Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 13, No. 2, 2024 

 

102 

A	Study	of	Production	Decision	Making	Problems	Based	
on	Sample	Testing	and	Cost	Optimisation	Models	
Xin Zhou1, * 

1Software Engineering Institute of Guangzhou, Guangzhou, China 
*Corresponding author: 2890091630@qq.com 

 

Abstract: In this paper, we provide an in-depth discussion on the optimization of sampling inspection and production decisions 
in the manufacturing process. In order to strike a balance between cost and accuracy, we propose a method based on two-stage 
sampling inspection. The method accomplishes the cost inspection with a better inspection cost the method effectively reduces 
the inspection cost while ensuring the inspection quality. Further, we propose a dynamic model-based production decision 
scheme. The scheme is able to minimize the production cost by continuously updating the estimate of the defective rate during 
multi-stage inspection and assembly sessions. Finally, we provide a detailed analysis and discussion of the production decision 
scheme combining sampling inspection and dynamic modeling using confidence intervals and Monte Carlo simulation methods. 

Keywords: Sampling and Testing, Two-stage Decision Making, Dynamic Modeling, Monte Carlo Simulation. 
 

1. Introduction 
In this paper, we conduct an in-depth study on the problem 

of sampling inspection and production decision optimization 
in the manufacturing process. We first propose an innovative 
two-stage sampling inspection method, which realizes 
effective monitoring of supplier part quality with lower 
inspection cost by comparing the estimation of defective rate 
and inspection cost [1,2]. Further, we introduce the concept 
of dynamic modeling, which provides more accurate data 
support for production decision-making and cost optimization 
by continuously updating the estimate of defective rate during 
multi-stage inspection and assembly sessions [3,4,5]. Finally, 
assuming that the estimated value of the defective rate is 
obtained by sampling inspection method, we synthesize the 
confidence interval and Monte Carlo simulation methods, and 
analyze and discuss in detail the production decision-making 
scheme combining sampling inspection and dynamic 
modeling [6,7,8]. Through these innovative researches, this 
paper provides new perspectives and methods for the 
optimization of sampling inspection and production decision-
making in the manufacturing process, which has important 
theoretical significance and application value. 

2. Sampling and Testing Program 
Based on Two-stage Sampling 

This paper focuses on decision making in the 
manufacturing process of spare parts, semi-finished products 
and finished products in an enterprise. In this chapter, we first 
consider the sampling and testing scheme. We have obtained 
data on the quality of the company's spare parts from our 
research. The supplier claims that the defective rate of a batch 
of spare parts will not be more than 10%, and the enterprise 
is going to decide whether to accept this batch of spare parts 
purchased from the supplier by using the sampling and testing 
method, and the cost of testing will be borne by the enterprise 
itself. By using the sampling method to test the spare parts, 
this paper conducts a hypothesis test on the defective rate. We 
define the following two hypotheses: 

The original hypothesis 𝐻 : Substandard rate Substandard 

rate 𝑞 10%, i.e., it is assumed that the substandard rate of 
the batch does not exceed 10%.  

Alternative assumption 𝐻 : defective rate 𝑞 10%, i.e., 
assume that the defective rate of the batch is more than 10%. 

Decide whether to accept the lot of spare parts or not at the 
given level of significance through a statistical test. If the test 
is significant and the alternative hypothesis is valid, the lot of 
spare parts is rejected; otherwise, the lot of spare parts is 
accepted. 

In order to strike a balance between cost and accuracy, a 
two-stage sampling and testing strategy is used. The core idea 
of this method is to increase the confidence level through 
staged sampling and gradually increase the sample size to 
reduce unnecessary testing costs. 

In the first stage, we set the initial sample size 𝑛 50, 
from which we detect the number of defective products 𝑘 , 

and calculate the initial defective rate 𝑞 . If 𝑞 15%, 

the batch of parts will be rejected directly; if 𝑞 5%, the 
batch of parts will be accepted directly; if 𝑞  is between 5% 
and 15%, the batch will enter the second stage for further 
testing. 

In the second stage, the sample size is expanded to 
𝑛 138, which is the minimum sample size required at the 
95% confidence level. The number of defective products 
detected in the second stage is 𝑘 , and the total number of 
defective products is 𝑘total 𝑘 𝑘 , thus the total defective 
rate can be expressed as: 

 

 
total?

total?
1 2

k
q

n n



             (1) 

 
Based on the accumulated sample data, a final decision is 

made whether to accept or reject the batch of spare parts. 
For ease of calculation, it is assumed that the number of 

defective products follows a binomial distribution, and when 
the sample size is large, according to the central limit theorem, 
the binomial distribution can be approximated by a normal 
distribution. In this case, the expectation of the number of 
defective products is 𝜇 𝑛 ⋅ 𝑞 , and the variance is 𝜎



 

103 

𝑛 ⋅ 𝑞 ⋅ 1 𝑞  . In order to check whether the sample 
defective rate meets the nominal value, the defective rate can 
be standardized and transformed into the z-value of the 
standard normal distribution with the following formula: 

 

 1

/

q q
z

n


                  (2) 

 

where 1q  is the defective rate of the sample and q is the 

supplier's nominal defective rate. 
We begin by discussing the acceptance and rejection 

criterion for the first stage, at the 95% confidence level. 
Assuming a sample size of 𝑛 50 and a nominal defective 
rate of 𝑞=0.10, the desired number of defective products 𝜇
50 0.10 5 , with the standard deviation, 𝜎
√50 0.10 0.90 2.12. When the detected defective rate 
is 𝑞 0.15 , the number of defective products 𝑘 50
0.15 7.5 , and we can calculate the normalization 𝑧

. .
.

√

1.67, so the batch of parts will be rejected. 

If the defective rate 1q   is greater than 15% in the first 

stage of inspection, it is necessary to go to the second stage 
for additional sampling until the total sample size reaches 
𝑛 138 . Based on the properties of the binomial 
distribution, we calculate the critical number of rejected 
defective parts in the cumulative sample: the total sample size 
is 𝑛 138 , which corresponds to a critical number of 
defective parts of 𝑘

reject 
20 at the 95% confidence level, 

and if the total number of defective parts detected, 𝑘
total 

20, then the batch of parts will be rejected. 
The second stage was designed based on 138 samples to 

ensure more reliable detection results at higher confidence 
intervals. With an additional 88 samples (totaling 138 
samples), companies are able to more accurately determine 
the lot defect rate. A reasonable interval is provided between 
the rejection and acceptance criteria for the number of 
substandard products, and companies are able to avoid 
erroneous judgments due to small sample fluctuations, 
especially for borderline substandard rate situations. 
Minimization of testing costs and high accuracy of decision-
making are ensured through two-stage sampling. 

3. Dynamic Model-based Production 
Decision-making Program 

We have collected data on the production process of a firm 
that produces a certain best-selling electronic product and 
needs to purchase the parts and components separately, and 
assemble the parts and components into a finished product at 
the firm. In the production process, the enterprise needs to 
make decisions for each link, and the specific decisions 
include whether to perform quality inspection on parts 1 and 
2, whether to perform quality inspection on the finished 
product, whether to disassemble the finished product when it 
fails, and whether to disassemble and handle the returned 
nonconforming products again, etc. We construct a dynamic 
optimization model to optimize the production process in 
multiple stages. 

We construct a dynamic optimization model to 
continuously update the estimate of the defective rate during 
the multi-stage inspection and assembly sessions, and make 
decisions based on the inspection results of each cycle. In 

order to clarify the decision-making in each production stage, 
we introduce the following strategy variables: 

𝑥 , 𝑥 ∈ 0,1 : Indicates whether or not Part 1 and Part 2 
are detected, respectively, with 1 indicating that detection is 
performed and 0 indicating that detection is not performed.  

𝑦 ∈ 0,1 : Whether or not to test the finished product, 1 
means to test, 0 means not to test.  

𝑧 ∈ 0,1 :  Whether to dismantle substandard finished 
products, 1 means dismantling, 0 means no dismantling. 

𝑤 ∈ 0,1 : Whether or not to process the returned 
nonconforming products, 1 means process, 0 means no 
processing. 

Different costs are incurred under different decision-
making stages, consisting of five main components. For Parts 
1 and 2, the cost of parts inspection in each cycle is calculated 
as follows:  

 
               
part_detect? 1 1 1 1 1 2 2 2 2 21 1n n n n n n n

d b d bC x C x q C x C x q C            (3) 
 

where 𝑥 , 𝑥  are the decision variables for inspection 
of parts 1 and 2 in the nth loop, respectively. The cost of 
finished product inspection and assembly consists of the 
following components:  

 
          assembly? assembly? product? assembly? market?1n n n nC C y C y q C C        (4) 

 
where 𝑦  is the decision variable for whether or not to 

test the finished product in the nth loop. If the finished product 
is not tested, the defective product will enter the market and 
incur return and exchange losses. If the product fails testing, 
the firm can choose to disassemble it, and the disassembled 
parts will go through testing and assembly again. For the nth 
cycle, the total cost of disassembly is calculated as follows: 

 
       

dismantle srework ? 1 test1 2 te t ?as l 2 semb y
n n n nC z C q C q C C       (5) 

 
For nonconforming products that are returned, companies 

have the option of disassembling and reusing them. The cost 
of this treatment can be expressed as:  

 
       
return disassembl s ?e as m  1 te yst1 s2 te t?2 e bl
n n n nC w C q C q C C       (6) 

 
At the end of each round of the cycle, the defective rate will 

be reduced, and the specific decreasing model is as follows:  
 

      1
1 1 11n nq q               (7) 

 

      1
2 2 21n nq q               (8) 

 

 
     2
pradact? product? prodecel?1n nq q          (9) 

 
The total cost model works by cumulatively summing the 

costs of each of the inspection, assembly, rework, and return 
processing in each round of the cycle, using the following 
formula: 

 
        all 1 test?  rework rassembly eturn?
n n n nN

nC C C C C       (10) 

 



 

104 

where N is the total number of rounds in the loop and the 
termination condition is one of the following two conditions: 

𝑛 𝑁: Reach preset maximum number of cycles. 

𝑞 , 𝑞 , 𝑞produc 𝜖 :: Decrease in defective rate to below 

the set threshold 𝜖. 
The enterprise also produces a best-selling electronic 

product through a multiprocessing process, which involves 2 

processes and 8 parts, where parts 1,2,3 produce semi-
finished product 1, parts 4,5,6 produce semi-finished product 
2, parts 7,8 produce semi-finished product 3, and semi-
finished product 1,2,3 produce the finished product, with a 
decision-making process similar to that of a single-processing 
process. Table 1 below gives the 6 scenarios encountered by 
the firm in production with 2 processes and 8 spare parts.  

 
Table 1. Situations encountered by enterprises in production 

Scenario
Part1 Part2 Product Unqualified products 

Rate of 
defective

Price 
Inspection 

Costs 
Rate of 

defective
Price 

Inspection
Costs 

Rate of 
defective

assembly 
cost 

Inspection 
Costs 

Selling 
Price 

Replacement 
Costs 

Disassembly 
Loss 

1 10% 4 2 10% 18 3 10% 6 3 56 6 5 
2 20% 4 2 20% 18 3 20% 6 3 56 6 5 
3 10% 4 2 10% 18 3 10% 6 3 56 30 5 
4 20% 4 1 20% 18 1 20% 6 2 56 30 5 
5 10% 4 8 20% 18 1 10% 6 2 56 110 5 

6 5% 4 2 5% 18 3 5% 6 3 56 10 40 

 
By solving the production decision model based on the 

dynamic model, we obtained the results of decision variables, 
total cost and number of cycles for the six scenarios as shown 
in Table 2 below: 

 
Table 2. Dynamic model-based production decision-making scheme 

 
Inspection of 
spare parts 1 

Inspection of 
spare parts 2 

Inspection of 
finished products 

disassemble total cost 
Number of 

cycles 
0 False False False True 29.18 2 
1 False False False True 32.16 2 
2 False False False True 29.18 2 
3 False False True True 28.48 2 
4 False False False True 28.24 2 
5 False False False False 4.20 1 

 
In all cases, Part 1 was not tested. This suggests that in 

these cases, Part 1 has a low defect rate and the cost of testing 
is too high, so it may be preferable to go directly to assembly. 
In Scenarios 2, 3 and 4, Part 2 was selected for testing. This 
suggests that in these scenarios, Part 2 has a high defect rate, 
and that testing would help reduce subsequent assembly and 
rework costs. Although testing increases the initial investment, 
in the long run it can effectively reduce rework and product 
obsolescence costs, so it is a reasonable strategy to choose 
testing. In Scenarios 1, 3, and 5, finished product inspection 
is enabled. Its main purpose is to prevent defective products 
from reaching the market, thus reducing the cost of 
subsequent exchange and compensation. In most scenarios, 
dismantling of nonconforming products is chosen. The reason 
for this is that in these cases it is more economical to 
dismantle and reuse the parts than to discard or re-purchase 
them. In most cases, 2 cycles were performed, indicating that 
after the initial inspection and measures, the defective product 
rate was still high and the system needed more cycles to 
further reduce the defective rate and reach the target threshold. 

4. Production Decision-making Scheme 
combining Sampling and Dynamic 
Modeling 

In this chapter, we consider that the defective rate 𝑞  is no 
longer given directly, but is derived by sampling parts, semi-
finished products, and finished products. We take a random 
sample of products at each production stage and derive the 

overall defective rate based on the proportion of defective 
products in the sample. 

The defective rate �̂�  is derived by sampling under a large 
sample normal distribution. Assuming that the number of 
defective products found in the sample is 𝑋  and the sample 
size is 𝑛 , the formula for estimating the defective rate is: 

 

 ˆ i
i

i

X
q

n
                  (11) 

 
where 𝑋  represents the number of defective products in 

the sample and 𝑛  is the total number of samples. 
In order to ensure that the estimate of the defective rate has 

a high confidence level 𝛼, we calculate the required sample 
size based on the set confidence level α and the allowed error 
𝐸. The formula is as follows: 

 

 
 2

/2
2

ˆ ˆ1i i
i

N q q
n

E
  

          (12) 

 
where 𝑁 /   is the critical value of the standard normal 

distribution at confidence level 𝛼. With the initial estimated 
defective rate �̂�  and the set error 𝐸, the required number of 
samples 𝑛  can be calculated. 

Since the defective rate is estimated by sampling and 
testing, the effect of sampling error needs to be considered in 
the defective rate decrement process. We set the value of the 



 

105 

decreasing defective rate as the estimated value of the 
defective rate based on the previous sampling. The formula 
for decreasing the defective rate is:  

 

  ( 1) ( )ˆ ˆ 1k k
i iq q                (13) 

 
where �̂�  is the estimate of the defective rate in k cycles, 

and 𝛼 is the decrement ratio. 

The defective rate �̂�   after each decrement will be 
used as the basis for the next round of sampling and testing. 
It can be seen that by this method, a dynamic updating process 
is formed under the decreasing substandard rate. This is 
different from the previous model with a fixed decreasing rate 
of substandard products. The substandard rate �̂�   in the 
detection decision equation is to be updated after each 
sampling. the detection decision is given by: 

 

 ˆi i ie q c               (14) 

 
Decisions in each loop will vary due to changes in the 

defective rate. 
The process is executed in a loop, with each new sampling 

result used to update the estimate of the defective rate. This 
dynamic updating mechanism can respond to the changes 
occurring in the production process in a timely manner, thus 
improving the efficiency and effectiveness of quality 
management. In the fixed defective rate model, the inspection 
decision is based on the known defective rate. However, in 
the sampling model, the defective rate is estimated by 
sampling, so the impact of the estimation error on the decision 
needs to be considered, and we use Monte Carlo simulation 
to reduce the impact of the error. By solving the production 
decision model incorporating sampling checks, we obtain the 
results for the decision variables, total cost and number of 
cycles for the six scenarios, as shown in Table 3 below: 

 
Table 3. Comparison of power load forecasting of 403 line 

 
Inspection of 
spare parts 1 

Inspection of spare 
parts 2 

Inspection of 
finished 
products 

disassemble total cost 
Number of 

cycles 

0 False False False True 30.799186 2 
1 False False True True 34.241151 2 
2 False False False True 29.917576 2 
3 False True True True 29.620153 2 
4 False True False True 29.515179 2 
5 False False True False 4.328443 1 

 

5. Conclusions 
In this paper, we explore the problem of optimizing 

sampling inspection and production decisions in the 
manufacturing process. First, we propose a method based on 
two-stage sampling inspection, which aims to reduce 
inspection costs while ensuring quality control. Further, the 
article proposes a dynamic model-based production decision 
scheme to optimize the production cost by continuously 
updating the defective rate estimate. Finally, the article 
constructs a model for updating the defective rate in 
conjunction with sampling, = which assumes that the 
defective rate is obtained through sampling, and dynamically 
updates the defective rate estimate to adapt to the changes in 
the production process and improve the efficiency of quality 
management. The research in this paper provides a new 
perspective and method for manufacturing enterprises to 
effectively control production costs, improve production 
efficiency and product quality, which has important 
theoretical and practical application value. 

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