## Measurement System Analysis

Once the critical parameters have been selected, and specification limits have been set, it seems reasonable that the next steps might be to set things up so that progress towards achieving expectations can be monitored. As discussed in Chapters 7 and 9, the critical parameters have been defined in measurable terms. The next logical step is to set up the measurement systems and determine whether each is capable of measuring appropriate critical parameters.

Figure 10.10 summarizes the purpose, results, and outputs from measurement system analysis, focusing on MSA for critical parameters that are continuous rather than discrete. There are several indices used to determine if the measurement system is adequate for the purposes of optimization and validation of the critical parameters, including assessments of stability, linearity, accuracy, and measurement error. MSA is discussed further in Chapter 16.

Figure 10.10 Summary for measurement system analysis

The assessment and estimate of measurement error is a key, recurring topic in DFSS, and this is an appropriate point to begin that discussion. The measurement error is one of several "noises" that can be flowed down, as discussed later in this chapter, and that will be encountered along the way as the design team uses approaches such as design of experiments (DOE) and response surface methodology (RSM).

This aspect of the flow-down process is illustrated in Figure 10.11, which starts with the concept of squared deviation from the target. If the target is the desired value, as discussed in the previous section, then one can define a statistical index, the second moment about the target, which can represent the degree of customer satisfaction. The ideal case, in which every product is exactly on target with no variation, would have a value of zero for this statistical index. As further illustrated in Figure 10.11, this squared deviation from (or second moment about) the target corresponds to the Taguchi Loss Function for a target-is-best situation. A useful aspect of this equivalence is that the deviation from the ideal situation can be partitioned into two parts: the degree to which the deviation is a result of the average being off-target, and the variance about the mean. This variance can be further partitioned into variance as a result of the measurement system (discussed here) and variance as a result of manufacturing variation and variations in usage and environment (including system interactions).

Figure 10.11 Partitioning of squared deviation from the target, including variance associated with the measurement system

MSA for continuous parameters provides an estimate for the variance caused by the measurement system, and compares it to the tolerance in terms of the precision to tolerance ratio (P/T ratio), and to the total observed variance in terms of the GR&R ratio (gauge repeatability and reproducibility). The P/T ratio is defined as six times the standard deviation of the measurement system divided by the difference between the upper and lower specification limits. The GR&R ratio is defined as the standard deviation of the measurement system divided by the total observed standard deviation, combining sources of variation including measurement error, variation from manufacturing, variation from how the customers use it, variations from the environments where the product will be used, and variations in how the interactions among the subsystems and the product with other systems affect the parameter.

If the measurement variance consumes too much of the tolerance window, or obscures the ability to assess the other sources of variation, then the measurement system is not acceptable. For many situations, the rule of thumb is that both the P/T ratio and the GR&R ratio should be less than 30 percent; for other situations, a rule that both should be less than 10 percent is imposed. Acceptable values for the P/T ratio derive from statistical analyses that indicate that a P/T ratio more than 30 percent corresponds to a very high risk of incorrectly passing bad parts or incorrectly rejecting good parts.

The measurement system for critical parameters at the system or product level will generally link to the test and evaluation plan for the product, as illustrated by the arrow to the deliverable initial critical parameter (CP) test plan in Figure 10.2. This deliverable is a starting point for the verification of capability discussed in Chapter 16, and summarized in Figure 10.12. Clearly, the preparation of the measurement systems to be used for verification do not need to wait, and should not wait, but should be initiated with the initial measurement systems analysis effort.

Figure 10.12 DFSS flowchart, drilled down to detailed flowchart that includes actions for improving the measurement system if MSA results are unacceptable

If the GR&R or the P/T ratio, or both, fail to meet acceptable guidelines, then there are a series of actions for improving the measurement system that are summarized in Figure 10.12 and discussed in Chapter 16.