process capability index Cpk. Cpk is yield-based and is isolated from the target T. This is unable to explain the process centring with the regular tolerances with more problems to irregular tolerances. The authors launched a new index, Cp”k which is highly superior to the current Cpk. The statistical characteristics of Cp”k was then investigated with the assumption of normal distribution of the process.
• A new process capability index for non-normal distributions, Chen, J. P., & Ding, C. G. (2001). International Journal of Quality & Reliability Management, 18(7), 762-770. There have been many proposals of process capability indices in the measurement of process performance. In this paper, there is an examination of Cp, Cpk, CPM, and Cpmk and their overviews, the distribution which considers process variance, the variation of process average from the objective value and fraction of nonconformity. Spmk shows the proportion of nonconformity a previously developed index, also considers the variability and parting from the objective amount which is illustrated by abnormal procedures.
• Variable sampling inspection for resubmitted lots based on process capability index Cpk for normally distributed items, Aslam, M., Wu, C. W., Azam, M., & Jun, C. H. (2013). Applied Mathematical Modelling, 37(3), 667-675. This paper develops a variables sampling review strategy for the resubmitted lot based on the process capability index Cpk for the normally distributed objects with unknown average and variance. The paper also describes the importance of the suggested resubmitted sampling strategy over variable unit sampling. Table of proposed strategy factors for the selected values of acceptable quality levels (AQL), limiting quality levels (LQL), producer’s α-risk and consumer’s β-risk are given out and examined with the assistance of appropriate example.
• Distributional and inferential properties of process capability indices, Pearn, W. L., Kotz, S., & Johnson, N. L. (1992). Journal of Quality Technology, 24(4), 216-231. The author performs a study and judgmental comment regarding some necessary concepts and perceptions appropriate to the procedure involving process capability indices (PCIs). The intention is to add on the thoughtful and explanation of these indices if need of their use arises, The authors treat them as main indicators of anticipated part of noncompliant objects and not as having some association to neutral loss curve as is the most current trendy perspective. There are suggestions of Current PCIs associated with nonconforming objects.
• Variables sampling inspection scheme for resubmitted lots based on the process capability index Cpk, Wu, C. W., Aslam, M., & Jun, C. H. (2012). European Journal of Operational Research, 217(3), 560-566. This paper develops a variables sampling review strategy for the resubmitted lot based on the process capability index Cpk for the normally distributed objects with unknown average and variance. The paper also describes the importance of the suggested resubmitted sampling strategy over variable unit sampling. Table of proposed strategy factors for the selected values of acceptable quality levels (AQL), limiting quality levels (LQL), producer’s α-risk and consumer’s β-risk are given out and examined with the assistance of appropriate example.