Determining Dimensional Capabilities From Short-Run Sample Casting Inspection A.A. Karve M.J. Chandra R.C. Voigt Pennsylvania State University University Park, Pennsylvania ABSTRACT A method for determining dimensional capability guidelines for use by casting customers, which incorporates sample casting inspection uncertainties, has been developed. Tooling validation practices have traditionally been based on the dimensional inspection of a few sample castings. Sampling an insufficient number of castings can introduce significant dimensional error into casting feature estimation. This paper reviews the nature and causes of these inspection errors during tooling validation studies. The foundry s dimensional capabilities dictate the minimum number of castings that need to be inspected in order to make confident pattern adjustments. In practice, the willingness to sample the required number of castings for tooling evaluation often depends upon the projected casting production quantities. When an insufficient number of castings are dimensionally inspected, sampling uncertainty errors are introduced. These errors need to be incorporated into the overall estimates of the foundry s dimensional capabilities, to account for the additional uncertainty introduced by insufficient sampling. A simple, practical method for determining these dimensional capability guidelines is presented. The impact of inspection strategies on the development of separate dimensional tolerance specifications for long- and short-production-run castings is also discussed. INTRODUCTION Traditional new tooling validation practices involve pouring sample castings and inspecting them for dimensional conformance to customer tolerance specifications. Decisions to modify the pattern equipment are then made, based on the dimensional surveys of these sample castings. For short-run castings, it is common practice to dimensionally inspect only one sample casting prior to making the necessary tooling adjustments. For long-production-run castings, multiple sample castings are inspected before making tooling adjustments. This sample casting inspection cycle is often repeated, if dimensional discrepancies are noted, until the casting(s) produced are dimensionally acceptable. Although unstructured pattern approval methods have been in use for a long time, increasingly tight casting dimensional tolerances and pressures to reduce new casting lead times raise questions about the validity of these arbitrary methods. Short lead times make repeated sampling and pattern correction cycles difficult, if not impossible. By understanding the dimensional uncertainty associated with traditional pattern approval methods, improved sampling strategies that minimize dimensional uncertainty can be developed. Similarly, when cost and lead time demand limited sampling, it is possible to estimate the resulting uncertainty and incorporate it into realistic estimates of foundry dimensional capabilities for use by casting customers. Statistically-based pattern approval procedures are an attractive option, in light of the above arguments. Statistical methods can be used to decide how many castings need to be inspected. If this minimum number of castings cannot be inspected, these same methods can indicate the extent of the sampling error that should be incorporated into dimensional capability estimates. This requires a more rigorous sampling than may be required for statistically sound pattern approval methods. Although this is an additional up-front cost, there are significant costs associated with pattern approval practices based on insufficient sampling. These invisible downstream errors result in unnecessary pattern change iterations, which consume a significant amount of lead time. It has been estimated that up to 60% of the lead time for new part production in sand foundries is consumed by dimensional issues. Investing in statistical pattern approval methods, up front, can lead to significant cost savings down the road, not to mention the intangible advantages of increased customer satisfaction. CASTING SAMPLING ISSUES FOR TOOLING VALIDATION Casting part prints generally have a large number of dimensional call-outs. Some of these dimensions may be considered to be more critical than others, by the customer. However, most, if not all, have upper and lower dimensional tolerance limits specified on either side of the nominal dimension, as shown in Fig. 1. The designer/ patternmaker first applies the necessary shrink rule to all casting features, which dictates the required pattern dimensions. The pattern is built and sample casting(s) are poured. These sample castings are dimensionally inspected. The resulting dimensional distributions for each feature, due to casting process variability, are also shown in Fig. 1. The mean of each of the dimensions measured on the sample castings, as well as the process variation, can be determined. In the situation where only one sample casting is poured, its dimensional value is assumed to be the mean of the expected casting feature dimensional distribution, and no estimates of process variability can be obtained. Fig. 1. Aspects of dimensional control of a casting feature. AFS Transactions 98-115 699 AFS Library Copy: 19990138A.pdf, Page 1 of 5 Pages, Provided to User for Internal Use and Not Public Redistribution or Resale.
Now, as seen in Fig. 1, the mean of the casting measurements for each feature could be offset from the target values, i.e., the nominal dimensions. This can occur for a number of reasons. In many cases, a uniform pattern allowance is applied to all features of the casting while building a pattern. If each feature shrinks by an amount equivalent to the pattern allowance applied, the mean of the casting dimensions would coincide with the nominal dimensions. But not all features shrink the same. This difference in shrinkage behavior can cause the mean of the casting dimensions to be offset from the nominal dimension. Furthermore, measurement or pattern production errors in the pattern shop can cause the pattern to be made to the wrong size. This can also cause the mean of the casting measurements to be offset from the nominal dimension. To ensure that all the production casting dimensions conform to print, two aspects of dimensional behavior need to be controlled for a feature. First, the process variation (6σ) should be controlled so that it is less than the total customer tolerance. Second, the mean of the casting dimensions should coincide with the nominal dimension, as closely as possible. This centered process will ensure that the dimensions of the castings produced will lie within the tolerance limits, even if the process variation is a significant percentage of the tolerance allowed. Therefore, to center this process, the mean of the casting dimensions should be moved to coincide with the nominal dimension. This is achieved by moving or adjusting the pattern dimension (in the appropriate direction) by an amount that is equivalent to the offset between the nominal dimension and the mean of the casting dimensions. For high-production patterns, where pattern wear is anticipated, the pattern dimensions may be adjusted to produce castings not centered on the nominal dimension. However, the same principles apply. The basic question that a foundry should seek to answer during the pattern approval process is: How many castings should be sampled to confidently make decisions about necessary pattern adjustments? This question is valid because the calculated mean of the sample casting dimensions (from which the pattern adjustments are made) depends on the number of castings sampled. If the number of castings sampled is too small, the true value of the mean of the casting dimensions cannot be confidently determined, and incorrect or unnecessary pattern adjustments will be made. Thus, the foundry should determine this minimum inspection sample size that will enable pattern adjustments to be made confidently. The implementation of statistical pattern approval methods developed here can provide this estimate. A prerequisite for effective casting dimensional control is that the process variation (6σ) must be less than the total allowable tolerance, as shown in Fig. 2. The relation between process variation and the total customer tolerance can be expressed in terms of a process capability ratio. Process capability ratio can be defined as the amount of variation observed per amount of variation allowed. This ratio essentially describes how capable the manufacturing process is, with respect to the allowed limits of variation. In Fig. 2, the process variation (6σ) is the observed variation and the total customer tolerance is the allowed variation. Thus, the process capability ratio would be Process capability ratio = Process variation (6σ) Total customer tolerance Statistically based pattern approval methods use this process capability ratio to predict the statistically minimum sample size needed to confidently make decisions about pattern adjustments. One such statistical pattern approval method is described in the next section. Statistical Determination of Sample Size for Tooling Validation Before a foundry can employ statistical pattern approval methods, it is important to quantify and control the measurement system variation. All decisions made from dimensional inspection information must be questioned, if significant measurement system errors are introduced during part inspection. Methods and procedures to quantify and control measurement system variation have been previously described 1 and are incorporated into commonly used production part approval processes (PPAP). 2 The importance of capable measurement systems cannot be overemphasized. When the foundry is confident that capable measurement systems are being used, it will be in a position to apply statistical pattern approval methods to make proper decisions about tooling adjustments. The number of castings that need to be sampled can be determined on the basis of the foundry s dimensional capability, with respect to the allowable dimensional tolerance. The process capability ratio, as described earlier, can be used to predict the minimum number of castings that need to be inspected. Detailed statistical calculations for determining minimum inspection sample sizes have been described earlier by the authors. 3 This previous work is the basis for additional guidelines developed here for improved dimensional capability estimates. Table 1 shows the minimum number of castings (N) that need to be inspected for pattern approval, as a function of the process capability ratio. If less than this minimum number are inspected (at Table 1. Statistically Determined Minimum Number of Sample Castings (α = 0.05 and β = 0.05) Fig. 2. Comparison between total customer tolerance (allowed variation) and process variation (actual variation). 700 AFS Transactions AFS Library Copy: 19990138A.pdf, Page 2 of 5 Pages, Provided to User for Internal Use and Not Public Redistribution or Resale.
each pattern adjustment cycle), uncertainty is introduced into the estimate of the mean of the sampled distribution that cause statistically significant errors in the pattern adjustment estimates. Table 1 indicates that, as the process capability ratio decreases, a smaller number of castings needs to be inspected. This is due to the fact that a small process capability ratio indicates that the process is much more capable than the allowable limits. Therefore, small errors in mean estimation, from small numbers of sampled castings, do not result in production casting dimensions outside of the customer s tolerance limits. Higher process capability ratios dictate that a larger number of castings needs to be inspected to confidently make decisions about pattern adjustments. For example, when the process capability ratio is equal to or greater than 0.6, Table 1 indicates that at least 44 castings need to be dimensionally inspected to obtain an accurate estimate of the mean casting dimension for proper tooling adjustment. The results in Table 1 are based on the assumption that α and β errors are 0.05. 3 These α and β values indicate a 95% confidence level and are typically used for decisions making. For increased confidence levels in decision making, the minimum number of castings to be inspected can increase significantly. For example, when a 99% confidence level is used for decision making (α and β = 0.01), the minimum number of castings that need to be inspected increases. 3 The required number of castings to be inspected for a process capability ratio of 0.1 increases to N = 2, while, at a process capability ratio of 0.6, N increases to 87. In the rest of this discussion, the more typical 95% confidence level value for N will be used for the further development of short-run sampling strategies. When the process capability ratio is high, the number of castings that need to be inspected for a 95% confidence level in decision making can be as high as 44. These sample casting inspection demands are not compatible with short-run production constraints. In many cases, short-run castings can have production order volumes that are much less than 44. In these cases, the foundry will not be willing or able to inspect this large number of sample castings. Also, when large and/or complex castings are involved, economic factors can prevent the foundry from sampling such a large number of castings. Small sample inspection analyses methods must be developed for these important casting applications. Fig. 3. Additional component of dimensional variability estimates due to sampling uncertainty errors. In this situation, an additional component of dimensional error can be incorporated into the estimates of casting dimensional variability. This compensates for the uncertainty introduced into the prediction of the mean, due to inadequate sampling. A simple and effective statistical scheme can be used to incorporate this additional uncertainty introduced, due to inadequate sampling into the dimensional variability estimates, as described in the next section. These improved dimensional capability estimates are a better reflection of a foundry s capabilities, from the customer s perspective. Uncertainty Errors Introduced Due to Inadequate Sampling Of great concern are the large numbers of sample castings that need to be dimensionally inspected when the process capability ratio is high. Unfortunately, this is not an uncommon occurrence. However, the sampling uncertainty errors introduced from inadequate sampling can be readily quantified, and should be incorporated as an additional component of casting feature variability. This adjusted dimensional capability, which includes sampling uncertainty errors, is a better measure of the foundry s true dimensional capabilities when inadequate sample casting inspection procedures are employed. This additional component of uncertainty is necessary when not enough sample castings are inspected to confidently predict the mean of the casting dimensions. This is shown schematically in Fig. 3. The shaded region in Fig. 3 indicates the uncertainty introduced into casting measurement for tooling validation, due to inadequate sampling. This uncertainty inflates the process variation (dimensional variability) estimates. Thus, the true dimensional capability is influenced by process control that determines variability, and inspection uncertainty, which is influenced by sampling strategies. This additional sampling uncertainty can be expressed as a multiplier of the casting feature variability. This multiplier would be 1.0 when the number of castings inspected are equal to or greater than the minimum number of castings (N) from Table 1. The multiplier is greater than 1.0 when less than the minimum number of castings (N) are measured. First, the uncertainty error of estimation will be expressed as a function of the sample size. The sample mean, X, obtained from a sample batch of size, n, is the best estimate of the unknown mean, µ. As the sample size, n, increases, the sample mean, X, becomes closer to the true mean, µ. In other words, as n increases, the sampling uncertainty error, which is defined as the absolute value of the difference between X and µ, decreases, and vice versa. Because the X value obtained from a given sample varies from batch to batch, we have to include the confidence level (or the probability of confidence) in the relation between the error of estimation and the sample size. Let e be the maximum absolute error between X and µ: e = X µ (1) For a sample size of n and for a true process standard deviation of σ, the inspection uncertainty error, e, can be expressed as: e = Z α/2 σ / n (2) where Z α/2 is the standard normal variable for a chosen (1 α) confidence level. For example, if the confidence level specified by the user is 1 α = 0.95 (95%), then α = 1 0.95 = 0.05 and α/2 = 0.025. From normal distribution tables found in any statistics textbook, Z 0.025 = 1.96. The uncertainty error, e, can then be used to develop an uncertainty error multiplier for the 6σ. For example, let us assume that the estimate of the standard deviation of the casting feature dimensional variability (σ) is 0.10 AFS Transactions 701 AFS Library Copy: 19990138A.pdf, Page 3 of 5 Pages, Provided to User for Internal Use and Not Public Redistribution or Resale.
inches, and the confidence level specified by user is 1 α = 0.95. If the measured sample size (n) is 9, then the maximum deviation between the true mean (µ) and the sample mean is: e = 1.96 0.10 / 9 = 0.065 inches (3) This is the amount of uncertainty introduced into the measurement, if the sample size is 9. Let the mean of the sample casting dimensions (X ) be 2.25 inches. This means that there is a 95% chance that the true mean (µ) could lie anywhere between 2.25 0.065 = 2.185 inches and 2.25 + 0.065 = 2.315 inches; i.e., 2.185 µ 2.315. If the measured sample size is increased from 9 to 16, then the uncertainty error will decrease to: e = 1.96 0.10 / 16 = 0.044 inches (4) keeping σ and (1 α) the same as in the earlier example. Next, this error formula (Eq. 2) will be incorporated into the process variability 6σ (±3σ) range, which covers 99.73% of the dimensional values. Assuming that the true mean (µ) is located to the right of the sample mean (X ), the total possible spread of the characteristic is shown in Fig. 4. It can be seen that the total dimensional capability estimate is: Dimensional Capability = 6σ + e = 6σ + Z α/2 σ / n = 6σ [1 + Z α/2 / 6 n] (5) In this expression, this sampling uncertainty multiplier [1 + Z α/2 / 6 n] is the multiplier for the 6σ process variability that accounts for the uncertainty errors due to inadequate sampling. As the equation indicates, the uncertainty multiplier is dependent upon the sample size, n, and the α error. An α value of 0.05 (decision making confidence of 0.95 or 95%) is most widely used. Table 2 shows the values of this multiplier for various sample sizes. In this table, the values are calculated for the minimum sample sizes developed in Table 1. The multiplier values in the table indicate by how much the dimensional variability estimates should be inflated to account for the sampling uncertainty errors. For example, if the minimum number of sample castings that a foundry should inspect is 44, and if the foundry inspects only 2 castings, Table 2 indicates that the foundry should multiply its dimensional variability estimates by 1.23 (123%) to account for the additional uncertainty errors introduced, due to inadequate sampling. These statistical concepts will be better illustrated with an example as described in the next section. Statistical Concept Example Let us assume that the foundry builds a pattern for a new casting order. A 4-inch long feature on the casting is considered critical by the customer and has a total allowable customer tolerance of 0.7 inches (±0.35 inches). Initially, the foundry estimates that it is capable of holding a total tolerance of 0.45 inches for this feature. This estimate could be based on previous dimensional capability studies that the foundry has carried out on similar castings and features in the past. Thus, dimensional variability (actual variation) = 0.45 in. Then, the foundry s process capability ratio for this feature can be calculated: Process capability ratio = Observed variation (6σ) Allowed variation = 0.45 0.70 = 0.642 Based on a process capability of 0.642, Table 1 indicates that the foundry should sample 44 castings (N = 44), in order to be able to make confident decisions regarding adjustments to the pattern. Let us assume that, based on a number of factors, the foundry management and engineers decide that only three castings can be sampled during initial tooling validation studies. Thus, the sample size n = 3. Now, in order to compensate for the uncertainty introduced by sampling three castings instead of the minimum required number of 44, the dimensional variability estimate of 0.45 inches should be inflated by the appropriate dimensional variability multiplying factor. Table 2 indicates that, if 44 castings need to be sampled, and only 3 are sampled, the required process variability multiplier is 1.18. (This is calculated from Equation 5 for n = 3 and based on an α level of 0.05.) Thus, the modified dimensional capability estimate, based on the foundry s dimensional capabilities and sampling uncertainty error, equals 1.18 0.45 inches = 0.531 inches. This is the value of overall dimensional capability that the foundry can confidently quote to the customer and use when making decisions about their ability to meet the allowable tolerance on the feature. This is a more accurate picture of the foundry s dimensional capabilities, as it incorporates actual process variability, as well as the additional error due to inadequate sampling. In this case, the foundry is still capable of producing acceptable castings, as the adjusted variability is less than the allowed tolerance. This may not always be true. In some cases, after compensating for the additional inspection uncertainty error, the adjusted value of dimensional capability may exceed the allowable tolerance, even though process variability is within the allowable tolerances. In this situation, the allowable tolerance should be re-examined and changed, if possible. Table 2. Dimensional Variability Multiplying Factors Fig. 4. Schematic representation of total dimensional capability, including sampling uncertainty errors. 702 AFS Transactions AFS Library Copy: 19990138A.pdf, Page 4 of 5 Pages, Provided to User for Internal Use and Not Public Redistribution or Resale.
DISCUSSION The overall dimensional capabilities of a foundry are influenced not only by its process capabilities, but also by its inspection thoroughness sampling enough castings at each pattern validation stage to make accurate tooling adjustments. If the anticipated production volumes are small, or if only a few castings are to be produced, the foundry will not be willing to or able to sample enough castings. Then, the foundry s process capability estimates should incorporate a sampling uncertainty multiplier, as developed in the previous sections of this paper. An important part of these calculations is the dimensional variability estimate required to calculate the process capability ratio. This estimate must be made before any new castings are poured. It is difficult to predict the actual dimensional variability of the casting features unless accurate historical data is available. Therefore, it is recommended that the foundry use historical dimensional variability values from similar castings and features that the foundry has evaluated in the past. Developing a comprehensive ongoing database of dimensional variability for different casting and feature types (along with the corresponding process and design variables) can help the foundry in estimating the dimensional variability values needed to design appropriate sample casting inspection strategies. Therefore, in many situations, more rigorous sample casting inspection may be necessary to reduce inspection uncertainty errors and conform to tight customer dimensional tolerances. The customer must be made aware of these sampling requirements and the casts associated with adequate sample casting inspection. A foundry s dimensional capabilities, as quoted to a customer, are not fixed and determined solely by the degree of process control practiced by the foundry. Rather, they depend on process variability and sampling uncertainty, which can be minimized only by adequate sample casting inspection. This sliding scale of foundry process capabilities, depending on the customer s willingness to pay for sufficient tooling adjustment sampling, is the basis for quoting different short-run and long-run dimensional capabilities to the customer. Short-run casting production generally prevents the sampling of a statistically adequate number of castings. This introduces significant inspection uncertainty errors that effectively reduce short-run dimensional tolerance capabilities. Published dimensional tolerance standards vary in their approach to short-run and long-run casting dimensional tolerances. The most comprehensive dimensional tolerance standard for castings is the ISO 8062 standard developed by the International Standards Organization, which outlines a system for dimensional tolerances to be specified on raw castings. This standard specifies 16 casting tolerance grades (CT grades) from CT 1 through CT 16. Depending on the alloy being cast, and the casting process, the standard recommends ranges of tolerance grades. The actual tolerance value within a grade is based on the feature size. The current ISO 8062 standard also specifies different tolerance grades for short-run and long-run production sand castings. 4 For example, grades CT 13 15 are recommended for short-run steel castings produced in nobake molds (hand molded), while narrower grades CT 11 14 are recommended for their long-run counterparts. The short-run CT grades specified are broader than the long-run tolerance grades by a factor of 1.2 1.4, depending on feature size. This is in general agreement with the multiplier of 1.32 developed earlier for the case when the process capability ratio exceeds 0.6 and only a small number of castings are sampled. The additional sampling uncertainty error typically introduces short-run castings where sample casting inspection is more likely to be inadequate as a sufficient justification for separate wider short-run casting tolerance specifications. Another important aspect of sampling uncertainty that can be developed from this analysis is the minimum magnitude of a pattern dimensional change that can be made with confidence. Traditionally, after the sample castings have been dimensionally inspected, the mean of the dimensions is calculated. Depending on how far this mean value is from the target nominal dimension, pattern features are adjusted so that the process mean will be centered between the tolerance limits. Inadequate inspection strategies limit the resolution of this pattern adjustment that can be made with confidence. The highest multiplier shown in Table 2 is 1.32. This indicates that a region of sampling uncertainty corresponding to (0.32 Process Capability) is introduced into the measurement as a result of inadequate sampling. The true casting mean dimension lies somewhere in this region, as seen in Fig. 3. Since the exact value of the mean is unknown, any pattern adjustment of less than (0.32 Process Capability) cannot be made with confidence. It would be a shot in the dark. This lack of resolution for pattern adjustments can be significant, depending on the foundry s process capability, and can make it difficult for the foundry to center the process between the tolerance limits, even after repeated sampling and pattern adjustment cycles. This more complete sampling also increases the resolution of pattern adjustments. SUMMARY The nature of the dimensional errors introduced by inadequate sampling procedures during new tooling validation has been discussed. The foundry s process capabilities dictate the minimum number of castings that should be inspected during new tooling validation, in order to confidently make tooling adjustment decisions. When the statistically determined minimum number of castings cannot be inspected (for economic or production reasons), a simple and effective scheme has been developed to incorporate this additional sampling uncertainty into improved dimensional capability estimates. This sampling uncertainty error has been presented in the form of a multiplier to be factored into the process capability. This multiplier should especially be considered when setting tolerance grades for short-run castings, where sampling uncertainty errors due to insufficient inspection are likely. Inadequate sampling also limits the resolution with which pattern adjustments can be made. REFERENCES 1. Karve, A.A., R.C. Voigt, L.A. Potter, Use of Measurement Equipment for Casting Dimensional Inspection, AFS Transactions, vol 105, p 971, 1997. 2. Automotive Industry Action Group, Production Part Approval Process, 1st edition, AIAG, 1993. 3. Potter, L.A., R.C. Voigt, F.E. Peters, J. Lies, M.J. Chandra, A Statistically Based Pattern Approval Process, AFS Transactions, vol 104, p 307, 1996. 4. ISO TC-3, ISO 8062 - System of Casting Tolerances and Machining Allowances, 2nd edition, International Standards Organization, 1994. AFS Transactions 703 AFS Library Copy: 19990138A.pdf, Page 5 of 5 Pages, Provided to User for Internal Use and Not Public Redistribution or Resale.