The Capability Index

While Six Sigma was originally used for developing continuous improvement programs related to production, many of its management structure components and methodologies can be applied to safety performance improvement in the workplace. One useful measure frequently used in Six Sigma is the capability index. 

The Capability Index (Cpk) is a measure of how well the process is in control—or otherwise stated, free from defects or departures from what can be statistically considered normal. The Capability Index tells you if the process is truly capable of meeting the outcomes requirements. 

The Capability Index (Cpk) is calculated as follows (Juran and Godfrey 1999,22.18)  

To calculate the Capability Index for a process, information about the process must be known including an acceptable average, upper and lower specification limits, and the knowledge that the process is in statistical control. An acceptable value for Zj and Zyis 3.00 or greater. Referring to a Z table, a Z value of 3.00 indicates that approximately. 1% of the readings will be above the upper limit while a Zj value of 3.00 indicates that approximately. 1% of the readings will be below the lower limit. 

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Following production launch, process capability and performance should be measured continually in order to demonstrate that your processes remain capable and the capability index continues to rise. Appropriate action should be taken on characteristics that are either unstable or non-capable. Action plans should be implemented to contain process output and continually improve performance. 

The capability index for a stable process defined as the quotient of tolerance width and process capability where process capability is the 6a range of a process s inherent variation. 

The capability index which accounts for process centering for a stable process using the minimum upper or lower capability index. 

Lin, H.C. and Sheen, G.J., An approximation approach for making decisions in assessing the capability index C-pk from the subsamples. Comm. Stat. Sim. Comp., 34, 191, 2005. 

Process Capability Indices Also known as process capability ratios, these provide a measure of w hether an in-control process is meeting its product specifications. Suppose that a quality variable x must have a volume between an upper specification limit (USL) and a lower specification limit (LSL) in order for product to satisfy customer requirements. The capability index Cp is defined as... 

Use of the Cp estimate assumes that the process is centered at or near the exact middle of the specification range of a two-sided specification. This represents the best capability achievable. The Cpk estimate assesses the process with no assumption that the process is centered, and this estimate is always equal to or less than the corresponding Cp estimate. If the distribution of the output is close to a normal distribution and the capability index is based on a large and representative sample size, then acceptance values can be established. Some industries define a process with Cp = 1 as barely capable and one with Cpk = 1-33 or higher as capable. (Note that a CpK = 1 represents a situation where the process is centered and the specification range is equal to 6dc)... 

Apply a transformation to the data to make the transformed data normal. If the distribution is skewed to the right, one might try a log, inverse, square root, or cube root transformation of the data to make the data normal. If the data are skewed to the left, an exponential, squared, or cubed transformation might be applied. A histogram can be applied before and after the transformation to assess the ability of the transformation to make the data normal. It is important to remember that the transformation must be applied to the USL and LSL, in addition to the data, before computing the capability index of interest. 

A tolerance analysis predicts the behavior of the output for a specified set of targets for the inputs. The tolerance analysis can be repeated for different sets of targets to identify the optimal targets. In this case study, we have an observed equation for the average, and we have a predicted equation for the standard deviation. These can be used to obtain an equation for the capability index Cp. We can then maximize this equation to identify the optimal targets for the inputs. 

Cp, Cpi, Process capability and the index of capability, respectively. Typically a process is considered capable when the capability or the capability index is 1.33 or greater. Cp = (upper — lower specification limit)/6s, where s = standard deviation of process output. Cpk (mean — nearer specification)/3s... 

The next step is to determine the Capability Index (Cpk) for the system. Cpk is calculated by using the smaller value of either the Upper Process Location (Z ) or Lower Process Location (ZJ. In this example, Z is the smaller value with -. 5, therefore, the Process Capability (Cpk) is calculated and interpreted as follows ... 

Since the result of -. 16 is less than 1.00, which is required, and 1.33, which is preferred, the Capability Index indicates that the air handling equipment is not capable of meeting the requirements of controlling the air contaminant levels to a sufficient level as defined by the process specifications. In approximately 37% of the samples would be expected to be outside of the desired upper and lower specification ranges. While the lower readings in this example would be beneficial, the readings above the upper level could pose increased risks. 

Capability and the index of capability respectively. Typically, a process is considered capable when capability and/or the capability index is 1.33 or greater. Chapter 26... 

Cpk is the process capability index, a statistical ratio that compares the process capability to the product-tolerance band. The capability index indicates whether the process will produce units within the tolerance limits. 

The function of viscosity additives is to improve the viscosity index so as to obtain multigrade oils. The problem is to use materials that, by only slightly increasing the low temperature viscosity, are capable of counterbalancing the decrease in viscosity when the temperature increases. 

The critical characteristic on each component was analysed, calculated from the analysis and the value obtained was plotted against the process capability indices, Cpk and Cp, for the characteristic in question. See Appendix V for descriptions of the 21 components analysed, including the values of Cp and Cp from the SPC data supplied. Note that some components studied have a zero process capability index. This is a default value given if the process capability index calculated from the SPC data had a mean outside either one of the tolerance limits, which was the case for some of the components submitted. Although it is recognized that negative process capability indices are used for the aim of process improvement, they have little use in the analyses here. A correlation between positive values (or values which are at least within the tolerance limits) will yield a more deterministic relationship between design capability and estimated process capability. 

Note that the squared relationship which was initially used to model the degree of difficulty in obtaining more capable tolerances for a given manufacturing route and product design is being returned by the power law. Similarly, a relationship between the process capability index Cp and q for the components analysed is shown in... 

A standard for the minimum acceptable process capability index for any component/characteristic is normally set at = 1.33, and this standard will be used later to align costs of failure estimates. If the characteristics follow a Normal distribution, Cp = 1.33 corresponds to a fault probability of ... 

The way of measuring capability is to carry out a capability study and calculate a capability index. There are two commonly used process capability indices, Cp and Cpk. In both cases, it is assumed the data is adequately represented by the Normal distribution (see Appendix I). 

The process capability index is a means of quantifying a process to produce compo nents within the tolerances of the specification. Its formulation is shown below ... 

Where a process is producing a characteristic with a capability index greater than 2.5 it should be noted that the unnecessary precision may be expensive. Figure 2 shows process capability in terms of the tolerance on a component. The area under each distribution is equal to unity representing the total probability, hence the varying heights and widths. 

A computer program is provided for ease of calculation and efficient use of the standard. This rational method of assessing hot environments allows identification of the relative importance of different components of the thermal environment, and hence can be used in environmental design. The WBGT index is an empirical index, and it cannot be used to analyze the influence of the individual parameters. The required sweat rate (SW. ) has this capability, but lack of data may make it difficult to estimate the benefits of protective clothing. 

The refractive index of a medium is the ratio of the speed of light in a vacuum to its speed in the medium, and is the square root of the relative permittivity of the medium at that frequency. When measured with visible light, the refractive index is related to the electronic polarizability of the medium. Solvents with high refractive indexes, such as aromatic solvents, should be capable of strong dispersion interactions. Unlike the other measures described here, the refractive index is a property of the pure liquid without the perturbation generated by the addition of a probe species. 

Current computerized analyses of adverse events still typically consist of a vast number of discrete, often personal, ad hoc processes that mimic paper and pencil methods. Some commercial-off-the-shelf (COTS) software tools (e.g., Adobe Acrobat , Microsoft Word , Excel ) do have the capability to search for specihc terms in electronic documents/case reports and do have navigational tools with hyperlinks and fullfull-text indexing that enable researchers to create their own hyperlinks. Some other COTS software tools (e.g., SAS , Excel , Access , IMP ) allow importation of electronic case report tabulations (ECRT) for more detailed analysis. 

By using the simulation model developed in Samsung Total we applied the ideas of pFoductivily enhancement successfiiUy to LDPE plant and accomplished considerable productivity incn e. The MWD as well as the melt index and density calculated by the simulation model convinced us of applying the ideas to commercial plant. The end user property prediction capabilities of the model will be refined further by integration of phj icxjchemical and statistical approaches and be one of the next potential research items. 

Metrics for this might include number of excursions from statistical process control, but one very useful metric for controllability is process capability, or more accurately, process capability indices. Process capability compares the output of an in-control process to the specification limits by using capability indices. The comparison is made by forming the ratio of the spread between the process specifications (the specification width ) to the spread of the process values. In a six-sigma environment, this is measured by six standard deviation units for the process (the process width ). A process under control is one where almost all the measurements fall inside the specification limits. The general formula for process capability index is ... 

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