3.1. Prior Distribution and Sampling Risk
In industrial practice, product quality characteristics are usually assumed to obey a normal distribution with known constants for both the mean and variance. Lam et al. [
25] study the problem of designing sampling schemes for a sequence of normally distributed variables, and Liu [
26] discusses the sampling test for sample means obeying a normal distribution at stage
k. However, the mean of product quality characteristics is often unknown in some cases, causing it to be assigned a random variable due to customer requirements for quality characteristics or processing. In actual production, the prior distribution of the mean values of the quality characteristics can be inferred based on the sampling data and information of the past product quality characteristics. In this paper, the normal distribution is used to describe prior information about quality characteristics.
Assume that in the production process, the product quality characteristics
obey a normal distribution with mean
and variance
. For the quality characteristic with upper specification limit
, the pass rate of the batch product is
For the sampling test using the hope-small characteristic, the producer is mainly concerned with whether to accept the null hypothesis, , where is the acceptable upper bound of the mean, while the user focuses on the alternative hypothesis, , where . When , the batch should be accepted, and when , the batch should be rejected. Traditional methods assume that the mean value is a fixed constant, but in actual production, the mean value of quality characteristics is sometimes not a fixed constant but a random variable due to the influence of many factors. If the sampling plan is designed according to the traditional method, it may cause a greater risk of misjudgment and bring greater loss to the producer or user. For this reason, a more scientific and reasonable sampling scheme needs to be designed.
The information of the mean
is contained in the sample mean
when a sample of capacity
is randomly selected from the batch of products. Assuming that the standard deviation of the quality characteristics is a known constant, the likelihood function can be constructed when estimating the mean value
using the sample information.
The purpose of batch sampling acceptance sampling is to distinguish between satisfactory and unsatisfactory batches correctly. Based on the sample observations, the null hypothesis is rejected when is sufficiently large. For sampling tests with upper specification limits, the lot will be accepted when . The inspection scheme can be described by the sample size and . A sampling acceptance scheme can be outlined as follows: (i) samples are randomly selected from the submitted batch; (ii) the observed value of the sample mean is calculated; (iii) is less than or equal to , the inspection lot is accepted; otherwise, the inspection lot is rejected.
Using to denote the probability that the batch is accepted, the information of can be described by the distribution of the sampling scheme and the sample mean when the mean is unknown, where obeys a normal distribution with mean and variance .
Therefore, the likelihood function can be expressed as
, i.e.,
The optimal design of traditional batch sampling schemes usually assumes that the mean value of quality characteristics is constant. In actual production, the mean value of quality characteristics is often a random variable due to factors such as equipment deterioration, raw material quality fluctuations, and stress variations. Therefore, it is more appropriate to use the prior distribution information of the mean value of quality characteristics. In this paper, the mean value of product quality characteristics is regarded as a random variable, denoted as
, and its prior probability density function and cumulative distribution function are denoted as
and
, respectively. Further, if the mean values of the quality characteristics obey the truncated normal distribution, i.e.,
, its probability density function is
Since the sampling scheme is to provide the necessary risk protection for both the producers and the users, posterior probabilities are used in order to integrate the sample information into the optimal design of the sampling scheme. For a given sampling scheme
, the user risk can be defined as
The producer risk can be defined as
Among them are
3.2. Sampling Plan Design
In the actual sampling, producers and users need to determine in advance the mutually acceptable risk range. If a batch is rejected, the producer risk should be controlled within . On the contrary, if a batch is accepted, the risk of the users should be controlled within , then the optimal sampling scheme should satisfy the requirements and .
The feasible domain of the sampling scheme can be expressed as
where
is the set of positive numbers and as a function of the sample size
,
is decreasing, while
is increasing. Therefore, if the sampling scheme
is feasible, for
, we have
where
and
are the upper and lower specification limits of the sample size
. As a function of
, the side bounds
and
are nondecreasing because
is nondecreasing and
is nondecreasing.
The feasible domain
can be further expressed as
When both and tend to infinity and tends to a fixed value , there exists and tends to zero, where . That is, both the producer and user risk are related to the sampling scheme and tend to zero, which indicates that the sampling feasible domain is nonempty.
3.3. Objective Function
The objective cost function is established to determine the optimal sampling plan, which is designed to minimize the total cost considering the sampling risk and quality loss. Assuming that
is the sampling cost of a single product,
denotes the cost of batch product rejection, and the quality loss function
of the received batch product denotes the cost of batch product acceptance, the total cost consists of the cost caused by batch product acceptance, rejection, and sampling inspection. Considering the randomness of the mean value
of the batch product, the mean of products can take any value from 0 to
. The closer the mean is to zero, the better. Therefore, the cost of batch product acceptance is proportional to the quality loss function of the hope-small characteristics. Combined with the impact of lot
on the total cost,
can be expressed as
The cost of lot rejection is assumed to be
, where
is a positive constant. Therefore, under the condition that
is known, the total cost corresponding to the sampling scheme
can be expressed as
The optimal sampling scheme needs to minimize the total expected cost
, so the objective function can be expressed as
where the expected cost of the sampling scheme can be expressed as
Thus, we can find the optimal sampling scheme by solving the following nonlinear programming problem
3.4. Optimal Sampling Scheme
In practice, it is difficult to find the optimal sampling scheme for the hope-small characteristic directly. However, the optimal sampling scheme can be better estimated by using algorithm optimization. Suppose that is a set of randomly selected samples from the distribution , then the following algorithm can be designed as shown in Algorithm 1 (usually considering the case where the variance is known).
Step 1. Select appropriate a priori parameters , based on the characteristics of product quality characteristics, and estimate the prior distribution of product mean by .
Step 2. Sample a set of estimates of a product lot from the truncated normal distribution and calculate the likelihood function of the sample means to estimate the probability of receiving a product lot.
Step 3. Give the target mean values of the product for the received and rejected batches, and , and the maximum allowable producer risk and user risk. Here it is required that and .
Step 4. Fixing sample size
n, conduct random simulations of the producer and user risk in combination with constraints
Filter the range of values of the canonical limit under the specified sample size n, where the initial range of is .
Step 5. Select the quality loss function of the hope-small characteristic and the unit cost loss function with and . Obtain the variation of the expected cost with respect to the upper normative limit for given sample size.
Step 6. Repeat the operations of steps 4 and 5 and compare the expected cost with different sample sizes to select the minimum expected cost and the corresponding optimal sampling scheme
. The solution of the optimal sampling scheme is introduced in an example below, and the effects of the model parameters on the total cost and decision variables are analyzed.
Algorithm 1. Optimal sampling scheme solution |
Input: Give the target mean values of and for the received and rejected batches, as well as the maximum allowable production-side risk and use-side risk . Select the appropriate a priori parameters and based on the characteristics of the product quality characteristics. is usually taken as 100. Sample a set of estimated samples of a product lot from a truncated-tailed normal distribution. |
1: for do |
2: for do |
3: if 4: Calculate the expected cost of the sampling scheme |
5: end if |
6: end for |
7: end for 8: Find the sampling scheme corresponding to minimum cost |
Output: Optimal sampling scheme . |