*1.1. Advantages of Prefabrication*

The use of precast concrete elements has many advantages compared to the production of individual elements on a construction site. According to [1,46], the advantages of the prefabrication process and prefabricated elements which establish the competitiveness of this solution in the construction market include:


Within the advantages of prefabrication, the key advantage is the high real quality of the product, which is guaranteed by standardized production methods and advanced quality control. The process of creating prefabricated elements with appropriate quality takes place through the implementation of basic management functions in production processes, i.e., planning, organizing, and controlling all of the activities in such a way that the result is a product that meets the assumed final needs. Quality is not only created during the final processing stage of the product on the production line but is also "created" in the preproduction, production, and post-production phases, determining the interdependence of a number of activities and leading to the creation of the final product. Repeatability throughout the cycle enables production development, constantly adapting it to customers' needs, and emphasizing the constant interpenetration of production and consumption [47]. Quality planning sets goals and quality requirements as well as requirements for the application of the statistical quality control system. The quality of a final product depends primarily on the method of its design (including the quality of input materials), the quality of the workmanship, or the conditions connected with the production technology, such as temperature, humidity, pollution, or shocks caused by operating devices.

Companies producing prefabricated concrete products on the construction materials market are expected to continuously increase their production capacity, maintain the highquality level of the offered products, and conduct a flexible pricing policy. Achieving these goals is possible through quality control at every stage of production, from the selection of input materials to ready-made prefabricated elements. To stay on the market, prefabrication plants must not only guarantee the high quality of the manufactured goods but also sell them at attractive prices for the customer. That dependence leads to the necessity of estimating the costs of controls related to the introduction of a given acceptance plan before its application. Therefore, manufacturing companies must carry out the acceptance control of a batch of products by a plan that guarantees the assumed level of quality with the lowest possible control costs. Then, it is essential to implement statistical control and an appropriate test plan for manufactured prefabricated elements in prefabrication plants. The goal is essential during the implementation of the quality control of prefabricated reinforced concrete elements, which, due to the complexity of climatic conditions, constitute the basis of modern construction in many European countries, including Poland. The production of prefabricated reinforced concrete structures requires the provision of the basic conditions which will ensure their reliability with minimal expenditure. This engineering and economic problem must be solved to improve the control system during the production process and the quality control of the output products to ensure that they meet the consumer requirements [30].

Currently, precast prefabrication plants assess the operational integrity of the output products in several stages specified in the applicable codes. First, the current control of individual quality indicators is carried out (for the properties of strength and deformation materials, data regarding geometric parameters, and reinforcement). The final product acceptance stage includes periodic tests to assess full-size elements. As described in [30], the selective control system is not always economically effective, does not consider the variability in the production process, and does not ensure the quality and reliability of all the elements in the tested batch. The need to move from selective to full factory control has been found in studies, e.g., [48].

The most rational solution is to evaluate the acceptance test plans used in ongoing inspections about specific parameters of the manufactured prefabricated elements. Consequently, the evaluation of the recommended test plans in the codes for precast elements of reinforced concrete structures based on statistical methods is gaining importance in research. The operational characteristic curves proposed in the article and the curves of average defectiveness after inspection, constructed for codes' acceptance plans, allow one to account for the influence of variability in process factors and controlled parameters of prefabricated elements (the number of elements to be tested or the defectiveness of manufactured elements) on the consumer properties of output products. Therefore, this article aimed not only to determine the discriminant power of the recommended attribute test plans in the codes for prefabricated elements but also to determine the effectiveness of the quality control of precast elements at the acceptance testing stage.

#### *1.2. Development Prospects for Prefabrication in Poland*

The economic market crisis in 2020 led to implementing methods that will improve the construction process. Above all, this is the result of the work carried out in parallel in many segments of the construction industry, which generates many problems. The most significant factors for construction companies were the limited availability of employees, the duration of construction investments, and the assurance of the appropriate quality of the facilities. They, along with emerging difficulties, solutions, and technologies limiting negative premises, have gained importance. Modern prefabrication is the solution to these challenges. Therefore, it is not surprising that an increasing number of actors in the construction market are looking more and more favorably toward prefabricated elements than in previous years; their increased use not only allows the implementation of works to be significantly accelerated but also allows the appropriate levels of quality and durability of the facilities to be guaranteed. Therefore, in the first stage of the research, to assess the condition and prospects for the development of concrete prefabrication in Poland, a

survey was carried out among investors, architects, and contractors. The research included 15 respondents from each target group. Based on the survey, it was found that prefabricated concrete elements represent a well-known and relatively commonly used solution in the implementation of most types of investments [49–52]. On average, it was shown that three out of four respondents implementing industrial and warehouse construction projects use prefabricated concrete elements. The responses obtained from the respondents also indicate the good popularity of this type of technological solution in particular segments of the construction industry (Figure 1). However, the widespread use of prefabricated concrete elements in a given construction segment does not necessarily translate into a large scale and a large share of this technology in the context of the entire investment (Figure 1).

**Figure 1.** The use of prefabricated concrete elements during the implementation of the investment.

The factors that determine the use of precast concrete elements in a construction site are shown in Figure 2. The distribution of responses differed depending on the group of respondents (Figure 2).

**Figure 2.** Factors that determine the choice of precast concrete technology.

The factor that determines the use of precast concrete technology is the reduction in construction times. This aspect was shown to be crucial for almost half of the respondents in this group (42%) (Figure 3). It was found that quality is a decisive factor for architects in choosing prefabricated technology. For contractors, work cost is a crucial factor.

**Figure 3.** The most important factor when choosing the technology of precast concrete elements in the opinion of investors.

Another important factor when choosing a prefabricated technology for investors is quality and technical parameters. The ease of assembly is another factor that determines the use of this technology. However, for investors, factors such as material costs and durability are practically just as important.

From the perspective of contractors, concrete prefabrication technology was very positively evaluated. As a result of the research, the contractors positively (good and very good) assessed precast concrete elements (Figure 4).

**Figure 4.** Evaluation of precast concrete elements—contractors.

When asked whether the share of contractors using precast concrete products will increase, up to 27% of contractors responded positively (Figure 5). This relatively high degree of popularization of this type of technology will reduce the potential for a further upward trend in the future.

**Figure 5.** Expected change in the share of contractors using precast concrete.

It was shown that the surveyed contractors have no doubts about the time savings that result from using prefabricated elements (Figure 6). In the opinion of the contractors, none of the elements mentioned in the study (Figure 6) can be realized faster using traditional technology. Only in the case of walls did a noticeable percentage of respondents (31%) believe that both technologies are comparable in terms of time. On the other hand, it was shown that the surveyed contractors have no doubts about the speed with which lintels, balconies, staircases, and stairs can be created. Everyone indicated that the use of prefabricated technology translates into time savings.

**Figure 6.** Implementation time—prefabricated technology and traditional technology—contractors.

The issue of the mentality and preferences of market participants remains a significant barrier to the dissemination of this technology in Poland, which results from bad experiences and associations with concrete prefabrication from the 1950s to the 1970s. However, as our research shows, favorable changes have occurred in this regard, largely due to the positive ratings from those who have already used this technology. The promotion of investments made of prefabricated elements is significant to the increase in awareness and the better perception of this technology, which we should expect an increasing level of in the future (especially in terms of housing construction). Research shows that the advantages of prefabrication, especially the shorter construction times and the stable quality of the manufactured prefabricated elements, are reflected in the use of this technology. Prefabricated product manufacturers have been modernizing their technological lines for years, increasing not only the number of products and solutions but mostly improving their quality parameters by implementing the statistical quality control procedures recommended in

the codes. Therefore, the next research stage is the analyses of the procedures' statistical quality control used in the quality assurance of the prefabricated elements produced. The results obtained from the questionnaire are similar to the research results in [52].

Quality assurance is based on the idea of preventing quality defects. It is the next step in developing quality concepts after quality control, which is based on statistical methods and developed research plans.

#### *1.3. Statistical Quality Control*

Statistical quality control (SQC) may include statistical acceptance control (SAC) and refer to statistical product acceptance (SPA) or statistical process control (SPC) focused on the production process [53–57], which may be combined with control charts [53], for example, Cusum charts (method C in the control of concrete compressive strength according to EN 206 + A2: 2021-08 [14]) (Figure 7).

**Figure 7.** Scheme of quality control.

The aim of SQC, in addition to ensuring the required level of the tested feature of products, is to reduce the cost of control and, in some cases, even to enable control.

SQC mainly deals with issues related to statistical methods of receipt of piece products and current control carried out during production on a random basis, i.e., a representative part of the examined whole.

These methods make it possible to reduce the number of erroneous judgments about the quality of tested batches of products, making the probabilities of rejecting a good batch (meeting the quality requirements) low enough (the first type of error) and accepting poor-quality products low enough (the second type error), as well as protecting against excessive shortages in production.

Acceptance control due to the measured and calculated parameters and the method of evaluation is divided into [48,57]:


The discriminatory power (effectiveness) of different sampling plans can be assessed by comparing how they perform their function at different possible levels of quality. The practical difficulty in finding the perfect random sampling plan is that you cannot change the laws governing random events. Therefore, when choosing collection plans based on batch testing with defective items, a decision should be made regarding what risks may be incurred in each case, and this is most often an economic decision. In the case of batch acceptance, according to an attribute assessment, the value of the risk of acceptance of batches containing a certain percentage of defective elements is given by the operating characteristic curve of a given acceptance plan.

#### *1.4. Statistical Acceptance Control According to the Attribute Assessment*

Statistical acceptance control methods define random sampling and provide the rules of a procedure to qualify the quality of finished product batches. On their basis, we can consider a batch good or defective. In the latter case, it is possible to lower the element's class or sort and reject the defective precast elements and in justified cases, destroy and recycle them.

One of the methods used is so-called acceptance plans. The simplest is a single (onestep) attribute assessment plan classifying each item as good or defective. The quality of the batch is then understood as the quotient *j = (n* − *k)/n*, where n is the number of simples in the batch with *k* defective items, and the batch defect is called *w = k/n*. A single test plan is a formal record specifying the size of a random sample taken at one time from a lot and the allowable k number of defective items in the sample; exceeding the allowable number leads to the lot being classified as bad. The number *k* is called the qualifying number, and the plan is denoted by the symbol *kn*, where *n* is the number of test results for precast elements [48,57].

To create such a plan, the acceptable quality level (AQL) that meets the inequality *w* ≤ *AQL* is determined in each batch. The probability that, in an *n*-element sample taken from a batch with a defect in w, there will be at most *k* defective pieces, can be calculated from the Bernoulli formula (RB—Bernoulli distribution) for independent samples (1) [57]:

$$P\_{\mathbf{d}} = \sum\_{k=1}^{n} \binom{n}{k} w^k (1-w)^{n-k} \tag{1}$$

The acceptance probability (*Pa*) of a defective lot (*w*) in the *kn* plan is called the characteristic of a single plan, and the graph of dependence on (*w*) is called the operating characteristic curve (OC curve). An ideal would be a plan to ensure that all of the lots with defectiveness *w* ≤ *AQL* are accepted and rejected when *w* ≥ *AQL*. However, this is only possible with faultless 100% control (Figure 8).

**Figure 8.** OC curve: (**a**) at 100% control and (**b**) with producer's and customer's risks.

The better—or more selective—the reception plan is, the closer the OC curve plot is to that shown in Figure 8 (steeper).

When constructing the OC curves, it is possible to assume a different distribution of the analyzed features/parameters of the precast elements; therefore, Equation (1) can be approximated by the formulas [57]:


$$P(w) = \sum\_{k=0}^{n} \frac{\lambda \exp(-\lambda)}{k!} \tag{2}$$

where *λ* = *nw*. - Gaussian distribution—RG (3):

$$P(w) \approx \varphi\left(\frac{k - nw + 0.5}{\sqrt{nw(1 - w)}}\right) \tag{3}$$

where *ϕ*(*x*) = <sup>√</sup><sup>1</sup> 2*π x* −∞ exp- −*t* 2 2 *dt* is a function of the normal distribution N(0,1).


$$N\left(nw,\sqrt{nw(1-nw)}\right)\tag{4}$$

While for a random variable *<sup>k</sup> <sup>n</sup>* → the distribution of RN has the form (5):

$$N\left(w, \sqrt{\frac{w(1-w)}{n}}\right) \tag{5}$$

Formula (5) can be used when the condition is met *nw*(1 − *w*) > 4.


$$P(w) = \binom{n-1}{k-1} w^k (1-w)^{n-k} \tag{6}$$


$$P(w) = \frac{n - 1^{\left(\frac{1}{w} - 1\right)} \mathfrak{e}^{-\frac{(n-1)}{2}}}{2^{\frac{1}{w}} \Gamma\left(\frac{1}{w}\right)}\tag{7}$$


$$P(w) = \frac{2w(n-1)^{\left(\frac{n-2}{2}\right)}e^{-\frac{2w(n-1)}{2}}}{2^{\frac{\mu}{2}}\Gamma(2k)}\tag{8}$$

The main purpose of the random acceptance of products is to determine whether the batch from which the collected element comes meets the previously assumed quality requirements and, thus, whether it can be considered compliant with these requirements. If the tested element does not meet the requirements, the entire batch should be considered non-compliant, and the procedure of dealing with a non-compliant product should be implemented. Due to the evaluation of the examined features, the acceptance inspection can be performed by an attribute or by a variables inspection. Only the measurable features are assessed during the acceptance test, e.g., size, density, and strength, using the variables inspection. A batch of products is considered compliant if the so-called quality statistic does not exceed the qualifying constant *k*, depending on the specified value of the acceptable quality level (*AQL*) and the number of samples—elements taken for testing [58,59].

According to the recommendations of the code ISO 2859-0: 2002 [18], non-conformities (defectiveness) should be classified in terms of their validity. Typically, a division of nonconformities into more significant—class A—and less significant—class B—is used. More important non-conformities will be controlled more strictly. By definition, the *AQL* is "a quality level that corresponds to the worst tolerated average process level" [18]. The *AQL* is a parameter of the process scheme because the *AQL* value, together with the letter sign of the sample size, allows one to determine the test plan and the control scheme. It is assumed that the average process level should be less than or equal to the *AQL* value in order to not reject too many batches produced.

During production control, a prefabricated element for testing is taken from each manufactured batch. The batch is considered compliant if the number of non-compliant units does not exceed the qualifying number *Ac* (9) [18,19,57]:

$$k \le A\_{\mathcal{E}} \tag{9}$$

where *k*—number of non-compliant items per 10 items taken for testing;

*Ac*—constant qualifying or disqualifying number of non-conforming items.

The control plan and control procedures using variables inspection define, e.g., code ISO 3951:1997 [20,21], complementary to ISO 2859-1:2003 [19]. In order to use variables inspection in the sampling procedure, the following conditions must be met:


All of the above-mentioned requirements are met for the production of precast concrete products (e.g., concrete paving stones, concrete paving slabs, and concrete curbs).

To determine the acceptance test plan, the values of the acceptable defectiveness are used, i.e., the AQL limit of acceptable quality, amounting to 0.1 (analogous to ISO 2859-1:2003 [19]). Depending on the severity of the non-conformities, different levels of AQL are used, e.g., significant non-conformities (e.g., the strength of paving slabs and the strength of curbs) are classified as class 1 and assigned higher values than less significant non-conformities (e.g., the dimensions of slabs and the dimensions of curbs).

Acceptance tests are carried out assuming a certain level of control, strictly related to the size of the batch and the number of samples taken. The code ISO 3951:1997 [21] provides three general levels of control: I, II, and III; and four special levels: S-1, S-2, S-3, and S-4. If there are no specific provisions in the relevant standards, level II data are usually used. Where stricter control is required, level III should be used, and when less severe control is needed, level I should be used. Special levels of control are used when a small number of test specimens and a high risk of random control (e.g., in a destructive strength test) are required simultaneously. For example, such situations occur when inspecting precast concrete products. The number of test samples can also be reduced by applying transition conditions between the different control levels.

Concerning the adopted test plans, it is possible to determine the dependence of the average defectiveness after the inspection of the defect and before the inspection, i.e., an actual defect in manufactured prefabricated elements (with a decreasing quality level). This relationship is called the average defectiveness curve after inspection (AOQ curve average outgoing quality curves [56,58,59]. These curves can present the risk associated with acceptance plans for the quality control of the concrete used for production and the ready-made prefabricated elements.

Average outgoing quality limit (AOQL) can be treated as a criterion for the selection of a research plan, thus verifying the correctness of the adopted acceptance plans. Furthermore, the AOQL contains very important information for the recipient/consumer regarding the maximum defectiveness that can be expected with successive receipts of a number of batches over a long period of time based on the adopted acceptance inspection criteria. The practical significance of AOQL is that its value can be equated with the quantile level for the verified property (feature), and thus reflect the quality of the manufactured precast elements.

In AOQ charts, an extreme expresses the AOQL value. The coordinates of the AOQ plot can be determined based on the operational characteristics curve (OC) according to the formula [56,57]:

$$\mathbf{A}\mathbf{O}\mathbf{Q} = \boldsymbol{w} \cdot \boldsymbol{P}\_a \tag{10}$$

where AOQ—the average defectiveness after inspection;

*w*—the defectiveness before inspection;

*Pa*—the probability of acceptance of the batch of precast elements with defective *w.* When verifying the correctness of the compliance criteria and the quality of the batch of precast elements, one can refer to two defectiveness values: 5% (AQL—acceptable quality level) and 10% (LQL—limiting quality level), for which L. Taewre [60] proposed formulas for boundary curves for three areas: controlled, unsafe, and uneconomical. In the case of concrete quality control, the determination of the impact of compliance control on the assumed concrete class requires the determination of the statistical quantile of the concrete compressive strength distribution after the compliance control. The quantile estimation for the compressive strength requires the determination of the concrete's defectiveness after a compliance check. Such inference can be made using AOQ (average outgoing quality) curves (Figure 9).

**Figure 9.** Diagram of the relationship between the defectiveness before and after quality control for compressive strength of concrete.

AOQ curves are also constructed for acceptance plans, i.e., for plans used in the acceptance of precast concrete elements. AOQ curves are curves obtained for the state after inspection; therefore, these curves make it possible to assess the discriminant power of acceptance criteria or compliance criteria in the case of concrete quality. Using AOQL for compliance control, the concrete compressive strength can be identified with the quantile defined for the concrete class, i.e., the quantile of the characteristic strength. According to EN 206 + A2: 2021-08 [14], if the industry standards do not define otherwise, the value of the permissible defectiveness after the inspection performed can be assumed to be at the level of 0.05 (11):

$$AOQL = 0.05\tag{11}$$

#### **2. Materials and Methods**
