An injection molding engineering system is composed of the following sub-systems: plastic material, part structure design, injection molding machine, injection mold, and molding process condition. In practice, the performance evaluation of an injection molding engineering system can’t be done until the injection mold is developed, manufactured, and assembled; and a molding process condition is worked out to go with it. Therefore, in reality in the industry, the performance evaluation of an injection molding engineering system refers to more about for newly developed injection mold and molding process condition, assuming that the other three sub-systems – plastic material, part structure design, and injection molding machine – are fixed and can’t be changed. And the process flow of the performance evaluation is also considered the qualification process of a newly developed injection mold with a specific molding process condition for the decision whether the mold should be approved capable or not to release for mass production.
Herein, we use a simplified example to directly demonstrate how to evaluate the performance of an injection molding engineering system; or saying, how to evaluate the performance of a process run by an injection mold going with a molding process condition, assuming that the plastic material, part structure design, and injection molding machine has been selected and fixed.
Example: An injection molding part is produced by a one-cavity mold. The performance of the injection molding process is evaluated on the critical dimension [20.00 +0.10/-0.10 mm] of the molded part. Four processes are developed, and 32 pieces of the sample are collected for each. The critical dimensions on all samples are measured and recorded as shown in Table 1. Question: Which process performs the best?
Method 1: Use the average (μ) of the raw data and infer that the process which produces the smallest difference between its average and the target value, 20.00 mm, in this case, performs the best. – Not suggested at all.
This method might be the primary one most engineers ever use for engineering decisions at the beginning stages of their careers. This method is not suggested at all because it does not take the raw data’s distribution upon the target value into account, leading to misinterpreting the performance of processes and making ineffective decisions in process selection.
As shown in Table 2 for the mentioned example, if only review raw data’s averages of the four processes, or orally discuss only according to the averages as summarized information, engineers and managers could jump to such a conclusion that both Process 1 and 3 are the most well-performed because their averages are exactly the same as the target value. However, when looking into the raw data piece by piece, it appears to be quite different. As shown in Table
3, for each process, the molded parts whose critical dimensions are within specification (meaning “pass”) are marked with green cells, and their numbers are counted. The pass rate of each process is defined and calculated by dividing the “pass” number counted by the sample size,32 in this case, of each process. The result of pass rate calculation suggests that <1> none of the parts produced by Process 1 is within the specification, that the average is exactly the same as the target value is just a coincidence; <2> Although the average is shifted from the target value, Process 2 is better than Process 1 since it produces more “pass” parts; <3> Process 4 is the worst because not only the average produced has the largest difference from the target value, but also none of the molded parts’ critical dimensions is within the specification, and <4> Process 3 performs the best because its pass rate is the highest as well as its average is exactly
the same as the target value. Therefore, Process 3 should be the best choice among the four. Nevertheless, it doesn’t seem optimistic at all from the mass production point of view; because its pass rate, the highest among the four, is still as low as 46.88%. Should we give up all four processes and make more efforts in searching for other processes?
developed processes
Method 2: Calculate the Variance (V) and Standard Deviation (SD, σ) of the raw data and infer that the process which produces the smallest variance or standard deviation performs the best. – A good step forward, but not enough.
The raw data’s value distribution is taken into account, which is not in the method using the average introduced above. And it starts involving some basic knowledge and assumption about statistics. It assumes that the distribution pattern of the raw data measured is the so-called normal distribution (as shown in Figure 1), meaning that the value distribution (the relationship between a value and its occurrence frequency) centers at the average of the sample group. With the value right at the average, its occurrence frequency is the highest; with the values deviate more and more at both sides from the average, their occurrence frequencies become lower and lower. As a result, the distribution curve looks bell-like.
Under this assumption, the so-called variance (V) of the raw data produced by each of the four processes is calculated
by the formula expressed as below.
Variance (V) is the sample group’s representative value that shows the width of the group’s distribution upon its average value. The bigger the value of variance, the wider the distribution of the individual measured values upon the average value of the sample group. On the other hand, the smaller the value of variance, the narrower distribution. When looking into the formula, it’s found that every individual measured value’s difference from average is squared, becoming a
positive number. By such an operation, even if the averages of several sample groups are all right at the target value, how wide/narrow the values of the sample groups each distributes can still be quantified, compared, and evaluated so that their performances can be differentiated. Additionally, such an operation help avoid mistaking an unstable process, which produces dimensions with wide distribution, for a good one just because the process coincidentally gets the sampled dimension average right the same as, or very close to, the target value.
Following the variance (V) obtained, we can calculate the so-called standard deviation (SD, σ) by the formula expressed
as below.
Mathematically, the value of the standard deviation is the square root of the value of variance. Similar to variance, the standard deviation is another representative value of the sample group to show the amount of sample values’ deviation from their average value. The bigger the value of standard deviation, the bigger the group deviation from its average value (wider distribution). On the other hand, the smaller the value of standard deviation, the smaller the group deviation (narrower distribution).
Statistically, both variance and standard deviation can be used to describe the width of a sample group’s value distribution upon its average, as well as the amount that a sample group deviates from its average. However, in practice in the industry, the standard deviation is more often used. Additionally, the standard deviation is used to tell the uncertainty degree of a set of measured data, therefore, representing the precision performance (stable or unstable) of
a process.
Back to the example in this article and as shown in Table 4, the results of the variance and standard deviation analyses suggest that <1> Process 4 performs the best (most precise; most stable) because both its variance and standard deviation are the smallest among the four, even though its pass rate is 0%; <2> the precision performance of Process 3 ranks third, after Process 4 and Process 2, even though its pass rate is the highest among the four; and <3> Process 1
performs the worst (least precise; least stable) because both its variance and standard deviation are the biggest among the four, even though its average is right the same as the target value.
The data analysis interpretation and the resulting judgment in this performance evaluation are diametrically opposed to that use the first method introduced above. Process 4 should be selected for mass production because it performs the most stably. However, it seems even more pessimistic from the mass production point of view; because, literally, its pass rate is 0%. Should we give up all four processes and make more efforts in searching for other processes?
four developed processes
It’s a good step forward for engineers to learn and apply the concept of standard deviation in their data analysis works about process performance evaluation and process selection. However, it’s still not enough because how small the standard deviation produced should be (i.e., what precise performance is acceptable) is still unknown or not defined. Moreover, the accuracy performance upon the target value is not considered and evaluated yet by this method.
Method 3: Use Process Precision Index (Cp) to tell the capability potential and infer that the process with the highest Cp value performs the best – the key index to get on the right track.
With the second method introduced above, evaluating an injection molding process by its performed standard deviation, now it’s known that the smaller the standard deviation of a process, the better; because it means the process is performed more stably and precisely. But how small the standard deviation should be and is acceptable, or how precise should it be and is acceptable? If the process performance is evaluated upon a target dimension, the answer will depend on its tolerance specification. Having known that the standard deviation represents the amount of the dimension’s distribution of a sample group that deviates from its average value, it is imaginable that the same standard deviation performed by a process can be comparatively considered more precise (small enough) upon a target dimension with a wide tolerance band; but otherwise with a narrow one. When discussing the precision or stability of a process, it’s essential to know “upon what” the precision or stability is based. Herein, the so-called Process Precision Index (Cp) is introduced for this purpose.
Process Precision Index (Cp) is calculated by the formula expressed as below.
The numerator in the formula (USL – LSL) is the difference between the upper and lower limits of the target dimension specification. It is the tolerance bandwidth of the specification; therefore, upon it, the discussion about the precision or stability of a process involving standard deviation performance becomes meaningful. In the example in this article, the USL is 20.10mm, and the LSL is 19.90mm; therefore, the tolerance bandwidth allowed with the target dimension
20.00mm is (20.10 – 19.90) = 0.20mm.
With the specified dimension tolerance bandwidth, the Process Precision Index (Cp) of each process can be analyzed. As shown in Table 5, the analysis result tells that Process 4 performs the best (most precise; most stable) upon the critical dimension “20.00 +0.10/-0.10 mm” because of its Process Precision Index (Cp) value the highest among the four. Accordingly, Process 2 ranks second, and then Process 3; and Process 1 performs the worst suggested by the
lowest Cp value.
Cp value has further meaning behind the quality control point of view. Suppose the process’s data distribution of a quality characteristic can be shifted so that its average value is mapped right onto the target value. In that case, the potential pass rate and potential defect rate of the process can be estimated, as shown in Table 6, based on the theory of probability in statistics. It explains why Process 4 is the most preferred, even though the present pass rate is 0%. The highest Cp value suggests it has the most potential and possibility, among the four, at process capability producing parts with the lowest defect rate.
From the viewpoint of process precision/stability improvement, how high the value of Cp should be set as a criterion depends on how high the defect rate would be tolerated. Nowadays, it’s believed that most of the companies in the industry require Cp values of their processes to be higher than 1.33 to ensure the potential of the process capabilities for future mass production.
Method 4: Use Process Capability Index (Cpk) to finalize the evaluation of a process and infer that the process with the highest Cpk value performs the best – the decisive index to represent the capability of a process that takes precision and accuracy performances into account at the same time.
The third method above-introduced that uses Process Precision Index (Cp) only evaluates the precision/stability performance of a process upon a specified tolerance range, despite the fact that the process’s produced critical dimensions are accurate or not. A practical method used for evaluating the performance of a process must cover the precision/stability performance and accuracy performance at the same time; the process can’t be said capable or qualified for mass production until all the critical dimensions of the parts are produced precisely/stably against the specified tolerance band and accurately at or nearby the target value simultaneously. Herein, the so-called Process Capability Index (Cpk) comes to serve this purpose.
Process Capability Index (Cpk) is calculated by the formula expressed as below.
In the formula, the existence of average (μ) and its arithmetical operation with the upper and lower specification limits suggest that the evaluation for accuracy performance is considered in this method, as well as the evaluation for precision/stability performance, which is told by being still existent of standard deviation (σ).
Table 7 shows data analysis results on the Process Capability Index (Cpk) of the four processes. In case that the average and the target value of a process are the same, such as Process 1 and Process 3 in this example, the Cp and Cpk values will be the same as well. And then as what Cp value functions, now the Cpk value also becomes an index that can be used to estimate the corresponding pass rate and defect rate as shown in Table 6. In this example, it seems that Process 3 is the best because its present Cpk value is the highest among the four. However, its pass rate is estimated as bad as less than 68.3%, so it seems that all four processes should be given up at first glance.
With the second thought, by reviewing the Cp value of the four processes, it was found that both Process 2 and Process 4 have the potential to reach a pass rate as high as more than 99.9937%. How to turn that potential into reality? The answer is to modify the mold dimension by offsetting the amount of the difference between the target value and average value, which is (20.00 – 19.88 = 0.12mm) for Process 2 and (20.00 – 19.60 = 0.40mm) for Process 4, individually. Table
8 shows the simulation outcome of the estimated raw data, pass rate, and Cpk value (Cpk -1) resulting from the offset on mold dimension for Process 2 and Process 4 individually. Now without changing the process, the Cpk value of Process 4 is expected to improve greatly from “-8.8012” to “2.9337”, making it the best process among the four, not only the performance in precision/stability but also accuracy. In terms of the process capability improvement, which is measured by the Cpk value, most of the companies in the industry require Cpk values of their processes to be higher than 1.33 for profitable mass production, similar to the Cp value criterion.
four developed processes
Summary:
Many companies in the industry use Process Capability Index (Cpk) as the criterion to qualify and approve their newly developed molds and injection molding processes. Engineers and managers who focus only on this index might easily conclude that they have to change the design in mold or part structure as they see a bad Cpk value. In that situation, they might miss and let go of the potential good process, like Process 2 and Process 4 in the example in this article, just because they review and evaluate by Cpk performance only. Engineers and managers should always evaluate a process’s Cp (Process Precision Index) and Cpk performance together. For an excellent process performance, it’s the high Cp value the precondition for getting a high Cpk value. What it needs at first to achieve a high Cpk value for a process is a high Cp value. Do not feel frustrated when poor Cpk and pass rate performance present; look at Cp value;
where there is a high Cp value, there is a way to high Cpk performance. Do not directly select the process with the highest Cpk value for improvement and mass production; select that with the highest Cp value at first, improve its Cpk performance by the way introduced in Method 4, and get an even higher Cpk value for mass production.
Complement:
Regarding the example presented in this article, when asking which process should be selected for mass production to the people who haven’t learned about Cp and Cpk, they should be hesitant at first glance, and it would be quite unimaginable for them that Process 4 could be among their choices. However, interestingly, when comparing the process selection to an athlete selection (Candidate A and Candidate B) for further training to win the archery competition at the Olympic Games by a 10-shot test, as illustrated in Figure 2, most people can pick Candidate B immediately on intuition because the two candidates’ performances in precision/stability and the potential to achieve consistent accuracy are visualized, and the decision-making becomes easier. Similarly, Cp and Cpk are the tools that visualize the precision/stability and accuracy performances of a process, respectively, as illustrated in Fig. 3, to help engineers and managers make correct decisions in process selection.
About the Author:
Hank Tsai is the owner and consultant of Effinno Technologies Co., Ltd. in Taiwan, an injection molding training and consulting service provider. He has been an SPE member since 1995 and has more than 25 years of experience in the injection molding industry. He has expertise in injection molding technologies and practices, production efficiency management, part costing, troubleshooting, simulation, mold/process/machine performance evaluation, and process optimization by Taguchi DOE. Contact: hank.tsai@effinno.com.
