2.1 Case Study Scenario
This case study research was conducted in one of the semiconductor companies in Malaysia coded name: Company A. Two sets of manufacturing data of different semiconductor products coded as Product A and Product B were collected from Company A. The supplier manufacturing input and output chain of this company is shown by Fig. 1 below where the raw materials are assembled to semifinished products and inspection is performed at the end of the inline process and the subsequent endline processing stage comprises of furnishing the semifinished products like inserting the coding program to the chip and inspecting the completed products through final test and inspection.
In general, inline process and endline process are managed by two different departments, each with different key performance indicators. The inline process aims to produce large quantity of semifinished products with optimum manufacturing and quality cost before delivering the products to the endline process. The normal practice of 100% inspection of every unit at the inline process are costly because resources like inspection machines and manpower are needed to inspect all lots or batches of items to judge conformance to predetermined standards. Therefore, only at the endline process where every semifinished product or unit received from the inline process would be coded and the final 100% inspection would take place to confirm if the produced units are fit and function. Upon completion of the endline process 100% inspection, the good quality of finished products is deemed fit and function and are ready to be shipped to customers following normal endline process cycle time. Any lots containing produced items with certain fraction of defects detected during the endline process will be hold and subjected to thorough fit and functional test for higher quality assurance and will not fail at customer side. Nonetheless, the thorough tests would incur additional cycle time on top of the normal endline process cycle time and, thus, impacting the endline process delivery commitment to customer.
For example, the endline process has committed to deliver one lot or batch consists of 10,000 pieces (pcs) of Product A to customer by a certain date following the normal endline process cycle time. However, when a lot of Product A received from inline process has high number of defectives resulting in it categorized as bad lot, it be hold from shipped to customer.
If the supplier would like to shift its internal inline process inspection from 100% to sampling, given the endline process inspection is maintained at 100%, what is the risk of its endline process inspection detecting bad lot due to sampling inspection at inline process? The objectives of this case study research are,
 to propose a statistical approach to acceptance sampling inspection plan for attributes by incorporating probability distribution at the inline process of manufacturing industry.
 to measure the risk of the endline process detecting bad lot due to sampling inspection at inline process by using operating characteristics (OC) curves.
2.2 Research Conceptual Framework
This study uses the common rejection rule in acceptance sampling plan in that; a lot will be rejected for 100% inspection if found k ≥ 1 defective item in n sample size. Thus, it can be said that it is a sampling without replacement, where each sample unit only has one chance to be selected. Assuming the supplier would like to shift from 100% inspection to a certain sample size in its inline process (semifinished products). Hence, the 100% inspection data of inline process is available. In this study, the endline process inspection is being retained at 100% inspection as a gating to protect customer. Thus, the scope of this study is within supplier’s factory (i.e., between inline process and endline process).
The relevance of this assumption is for supplier to reduce its operation cost by shifting its inline process inspection from 100% to acceptance sampling. It is also assumed that the endline process will tolerate a certain maximum defective items fraction, pLTPD in a lot depending on the product type. Lots detected by endline process which p ≥ pLTPD are called bad lots, b. Generally, there are three types of inspections strategy to be chosen for the inline process,
 100% inspection of units in a lot,
 sampling inspection of n units in a lot or,
 no inspection at all of units in a lot.
Fig. 2 shows the conceptual framework of this study that includes of all three types of inspection strategy at the inprocess production line and its effect to the endline process in detecting bad lots. Note that k is for c =0, where c is the acceptance number of a sampling plan. For the first type of inspection plan, the inline process executes 100% inspection of units in a lot (n = N) that any bad lots will be detected and rejected at inline process. Thus, the probability of bad lots detected at endline process will be 0. On the other hand, if there is no inspection conducted at the inline process, all bad lots available will be detected at the endline process inspection. The probability of bad lots detected at endline process is the fraction of bad lots from total lots (Fr = b/ TN).
If sampling inspection plan with n sample size being implemented, the inline process inspection may detect and rejected some of the bad lots when n sample is randomly taken from each lot. While OC curves measures how well the acceptance sampling perform with the probability of acceptance, Pa at various values of p , pm denotes the “worthiness” of the acceptance sampling. pm is the median of defective items fraction of lot at inline process representing the location of 50% lots from population (Swamidass, 2000; Schilling & Neubauer, 2009). Worthiness in the sense that if at pm , more than 50% of the lots in the population will fail the sampling plan, it is wiser not to waste time developing sampling plan. Hence, it also can be called as process average. If the bad lots not rejected at inline process due to sampling inspection, endline process will detect it. Therefore, the probability of bad lots detected at end line process (Pc) will be in between 0 to Fr (fraction of bad lots from total lots).
Mathematically, the concept of determining the risk or probability of bad lots detected at endline process due to sampling inspection at inline process can be represented by the probability theorem and Venn diagram (Fig. 3) as below,
Set notation
Let TN denote the set of all lots with N lot size under consideration, universal set
Let event b = {all lots with x/N ≥ pLTPD},
x is the number of defectives in N lot size; x/N is p,
and P(b) is given by Fr, bad lots fraction (Fr = b/ TN),
Let event d = {lot with k = 0 found in n sample size taken from N lot size},
k is the number of defectives found from n; k ≤ x and n ≤ N,
The multiplicative law of probability states that the probability of the intersection of two events b and d is,
P(b∩d) = P(b) x P(db) = P(d) x P(bd)
Since the probability of the occurrence of an event b is unaffected by the occurrence or nonoccurrence of event d, events b and d are independent.
If b and d are independent, then
P(db) = P(d) and P(bd) = P(b)
Thus, risk or probability of bad lots detected at endline process due to sampling inspection at inline process is the intersection of two events b and d given by,
Pc = P(b∩d) = P(b) x P(d)
P(d) is the probability of lot with k = 0 found in n sample size taken from N lot size depending upon type of probability distribution of its operating characteristics curve. Therefore, P(d) is given as below,
P(d) = P(k=0, n, p)
Hence, the risk or probability of bad lots detected at endline process due to sampling inspection at inline process,
Pc = P(b∩d) = P(b) x P(d) = Fr x [P(k=0, n, p)]
Pc = Fr x P(k=0,n,p)
2.3 Data Collection
This case study research was conducted in one of the semiconductor companies in Malaysia coded name: Company A. Two sets of manufacturing data of different semiconductor products coded as Product A and Product B were collected from Company A. The data collection was done in two phases (Phase 1 and Phase 2). In this research, an existing or secondary inspection data obtained from the manufacturing company were used. The inspection is nondestructive. The data were then analyzed graphically by using Microsoft Excel 2013 software and SAS JMP 10 software to derive the OC curves.
In Phase 1, six months manufacturing data of its inline process inspection were collected from January 2015 to June 2015 for sampling plan development. In Phase 2, another two months of manufacturing data at its endline process inspection were collected from July 2015 to August 2015 to monitor the sampling plan implemented at the inline process. The manufacturing data comprises of TN (total population lots in the process production line), N (lot size which varies), x (defective items in lot N), p (defective items fraction of each lot, p = x/N), pm (the median defective items fraction of lot at inline process), pLTPD (maximum defective items fraction of lot that can be tolerated by endline process), b (bad lots quantity with p ≥ pLTPD) and Fr (bad lots fraction, Fr = b/ TN). It is worth to note that pm, the median represents the location of defective items fraction at 50% of the population lots. The collected manufacturing data are in nature, attributes and discrete. Table 1 below shows the summary of manufacturing attributes data of Product A and Product B.
Table 1. Summary of manufacturing attributes
This following section mainly describes the procedure formulated in Phase 1 and monitoring of the effectiveness of designated sampling plan at endline processing in Phase 2.
2.4 Procedure Formulation (Phase 1)
Procedure formulation is a common term used by semiconductor manufacturing industry to describe the analytical process of developing a set of steps called procedure. The procedure formulation outlines the steps to develop a statistical method of acceptance sampling plan for attributes by incorporating probability distribution at inline process of manufacturing industry. This study proposes seven steps in the procedure formulation in developing a statistical method of acceptance sampling inspection plan for attributes by incorporating probability distribution at inline process of manufacturing process.
1. Define the maximum defective items fraction of lot that can be tolerated by endline process, pLTPD.
Example: pLTPD for Product A is 0.009 (or 0.90%). This value is set by endline process, which means endline process will only accept if the lot contains fraction defectives of <0.009 (9 defectives item over 1,000 item).
2. Determine the median defective items fraction of lot at inline process, pm and bad lots fraction, Fr (based on its historical 100% inspection data at inline process) by using run chart or histogram.
Example: The median yield of Product A at inline process is 99.87%, thus the median of defective items percentage in a lot (pm) is 0.13% (or 0.0013) and the Fr is 0.0065 (or 0.65%).
3.Choose a sample size.
Say n = 200 units per lot is chosen as the sample size as the starting point (This selected n value is purely based on ISO 28691, general normal inspection Level II at AQL of 0.065% and lot size, N between 501 to 3200 units. Note that the selected n value for the starting point has no statistical basis).
4. Choose the appropriate operating characteristics curve (OC curve) by developing different OC curve distribution at various lot size.
Common probability distributions used for OC curves are hypergeometric and binomial. Construct the OC curves for both probability distributions with different N lot size to determine P (probability of a lot rejected at inline process when p ≥ pLTPD due to method of inspection).
Example: The method of inspection is sampling with sample size, n = 200 and k=c=0 (reject the lot if number of defectives found in sample size more or equal than 1). Compare the characteristics of these two distributions by using pm and pLTPD as the reference lines.
The guideline in choosing the OC curve based on literature review (Schilling & Neubauer, 2009; Samohyl, 2018):
 For very big lot size (infinite), choose binomial distribution since the effect of sample size is less significant
 For small lot size, choose hypergeometric distribution since the effect of sample size is more significant (sensitive to sample size).
Fig. 4 shows the OC curves comparing hypergeometric and binomial OC curves built using JMP software for n=200 and k=c=0 at different lot size, N.
As the percent or fraction defectives (p) increases, the probability of acceptance (Pa) decreases for both distributions and vice versa. This is logical since the chances to find a defective item during sampling is higher when the actual quantity of defective item in the lot increased. This theory is also explained by Schilling and Neubauer (2009). Interestingly, for lot size, N = 600 and percent defective item, p = 0.35%, the probability of acceptance (Pa) given by hypergeometric distribution is less than 0.5 (around 43%) and binomial distribution is more than 0.5 (around 51%). Since Pa = 0.5 is the indifference quality (IQ) which defines the boundary to evaluate whether the sampling plan is “meaningful” and “sensitive”, the choice of probability distribution is important to select a proper sampling. Hypergeometric OC curve shows that at p = 0.35%, the sampling plan is not meaningful since the Pa < 0.5 (around 43%). In contrast, binomial OC curve shows that the sampling plan is meaningful with Pa > 0.5 (around 51%). The gap of probability acceptance value between hypergeometric and binomial OC curves reduces as the lot size increases as shown in the OC curves for N = 1200.
Hypergeometric and binomial OC curve also can be developed through Microsoft Excel by using the function =HYPERGEOM.DIST() and =BINOM.DIST() respectively across different value of x defective items.
Table 2 shows the Excel function for k=c=0, n=200, N=800 at different value of x defective items. Meanwhile, Table 3 shows the Excel function for k=c=0, n=200 at different value of p fraction defective items. Since N is infinite, it is not one of variables in the BINOM.DIST() Excel function.
Table 2. Excel function for k=c=0, n=200, N=800 at different value of x defective items
Table 3. Excel function for k=c=0, n=200 at different value of p fraction defective items
n (sample size)

p=x/N (fraction defective in the lot)

x=Np (number of defectives in the lot)

P(k=0,n=200)

Excel function for hypergeometric probability

200

0.00%

0

1

=BINOM.DIST(0,0,200,FALSE)

200

0.13%

1

0.78

=BINOM.DIST(0,0.13%,200,FALSE)

200

0.25%

2

0.61

=BINOM.DIST(0,0.25%,200,FALSE)

200

0.38%

3

0.47

=BINOM.DIST(0,0.38%,200,FALSE)

200

0.50%

4

0.37

=BINOM.DIST(0,0.50%,200,FALSE)

200

0.63%

5

0.29

=BINOM.DIST(0,0.63%,200,FALSE)

200

0.75%

6

0.22

=BINOM.DIST(0,0.75%,200,FALSE)

200

0.88%

7

0.17

=BINOM.DIST(0,0.88%,200,FALSE)

200

1.00%

8

0.13

=BINOM.DIST(0,1.00%,200,FALSE)

200

1.13%

9

0.10

=BINOM.DIST(0,1.13%,200,FALSE)

5. Check if the selected sampling plan is both “meaningful” and “sensitive” based on the OC curve.
The sampling plan is “meaningful” if at its inline process pm (which represent the location of defective items fractions at 50% of the population lots), the probability of lot to be accepted, Pa is more than 0.5. This is because at Pa = 0.5, the inline process will have fair chances of the sampling to pass or fail. Since, whenever the sampling fail, the inline process needs to perform 100% inspection of all the units in the lot, the inline process would like to have more than fair chances of sampling passing for it to be meaningful. It is not meaningful to switch from 100% inspection to sampling plan if the sampling plan will cause very high lot rejection rate at inline process.
Likewise, the sampling plan is “sensitive” if at pLTPD, the probability of lot to be accepted, Pa is less than 0.5. This is because at Pa = 0.5, the inline process will have fair chances of sampling failing or passing. The endline process would want the inline process to have sampling plan that is sensitive to reject bad lot (p ≥ pLTPD). The sampling is not sensitive if the sampling plan at inline process, will cause very high bad lots being detected at endline process.
Example: Hypergeometric OC curve has been chosen in step 4 and OC curve for P(k=0, n=200) was developed for Product A.
Through this step, the quality of product being assessed and exerts pressure to the inline process to improve quality before switching inspection from 100% to sampling plan. In other words, if the quality of product is high, it is worthwhile to shift from 100% inspection to sampling.
6. Determine the risk (probability) of bad lots detected at endline process.
Example: By using Hypergeometric OC Curve with P(k=0, n=200, p=pLTPD = 0.009).
Pc = P(k=0, n, p= pLTPD) x Fr of Product A
Pc = P(k=0, 200, 0.009) x 0.0065
Pc = 14% x 0.0065 = ~ 0.09% (equivalent to the risk of endline process detecting 9 bad lots per 10,000 lots)
7. Set an agreed risk of bad lots detected at endline process, Pc as benchmark for further sampling size optimization on other product.
Pc ≤ P(k=0,n,p=pLTPD) x Fr
Example: Benchmark Pc = 0.09% of Product A if inline process would like to develop sampling plan for Product B.
Based on Product B inline process quality performance, pm=0.0012, tighter pLTPD = 0.0083, Fr=0.0164, if sample size n=200 is chosen, the Pc expected is,
Pc = 14% x 0.0164 = ~ 0.2296% (equivalent to the risk of endline process detecting 22 bad lots per 10,000 lots)
Assuming inline process and endline process agreed on a value of Pc=0.09%, a different sample size is needed for Product B since 0.2296% is greater than 0.09%. Step 2 to 6 is repeated with higher sample size, say n=315 units (based on ISO 28691, general tighten inspection Level III at AQL of 0.065% and lot size, N between 501 to 3200 units).
2.5 Trial Run
Based on the Procedure Formulation, the Hypergeometric OC curve of Product A and Product B are presented in Figs. 5 and 6, respectively by using the SAS JMP software.
Based on Product B inline process quality performance, pm= 0.0012, tighter pLTPD = 0.0083, Fr = 0.0164, if sample size n = 315 is chosen, the Pc expected is,
Pc = 5% x 0.0164 = ~ 0.082% (equivalent to the risk of endline process detecting 8 bad lots per 10,000 lots).
Trial run of acceptance sampling with sample size, n = 200 were conducted at inline process for 15 lots from Product A. The lots were chosen randomly from the production line. Only 15 lots were chosen because of operation restriction imposed by the company. Table 4 shows the trial run of acceptance sampling at inline process with different lots.
Table 4. Trial run of acceptance sampling at inline process with different products
Product

Lot

N
(Lot size)

k (number of defectives found during sampling)

x (number of defectives in the lot) after 100% inspection

p=x/N (fraction defective in the lot)

pLTPD (maximum defectives fraction in a lot that can be tolerated by endline process)

A

Lot 1

792

1/200

3/792

0.38%

0.90%


Lot 2

794

0/200

0/794

0.00%

0.90%


Lot 3

789

0/200

1/789

0.13%

0.90%


Lot 4

795

0/200

1/795

0.13%

0.90%


Lot 5

647

0/200

1/647

0.16%

0.90%


Lot 6

791

1/200

3/791

0.38%

0.90%


Lot 7

793

0/200

1/793

0.13%

0.90%


Lot 8

1134

0/200

0/1134

0.00%

0.90%


Lot 9

789

0/200

1/789

0.13%

0.90%


Lot 10

794

0/200

0/794

0.00%

0.90%


Lot 11

796

0/200

0/796

0.00%

0.90%


Lot 12

791

0/200

1/791

0.13%

0.90%


Lot 13

855

0/200

1/855

0.12%

0.90%


Lot 14

793

0/200

0/793

0.00%

0.90%


Lot 15

797

0/200

0/797

0.00%

0.90%

The trial run shows that 2 out of 15 lots (13%) rejected by the sampling plan even though it did not exceed the pLTPD. Thus, decision was made to implement the designated acceptance sampling plan in Phase 2.