Chapter 3. Process Capability Indices
4.3. Sampling Procedure and Decision Making
Both producer and consumer will lay down their requirements in the contract:
the producer demands that not too many ‘good’ lots shall be rejected by the sampling inspection, and consumer demands that not too many ‘bad’ lots shall be accepted. Therefore, selection of a meaningful critical value for capability test requires specification of an acceptable quality level (AQL) and a lot tolerance percent defective (LTPD) for the S value. The AQL is simply a standard Tpk against which to judge the lots. It is hoped that the producer’s process will operate at a fallout level that is considerably better than the AQL. In choosing a sampling plan attempts will be made to meet these somewhat opposing requirements. Thus, both producers and consumers may set their own safeguard line to protect their benefits.
In order to judge whether a given process meets the capability requirement, the first step is determine the specified value of the capability requirement and (or fraction of defectives AQL and LTPD), and the AQL
S
SLTPD
α
-risk, β -risk.Two kinds of risks are balanced using a well-designed sampling plan. That is, if production process capability with STpk =SAQL (in high quality), the probability of acceptance must be larger than 1− . And if the producer’s capability is only
α
with STpk =SLTPD (in low quality), consumer would accept no more than β . Then, by checking Table 4, we would obtain the sample size and the critical value based on given values ofS value is greater than the critical value pk , then the consumer will accept the entire product. Otherwise, we do not have sufficient information to conclude that the process meets the present capability requirement. In this situation, the consumer will reject the product.
c0
For the proposed sampling plan to be practical and convenient to use, a step-by-step procedure is provided as below.
Step 1: Decide the process capability requirements (i.e. set the values of
and ), and choose the AQL
S SLTPD
α
-risk, the chance of wrongly concluding a capable process as incapable, and the β -risk, the chance of wrongly concluding a bad lot as good one.Step 2: Check Table 4 to find the critical value (or acceptance criterion) and the required number of product units for inspection, ( , ), based on given values of
n c0
α
-risk, β -risk, SAQL and SLTPD.Step 3: Calculate the value of ˆS (sample estimator) from these Tpk inspected samples.
n
Step 4: Make decisions to accept the entire lot if the estimated ˆS value is Tpk greater than the critical value (c0 ˆT
S > ). Otherwise, we reject the entire pk
products. 0
c
Table 4. Required sample sizes (n ) and critical acceptance values ( ) for various c0
α
-risk andChapter 5. An Application
Silicon photodiodes are semiconductor devices used for the detection of light in ultraviolet, visible and infrared spectral regions. Because of their small size, low noise, high speed and good spectral response, silicon photodiodes are being used for both civilian and defense related applications. Depending on the requirement of any particular application, photodiodes can be made in any desired geometry, and provided in a special package with a filter for any special application such as Mouse, Remote control, Receiver module, Wireless communication, etc. Figure 5 shows a particular chip of silicon photodiodes.
Figure 5. A silicon photodiode chip.
Silicon photodiodes are constructed from single crystal silicon wafers similar to those used in the manufacture of integrated circuits. The major difference is that photodiodes require higher purity silicon. A cross section of a typical silicon photodiode chip is shown in Figure 6. The bulk N-type silicon is the starting material. A thin “P” layer is formed on the front surface of the device by thermal diffusion or ion implantation of the appropriate doping material. The interface between the “P” layer and the “N” silicon is known as a P-N junction. Small metal contacts are applied to the front surface of the device and the entire back is coated with a contact metal. The back contact is the cathode, and the front contact is the anode. The active area is coated with either silicon nitride, silicon monoxide or silicon dioxide for protection and to serve as an anti-reflection coating. The thickness of this coating is optimized for particular irradiation wavelengths.
The following case is taken from a manufacturing factory located in a science-based industrial park at Hsinchu, Taiwan, making various types of silicon photodiode chips. The particular silicon photodiode chip we investigate has multiple concerned characteristics including the chip length (L), chip width (W), chip thickness (T) and P bonding pad (P). The product specification limits for the L, W, T and P characteristics of the silicon photodiode chip are set at (USL, LSL)
= (35.984 mil, 34.016 mil), (35.984 mil, 34.016 mil), (12.784 mil, 10.816 mil) and (5.393 mil, 4.607 mil), respectively.
Figure 6. A cross section of a typical silicon photodiode chip.
Table 5. Sample data of the silicon photodiode chip characteristics.
chip length (L) (unit: mil)
35.022 35.410 34.914 35.207 34.796 34.672 35.173 34.638 34.662 34.843 34.613 35.145 34.928 34.664 34.769 34.744 34.981 34.372 34.819 34.638 35.086 35.156 35.038 34.974 34.315 35.007 34.772 34.931 34.915 34.699 35.151 35.062 34.909 35.232 35.410 34.712 34.993 34.711 35.195 35.130 34.866 34.855 34.804 35.241 34.935 34.961 34.673 35.081 35.699 35.075 34.630 35.064 34.627 35.039 34.560 35.187 35.297 35.404 35.360 34.803 35.115 35.237 34.721 35.052 35.198 34.665 34.752 35.202
chip width (W) (unit: mil)
34.998 35.168 35.437 35.154 34.748 35.241 34.947 34.739 35.670 35.026 34.700 35.203 35.538 35.190 35.253 35.034 34.949 35.279 34.938 35.024 34.831 34.781 34.914 34.620 35.080 35.033 34.987 34.281 34.837 35.080 35.118 35.057 34.988 34.487 35.109 34.958 35.155 35.192 34.972 35.104 34.893 34.897 34.878 34.695 35.075 35.395 34.815 34.840 35.231 35.013 35.493 34.456 34.669 34.756 35.696 35.138 34.858 35.010 34.899 35.313 35.006 35.262 35.225 34.926 34.964 35.041 34.744 34.924
chip thickness (T) (unit: mil)
11.783 11.606 11.978 11.585 11.665 11.578 12.002 11.671 11.472 11.955 12.074 11.928 11.826 11.637 11.657 12.034 12.019 11.684 11.953 11.841 11.729 11.642 11.855 11.454 11.501 12.183 11.786 12.299 11.754 11.833 11.975 11.766 11.701 12.072 11.635 11.953 11.703 11.941 11.961 11.642 11.901 11.996 11.600 11.869 11.793 11.567 11.748 11.770 11.594 11.429 11.627 11.799 11.473 12.040 11.686 11.408 11.821 12.083 11.810 11.831 11.795 11.833 11.968 11.835 11.944 11.897 11.825 11.853
P bonding pad (P) (unit: mil)
4.882 4.887 4.998 5.043 4.893 5.097 5.013 5.328 5.049 4.843 5.046 5.060 4.939 5.028 4.928 5.002 5.050 5.143 5.093 4.683 5.034 5.099 4.999 5.103 5.093 5.157 5.148 5.115 4.801 4.881 5.082 4.883 4.927 5.082 5.002 5.128 4.839 5.058 4.804 4.827 5.035 4.983 4.864 4.965 4.930 4.892 5.030 4.821 5.038 4.757 5.063 5.041 5.091 4.917 5.082 4.871 5.108 4.745 5.034 4.912 4.907 5.018 4.960 5.116 4.987 5.122 4.943 5.008
Once the characteristic data do not fall within the specification limits, the lifetime or reliability of the silicon photodiode will be discounted. In the contract, the performance requirement SAQL and SLTPD are set to be 1.33 and 1.00 with
α
-risk and β -risk both set to be 0.05. Then, the problem for the inspection practitioners is to determine the critical acceptance value and the required sample size for the sampling plan that provides desired levels of protection for both the producer and the consumer. Based on the proposed procedure and Table 4, the practitioners can acquire the critical acceptance value and inspected sample size as (n, c0) = (68, 1.1416). The required samples for inspection are randomly taken, and the observations are displayed in Table 5.Based on the observations, we calculate the sample estimate ˆS of Tpk S as Tpk follows. Table 6 presents the sample average (X ), sample standard deviation (j )
and j
S of each silicon photodiode chip characteristics.
ˆpkj
characteristics X j Sj S
chip length (L) 34.9487 0.2642 1.2202 chip width (W) 35.0136 0.2614 1.2531 chip thickness (T) 11.7960 0.1884 1.7405 P bonding pad (P) 4.9899 0.1170 1.1152
Therefore, in this case, the consumer would reject the entire lot, since the sample estimate ˆS = 1.0763 is smaller than the critical acceptance value 1.1416 Tpk of the sampling plan. The process yield is exactly 0.9988 for index ˆS = 1.0763. Tpk Note that for existing sampling plans, it is almost certain that any samples of 68 silicon photodiode chips will contain zero defective items. All the products therefore will be accepted, which obviously provide no protection to the consumer at all.
Chapter 6. Conclusion
Acceptance sampling plans are practical tools for quality assurance applications. It provides the producer and the consumer a decision rule for product sentencing to meet their needs. However, as the rapid advancement of the manufacturing technology and stringent customers demand is enforced, the manufactured items require values of several different characteristics for adequate description of their quality. The S index measures the overall process yield Tpk when the processes with multiple characteristics. Therefore, in this paper, we developed a variables sampling plan based on the process capability index S to Tpk deal with lot sentencing problem even when the lots or processes with multiple characteristics. We developed a method to determine the sample size required for inspection and the corresponding acceptance criterion, to provide the desired levels of protection to both producers and consumers. We tabulated the required sample size and the corresponding critical acceptance value for various producer’s risks, the consumer’s risks with the capability requirement AQL and the LTPD.
The results obtained in this paper are useful to the practitioners in making reliable decisions.
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