mathematics 11 02872

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Citation:Wang, C.-C.; Chang, Y.-S.

Dynamic Acceptance Sampling Strategy Based on Product Quality Performance Using Examples from IC Test Factory.Mathematics2023,11, 2872. https://doi.org/10.3390/

math11132872

Academic Editors: Qing Li, Bing Si and Renjie Hu

Received: 28 May 2023 Revised: 23 June 2023 Accepted: 26 June 2023 Published: 27 June 2023

Licensee MDPI, Basel, Switzerland.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://

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mathematics

Article

Dynamic Acceptance Sampling Strategy Based on Product Quality Performance Using Examples from IC Test Factory

Chien-Chih Wang^1,* and Yu-Shan Chang²

1 Department of Industrial Engineering and Management, Ming Chi University of Technology, New Taipei City 243303, Taiwan

2 BellWether Electronic Corporation, Taoyuan City 330010, Taiwan; [email protected]

* Correspondence: [email protected]; Tel.: +886-2-2908-9889

Abstract:Acceptance sampling plans are divided into attributes and variables, which are used to evaluate the mechanism for determining lot quality. Traditional attribute sampling plans usually choose the Acceptable Quality Level (AQL) for each stage based on experience but need practical guidelines to follow. Previous research endeavors have predominantly centered around statistical perspectives and emphasized the reduction of sample size or sampling frequency while allocating lesser consideration to cost factors and practical applications when formulating sampling decisions.

This study proposes a dynamic sampling strategy to minimize costs and estimate AQL values and sample sizes for each stage based on product quality performance to establish a more effective and flexible sampling strategy. The study verifies the scenario in an integrated circuit (IC) testing factory, considering multiple combinations of between-batch quality conditions, within-batch quality conditions, sampling method, and cost ratio, and conducts sampling inspection simulations. When quality changes, the dynamic strategy is activated to adjust AQL. Finally, based on the sampling errors and costs in the inspection results, a comparison is made with the traditional MIL-STD-105E sampling plan, confirming that the dynamic AQL sampling plan has significantly improved performance.

Keywords:dynamic sampling; sampling error; quality cost; IC test factory; case simulation MSC:62P30

1. Introduction

The topic of measuring quality has always been important and has generated a lot of discussion. Traditionally, 100% inspection is often used to ensure product quality meets requirements. However, this method is time-consuming, costly, and unsuitable for destructive testing. Dodge and Roming proposed acceptance sampling plans to address these issues in 1929 [1]. Acceptance sampling plans determine whether a batch of inspected goods is accepted or rejected by sampling the batch. They judge the quality of the sampled items based on the risks both the producer and the consumer can bear. Compared to 100%

inspection, acceptance sampling plans save time and cost, making them one of the critical methods buyers and sellers use to measure batch quality. Acceptance sampling plans help producers and consumers reach agreements on quality standards and provide a basis for resolving disputes. This makes them essential to the quality control process [2–6].

Depending on the sampling method, acceptance sampling plans can be divided into single sampling, double sampling, multiple sampling, and other unique sampling plans.

In practical applications, the MIL-STD-105E sampling plan is commonly used as the basis for quality inspection, and the acceptable quality level (AQL) for each stage is usually determined based on industry experience and customer standards. Current sampling plans, however, still need to adjust successfully to reflect the actual quality condition of the products. For example, in the inspection process of IC product testing factories, the commonly used sampling plan is MIL-STD-105E, and the AQL values used by each

Mathematics2023,11, 2872. https://doi.org/10.3390/math11132872 https://www.mdpi.com/journal/mathematics

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Mathematics2023,11, 2872 2 of 16

inspection station are usually fixed values given based on the manufacturer’s experience or the production or product nature [7]. Product quality is dynamic in actual products due to production process variations. Therefore, the AQL value should be adjusted according to the different quality conditions of the products each time they enter the inspection station. However, this needs to be addressed. In addition, defects are still possible during an inspection. Taking IC products as an example, this may result in appearance losses.

Therefore, although increasing the number of samples can reduce sampling errors, it can also increase the risk of product defects, resulting in costly failures.

Although much literature discusses methods to reduce the number of sampling samples or the frequency of sampling, most of them make sampling decisions from a statistical perspective [8–12]. It needs to consider cost factors. In the actual application of sampling plans, the unknown quality of products is also one of the influencing factors in determining the sampling strategy. Therefore, there is a need for further research and development of more effective and efficient sampling plans that can adjust to the dynamic nature of product quality. These plans should consider both statistical and cost factors. Researchers need to integrate cost-sensitive optimization algorithms into their sampling plans. These algorithms should consider the cost of sampling and the cost of false rejection or acceptance when determining the optimal sampling strategy. It will make the sampling plan more efficient and effective in identifying product defects.

Therefore, this study proposes a dynamic sampling strategy based on single sampling, which aims to minimize costs by balancing between reducing errors and minimizing costs, considering the quality status of the product, sampling methods, and potential costs of sampling inspection. The AQL (Acceptable Quality Level) values and sample sizes for each product station are calculated to establish a more effective and flexible sampling strategy.

This study uses an IC (Integrated Circuit) testing factory as an example of simulation verification. It focuses on the sampling strategy issue at the sampling inspection station of the IC product testing factory. It is assumed that the empirical process of this study conforms to the testing factory inspection process and incorporates various cost factors into the sampling plan for consideration, along with the current sampling plan and the dynamic strategy proposed in this study for analysis, comparison, and discussion.

2. Literature Review

An acceptance sampling plan uses probability theory to determine whether many lots meet quality standards. Govindarajan and Bebbington’s study found that acceptance sampling is cost-effective for consumers with less-than-perfect lot quality, and it is also helpful for consumers with perfect quality and inspection errors [13]. Govindaraju et al. developed a statistical method to determine the appropriate sampling frequency for auto-samplers [14].

They used a two-state Markov chain model to detect foreign matter contamination during production. To improve acceptance sampling, Luca has proposed utilizing quality history less selectively [15]. With this approach, a smaller sample size can be used for inspections based on attributes and variables. By analyzing sample data, it is possible to determine whether production lots should be accepted or rejected. For imperfect quality items, manufacturers must weigh the impact of full inspection versus sampling inspection on economic production lot sizing. Whether the inspection is online or offline determines the decision. Bose and Guha developed an online sampling inspection model and found that the unit inspection cost threshold determines whether complete or sampling inspections are performed [16]. As a result of online sampling inspection, inventory holding costs can be reduced, and optimal profits can be increased. As a result of statistical sampling uncertainty, Markowski et al. discuss a problem with attribute acceptance sampling and inspection errors [17]. To reduce the risk of classification errors, new sampling plans are proposed that overcome traditional limitations. Comparison results are provided, and regulations of the new methods are discussed. It is not uncommon for the variable under study to come from an unknown distribution in practice. Existing methods can only be used for a few inspections in this case.

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In the case of unknown distributions or populations, a new sign sampling plan is proposed. The parameters of the proposed sampling plan are determined based on the given risks. Using industrial data, the proposed method is applied [18]. An acceptance sampling inspection plan for Lindley distributed quality characteristics is presented by Saha et al. A power Lindley distribution approach and an exact approach are included in the method [19]. Based on a modified Lindley distribution model, Tashkandy et al. assess quality control through representative sample testing [20]. It analyzes different acceptance sampling strategies at the consumer’s risk and calculates optimal sample sizes.

A cost function is an essential consideration for developing an acceptance sampling plan. Facchinetti et al. used a generalized beta distribution to create acceptance sampling plans for attributes to minimize quality cost [21]. They proposed a method to identify the best plan based on the production process and quality costs, including a simulation to show how key parameters affect the optimal plan. Wu and Darmawan suggest modifying MSP that considers independence between stages and integrates with the third-generation capability index [22]. They establish a mathematical model with two constraints to obtain triple plan parameters that minimize the average sample number and satisfy specified risk-and-quality conditions. The proposed model offers better efficiency and protection to stakeholders than the existing plan.

In contrast to previous studies, this research considers the entire sampling process when making recommendations. This means we address the issue of loss and consider factors such as quality and sampling methods to optimize our recommendations for the procedure. This approach is more accurate and efficient than previous methods, which only considered the loss of samples. Considering the entire sampling process, our recommendations are better tailored to the given context and can lead to improved outcomes.

3. Methodology

3.1. The Relationship between Sampling Number and AQL

In practice, single sampling assumes thatNis the total number of lots inspected for a product before it enters the inspection process. Based on the experience value and customer requirements, the AQL value is set, and the sampling plan design is used to determine how many inspections should be conducted. Figure1shows the sampling inspection process after the inspection process results in a defective sample. If the number of defective samples is less than the AQL value, the lot is accepted, and if the number of defective samples is greater than the AQL value, the lot is rejected.

Mathematics 2023, 11, x FOR PEER REVIEW 3 of 15

for the variable under study to come from an unknown distribution in practice. Existing methods can only be used for a few inspections in this case.

In the case of unknown distributions or populations, a new sign sampling plan is proposed. The parameters of the proposed sampling plan are determined based on the given risks. Using industrial data, the proposed method is applied [18]. An acceptance sampling inspection plan for Lindley distributed quality characteristics is presented by Saha et al. A power Lindley distribution approach and an exact approach are included in the method [19]. Based on a modified Lindley distribution model, Tashkandy et al. assess quality control through representative sample testing [20]. It analyzes diﬀerent acceptance sampling strategies at the consumer’s risk and calculates optimal sample sizes.

A cost function is an essential consideration for developing an acceptance sampling plan. Facchinetti et al. used a generalized beta distribution to create acceptance sampling plans for attributes to minimize quality cost [21]. They proposed a method to identify the best plan based on the production process and quality costs, including a simulation to show how key parameters affect the optimal plan. Wu and Darmawan suggest modifying MSP that considers independence between stages and integrates with the third-generation capability index [22]. They establish a mathematical model with two constraints to obtain triple plan parameters that minimize the average sample number and satisfy specified risk-and-quality conditions. The proposed model offers better efficiency and protection to stakeholders than the existing plan.

In contrast to previous studies, this research considers the entire sampling process when making recommendations. This means we address the issue of loss and consider factors such as quality and sampling methods to optimize our recommendations for the procedure. This approach is more accurate and eﬃcient than previous methods, which only considered the loss of samples. Considering the entire sampling process, our recom- mendations are better tailored to the given context and can lead to improved outcomes.

3. Methodology

3.1. The Relationship between Sampling Number and AQL

In practice, single sampling assumes that N is the total number of lots inspected for a product before it enters the inspection process. Based on the experience value and cus- tomer requirements, the AQL value is set, and the sampling plan design is used to deter- mine how many inspections should be conducted. Figure 1 shows the sampling inspection process after the inspection process results in a defective sample. If the number of defec- tive samples is less than the AQL value, the lot is accepted, and if the number of defective samples is greater than the AQL value, the lot is rejected.

Figure 1. The single-sampling inspection process.

This paper uses the production defect rate, denoted as p, as a metric to measure the percentage of defective in the previous production. We can estimate the number of popu- lation defects (𝑁

^∗

) based on the number of defective samples as follows:

Figure 1.The single-sampling inspection process.

This paper uses the production defect rate, denoted as p, as a metric to measure the percentage of defective in the previous production. We can estimate the number of population defects (N_B^∗) based on the number of defective samples as follows:

N_B^∗= ⁿ^B

n ×N (1)

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Nis the lot of batch size,nis the sampling size, andn_Bis the number of sample defects.

Accordingly, the following results are obtained if the number of defective products in the batch isN_B^T.

(1). In the case of N_B^∗ < N_B^T, defects in the population may be misclassified as good products R₁ because the defective population is smaller than the actual number of defects.

R1=N_B^T−N_B^∗ =N_B^T− ⁿ^B

n ×N (2)

(2). If the case of N_B^∗ < N_B^T, good products in the population may be misclassified as defectsR2because the defective population is larger than the actual number of defects.

R2=N_B^∗−N_B^T = ⁿ^B

n ×N−N_B^T (3)

Sampling errors can be divided into two types as follows:

(1). Type I error (α)

α=P(The product is deemed defective after inspection|The actual product is good)

= _N−N^R² _T

B

(4)

(2). Type II error (β)

β=P(After inspection, the product is good|In actuality, the product is defective) = ^R¹

N^T_B (5) As shown in Figure2, the dynamic AQL sampling plan proposed in this study consists of the following steps.

𝑁

^∗

= 𝑛

𝑛 × 𝑁 (1)

N is the lot of batch size, n is the sampling size, and 𝑛 is the number of sample defects. Accordingly, the following results are obtained if the number of defective prod- ucts in the batch is 𝑁 .

(1). In the case of 𝑁

^∗

< 𝑁 , defects in the population may be misclassified as good prod- ucts 𝑅 because the defective population is smaller than the actual number of defects.

𝑅 = 𝑁 − 𝑁

^∗

= 𝑁 − 𝑛

𝑛 × 𝑁 (2)

(2). If the case of 𝑁

^∗

< 𝑁 , good products in the population may be misclassified as defects 𝑅 because the defective population is larger than the actual number of defects.

𝑅 = 𝑁

^∗

− 𝑁 = 𝑛

𝑛 × 𝑁 − 𝑁 (3)

Sampling errors can be divided into two types as follows:

(1). Type I error ( 𝛼 )

𝛼 = 𝑃(The product is deemed defective after inspection|The actual product is good)

= 𝑅

𝑁 − 𝑁

(4)

(2). Type II error (𝛽)

𝜷 = 𝑷 After inspection, the product is good In actuality, the product is defective = 𝑹

_𝟏

𝑵

_𝑩^𝑻

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As shown in Figure 2, the dynamic AQL sampling plan proposed in this study con- sists of the following steps.

Figure 2. Dynamic AQL Sampling Plan Simulation Program.

Considering that the total number of products entering IQC inspection is N, and the AQL of each inspection depends on the quality condition of the previous product, the number of samples taken will also vary with the quality condition. Assuming that N prod- ucts are entering the inspection process, the sampling plan is the same as the traditional

Figure 2.Dynamic AQL Sampling Plan Simulation Program.

Considering that the total number of products entering IQC inspection is N, and the AQL of each inspection depends on the quality condition of the previous product, the number of samples taken will also vary with the quality condition. Assuming thatN products are entering the inspection process, the sampling plan is the same as the traditional sampling plan when a product enters the sampling process for the first time. Based on the customer requirements, the experience value, and the number of samples, the AQL value is AQL1according to the experience value and customer requirements, and the number of samples to be taken isn_B₁according to the sampling plan.

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When the product enters the sampling process for the second time, the same AQL value of AQL₁and sample number as the first sampling are still adopted because of the unknown quality change. After the IQC inspection process, it is assumed that there arenB₂

defective products in the sample. When the product enters the sampling process for the third time, the change in product quality can be known from the second and first sampling results. WhennB₁ <nB₂, that is, the number of defects in the second sampling is smaller than the number of defects in the first sampling, the estimated product quality is improved, so the AQL₃value in the third sampling can be increased. The number of samples required for the third sampling can be reduced. On the contrary, the estimated product quality decreases whennB₁ >nB₂, that is, the number of defects in the second sampling is more significant than in the first sampling. Thus, the third sampling must have a lower AQL3

value, and more samples must be obtained for a third sampling. Similarly, the changes in product quality between the third and second samplings can determine the AQL₄value and the number of samples required for the fourth sampling. Doing so makes it possible to decide on the AQLsand sample numbersnSfor a product entering this inspection station for theS-th time.

The adjustment magnitude for defined AQLsis∆AQL_S

∆AQL_S= ^N

B∗_S₋₂−N_B^∗

S−1

N (6)

whereN_B^∗

S−2 is the estimated number of defects in theS−2 times,N_B^∗

S−1 is the number of defects in theS−1 time, andNis the total number of lots inspected for a product. Therefore, the AQLs for theS-th time are as follows.

AQL_S=AQLS−1+∆AQL_S (7)

The following relationship equation can be obtained from the AQL definition.

1−α≤P(D≤c|AQL) (8)

wherenis the number of samples taken per lot,cis the acceptance number (Ac), andDis a random variable defined as the number of defects.

Dis a binomial distribution, and as the sample size (n) tends towards infinity, D approx- imates a normal distribution. When employing the normal distribution as an approximation for the binomial distribution, the introduction of a continuous correction term enhances this approximation, yielding an improved estimate whereP(D≤C) ≈ P(D≤c+0.5)[23].

Using the normal approximation to the binomial distribution, µ = nAQL, and σ = pnAQL(1−AQL)are calculated as expected values and variances, respectively.Zαindi- cates the critical value where the right-tailed area under a standard normal distribution isα. Consequently, upon standardizing Equation (8), it can be expressed in the following form.

Za≤ ^c+0.5−nAQL

pnAQL(1−AQL) ⁽⁹⁾ According to Equation (9), the relationship betweennand AQL is as follows:

AQL²

n²+^hZ²_aAQL²−Z_a²AQL−2AQL(c+0.5)ⁱn+ (c+0.5)²≥0 (10) Upon conducting partial differentiation with respect to the variablenin Equation (10) and subsequently solving for it, the minimum value ofncan be computed as follows:

n^∗=

−^hZ²_aAQL²−Z²_aAQL₁−2AQL(c+0.5)ⁱ± r

h

Z²_aAQL²−Z²_aAQL−2AQL(c+0.5)ⁱ²−4 AQL²

(c+0.5)² 2

AQL² (11)

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A minimum number of inspection samples (n^∗) corresponding to a change in AQL value can be calculated using Equation (11).

Table1provides a comparative analysis of the newly developed dynamic AQL sampling plan and the traditional MIL-STD-105E sampling plan in terms of the study methodology.

Table 1. Comparison of the newly developed dynamic AQL sampling plan with the traditional sampling plan.

Traditional Adjustment Single

Sampling Plan Dynamic AQL Single Sampling Plan Determine the AQL value Fixed to a specific value based

on experience

A dynamic adjustment based on changes in product quality

Determine the sample size

Use the lot size and AQL to obtain the sample code, and then make inquiries to

the sampling table.

Determine the sample size for sampling by using the AQL value that is

dynamically calculated.

Consider the change in the estimated number of population defective (change

in inspection results)

Design for normal, reduced, and strict inspection

According to the AQL adjustment scale, the estimated number of defects in the

population is subject to change.

∆AQLS= ^N

∗ BS−2−N^∗

BS−1

N

3.2. Costs Evaluation Function

The cost of quality is an important factor for companies to consider when evaluating their processes. It is crucial to ensure that the inspection cost is minimized and that the failure cost is minimal. The cost of quality can also be used to benchmark performance and identify areas for improvement. In this study, quality is the cost of inspection and failure.

The inspection cost is the cost of inspecting a batch of products, and the failure cost is the cost of misclassifying a good product as defective and misclassifying a defective product as good. The cost per unit of a defective product misclassified as good isC1, the cost per unit of a good product misclassified as defective isC₂, and the inspection cost per sample unit isC₃. Accordingly, if the number of defective products misclassified as good isR₁and the number of good products misclassified as defective isR2, the failure cost isC1×R1+C2×R2, and the inspection cost isC3×n. The total quality cost isC1×R1+C2×R2+C3×n.

A total quality cost calculation will be used to evaluate the effectiveness of the dynamic AQL sampling plans. Companies can more accurately determine whether their AQL sampling plans are effective by evaluating the total quality cost. This study considers three simulation scenarios for an IC testing factory: batch quality, intra-batch quality, and cost ratio. We discuss the impact of changing the ratio of inspection costs and error loss costs on inspection results and the effectiveness of a dynamic AQL sampling plan in reducing inspection costs.

4. Simulation Analysis and Discussion

To determine the most cost-effective sampling strategy, inspection results of MIL-STD- 105E sampling and dynamic AQL sampling will be compared to observe the differences in cost performance and sampling error.

The incoming quality control (IQC) station in the IC product testing plant was selected for the simulation. According to the current IQC product testing plant, the sampling plan and the AQL experience value are set according to the traditional MIL-STD-105E (Inspection Level II) sampling plan. For a normal inspection, the sample code is K, the number of samples is 125 units, the acceptance (Ac) is 0, and the rejection (Re) is 1. For stringent inspection, the sample code is L, the number of samples to be taken for each sampling inspection is 200, the number of acceptances (Ac) is zero, and the number of rejections (Re) is one. For reduced inspection, the sample code is K, the number of samples to be taken for each sampling inspection is 50, the number of acceptances (Ac) is zero, and the number of rejections (Re) is one. It is generally assumed that the number of defects is a random variable. A total of 20 random numbers are simulated from the binomial

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distribution, each containing 1250 values representing 1250 products in each batch. A value of 0 means the product is good, and one means the product is defective. In addition, the AQL empirical value of 0.1% is used by the general IC testing factories in the IQC inspection stations, so the AQL value of 0.1% is also used in this simulation.

About the analysis costs. According to an average of 4000 batches per year, there are 25,000,000 ICs in a test house, so each batch averages 6250 ICs. For MIL-STD-105E inspection level II and normal inspection under the conditions, 125 samples are typically taken. The estimated working hours for an IC testing plant are presented in Table2. An average batch inspection time of 0.3421 h translates into an average unit time of 0.002737 h for each workstation.

Table 2.Inspection station working hours estimated.

Work Hours

Unit work hours per batch at IQC (Incoming Quality Control) station (s) 973.5504

Unit work hours per batch at FQC (Final Quality Control) station (s) 914.6188

Unit work per batch at OQC (Outgoing Quality Control) station 1 (s) 1437.921

Unit work per batch at OQC (Outgoing Quality Control) station 2 (s) 1600

Average unit work hours per batch (h) 0.3421

Unit Work Hours per IC Unit (h) 0.002737

Table3shows the current hourly price of logic and mixed-signal tests. Logic tests verify digital circuits, while mixed-signal tests check analog and digital components. These tests are crucial for ensuring the quality and functionality of integrated circuits during manufacturing and development. The test price ranges from 2500 to 7000 USD, with an average of 4750 USD. Because there are 11 workstations for IC testing, it is assumed that each workstation will cost 431.82 USD per hour to run the test. Since the average working hours per unit of IC at the inspection station are 0.002737 h, the inspection cost is

$1.18 per unit.

Table 3.Inspection cost analysis.

Item Inspection Cost

The minimum cost of the test 2500 USD

The maximum test cost 7000 USD

Cost of an average test 4750 USD

The average cost of a test station (in hours) 431.82 USD

The cost per unit of an IC 1.18 USD

The company’s finance department estimated that packaging costs account for 1.5 to 20% of the total cost of IC manufacturing. The average inspection cost for a single IC at a single inspection station is 1.18 USD, so the average inspection cost for a single IC at 11 workstations is about 13 USD. Thus, the unit manufacturing cost of an IC is about 65 USD.

Based on the above cost analysis, the results in Table4can be compiled, and the simulation for the IQC inspection station will be discussed. The unit loss cost to compensate when a defective product is misclassified as good is at least 65 USD, and the unit loss cost to reduce sales when a good product is misclassified as defective is at least 65 USD.

Table 4.A summary of the quality cost analysis.

Item Values

IQC inspection stations’ average working hours (h) 0.2704

Time spent inspecting each IC at the IQC inspection station (h) 0.002163

IQC inspection station inspection cost per IC 0.93 USD

The loss of the lowest unit of defective product is misclassified as good 65 USD The loss of the lowest unit of a good product is misclassified as defective 65 USD

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4.1. Analysis of Product between-Batch Quality

This paper discusses four distinct quality conditions that need to be taken into account when comparing batches: stable quality, no trend in quality, increasing quality, and decreasing quality. Product quality remains stable when the variation between batches is within the desired control limits, resulting in a consistent defect rate. There is no discernible trend in quality for the product from batch to batch, indicating that each occurrence of the product being tested in the facility does not exhibit the same defect rate. However, when the variation exceeds these limits, the quality is considered to be trending either upward or downward. An increase in quality suggests an improvement in the production process, while a decrease signifies a deterioration in the process. In addition to comparing the proposed dynamic sampling strategy in this study with the traditional sampling plan, we also propose the concept of a cost improvement rate to assess the impact of the defect rate on inspection results. The cost improvement rate can be calculated using the following formula.

cost improvement rate=

Total Cost of Traditional Sampling − Total Cost of Dynamic Sampling Total Cost of Traditional Sampling

×100% (12)

• Stable quality

Based on an empirical value of the inspection station, the AQL value should be 0.1%, so the defective rate of the products entering the station should range from 0 to 0.0001.

For cases of 0, 0.00025, 0.0005, 0.00075, and 0.001 defective rate products, we compared the dynamic sampling strategy with traditional sampling based on average sampling number, type I error, and type II error, as well as inspection cost. A simulation was used to simulate the conditions of 20 product batches using binomial assignment, and a T-test was applied to determine whether there was a significant difference. Table5shows the analysis results, with#and

Table 4. A summary of the quality cost analysis.

Item Values

4.1. Analysis of Product between-Batch Quality

This paper discusses four distinct quality conditions that need to be taken into account when comparing batches: stable quality, no trend in quality, increasing quality, and decreasing quality. Product quality remains stable when the variation between batches is within the desired control limits, resulting in a consistent defect rate. There is no discernible trend in quality for the product from batch to batch, indicating that each occurrence of the product being tested in the facility does not exhibit the same defect rate. However, when the variation exceeds these limits, the quality is considered to be trending either upward or downward. An increase in quality suggests an improvement in the production process, while a decrease signifies a deterioration in the process. In addition to comparing the proposed dynamic sampling strategy in this study with the traditional sampling plan, we also propose the concept of a cost improvement rate to assess the impact of the defect rate on inspection results. The cost improvement rate can be calculated using the following formula.

cost improvement rate =( Total Cost of Traditional Sampling - Total Cost of Dynamic Sampling

Total Cost of Traditional Sampling ) × 100% (12)

• Stable quality

For cases of 0, 0.00025, 0.0005, 0.00075, and 0.001 defective rate products, we compared the dynamic sampling strategy with traditional sampling based on average sampling number, type I error, and type II error, as well as inspection cost. A simulation was used to simulate the conditions of 20 product batches using binomial assignment, and a T-test was applied to determine whether there was a significant difference. Table 5 shows the analysis results, with ○_and✕ for the T-test result, respectively, with significant and non-significant differences at α = 0.05. According to the results, when the defect rate is zero, the cost improvement rate is the lowest, while when the defect rate is 0.1, the performance is the best. Therefore, the cost improvement rate of the dynamic AQL sampling is effective when the batch quality is stable and when the defect rate is greater than 0, regardless of the defect rate. The trend in cost improvement is modeled as y = 0.0065(defective rate)³ − 0.0833(defective rate)² + 0.4433(defective rate) − 0.4837 using the least squares method to construct a polynomial regression model with R² = 1. The higher the defect rate, the better the improvement.

Table 5. Simulation results of dynamic AQL sampling plan under batch-to-batch quality stabilization with different defect rates.

Evaluation Indicators Defective Rate

Traditional Sampling

Dynamic Sampling (in This Study)

Results of T-Test

Improvement Percentage

Average Sampling Number

0 87.50 99.11 ○ −13.27%

0.00025 143.75 121.10 ○ 15.76%

0.0005 147.50 115.53 ○ 21.67%

0.00075 170.00 106.34 ○ 37.44%

0.001 181.25 121.67 ○ 32.87%

for the T-test result, respectively, with significant and non-significant differences atα= 0.05. According to the results, when the defect rate is zero, the cost improvement rate is the lowest, while when the defect rate is 0.1, the performance is the best. Therefore, the cost improvement rate of the dynamic AQL sampling is effective when the batch quality is stable and when the defect rate is greater than 0, regardless of the defect rate. The trend in cost improvement is modeled as y = 0.0065(defective rate)³− 0.0833(defective rate)²+ 0.4433(defective rate)−0.4837 using the least squares method to construct a polynomial regression model with R²= 1. The higher the defect rate, the better the improvement.

Table 5.Simulation results of dynamic AQL sampling plan under batch-to-batch quality stabilization with different defect rates.

Evaluation

Indicators Defective Rate Traditional Sampling

Dynamic Sampling (in This Study)

Results of T-Test Improvement Percentage

0 87.50 99.11 # −13.27%

0.00025 143.75 121.10 # 15.76%

0.0005 147.50 115.53 # 21.67%

0.00075 170.00 106.34 # 37.44%

0.001 181.25 121.67 # 32.87%

Average Type I Error

0 0.0000 0.0000

Item Values

• Stable quality

For cases of 0, 0.00025, 0.0005, 0.00075, and 0.001 defective rate products, we compared the dynamic sampling strategy with traditional sampling based on average sampling number, type I error, and type II error, as well as inspection cost. A simulation was used to simulate the conditions of 20 product batches using binomial assignment, and a T-test was applied to determine whether there was a significant difference. Table 5 shows the analysis results, with ○ and ✕ for the T-test result, respectively, with significant and non-significant differences at α = 0.05. According to the results, when the defect rate is zero, the cost improvement rate is the lowest, while when the defect rate is 0.1, the performance is the best. Therefore, the cost improvement rate of the dynamic AQL sampling is effective when the batch quality is stable and when the defect rate is greater than 0, regardless of the defect rate. The trend in cost improvement is modeled as y = 0.0065(defective rate)³ − 0.0833(defective rate)² + 0.4433(defective rate) − 0.4837 using the least squares method to construct a polynomial regression model with R² = 1. The higher the defect rate, the better the improvement.

0 87.50 99.11 ○ −13.27%

0.00025 143.75 121.10 ○ 15.76%

0.0005 147.50 115.53 ○ 21.67%

0.00075 170.00 106.34 ○ 37.44%

0.001 181.25 121.67 ○ 32.87%

-

0.00025 0.0002 0.0007

Item Values

• Stable quality

0 87.50 99.11 ○ −13.27%

0.00025 143.75 121.10 ○ 15.76%

0.0005 147.50 115.53 ○ 21.67%

0.00075 170.00 106.34 ○ 37.44%

0.001 181.25 121.67 ○ 32.87%

-

0.0005 0.0003 0.0004

Item Values

• Stable quality

0 87.50 99.11 ○ −13.27%

0.00025 143.75 121.10 ○ 15.76%

0.0005 147.50 115.53 ○ 21.67%

0.00075 170.00 106.34 ○ 37.44%

0.001 181.25 121.67 ○ 32.87%

-

0.00075 0.0005 0.0006

Item Values

• Stable quality

0 87.50 99.11 ○ −13.27%

0.00025 143.75 121.10 ○ 15.76%

0.0005 147.50 115.53 ○ 21.67%

0.00075 170.00 106.34 ○ 37.44%

0.001 181.25 121.67 ○ 32.87%

-

0.001 0.0002 0.0010

Item Values

• Stable quality

0 87.50 99.11 ○ −13.27%

0.00025 143.75 121.10 ○ 15.76%

0.0005 147.50 115.53 ○ 21.67%

0.00075 170.00 106.34 ○ 37.44%

0.001 181.25 121.67 ○ 32.87%

-