### NSGA-II model for multi-objective optimization

In this paper, the flow of solving the multi-objective optimization model using NSGA-II is shown in Fig. 2. with the following steps:

Step 1: Coding procedure. The arrangement coding approach is used for coding in this paper. The organization coding of chromosomes is depicted in Fig. 3 assuming that each type of component X_{i} will have n_{i} types of construction techniques to select from and that there are m types of various component types on different floors.

Step 2: Make the population start out. The settings of genetic algorithm parameters directly impact its performance, and a reasonable choice and control of these parameters enable the genetic algorithm to search for the optimal solution along the most favorable path^{57}. The parameter settings for genetic algorithms are typically determined based on the characteristics of the problem and the search space for solutions^{58,59}. These parameters primarily include the initial population size (N), the crossover rate (p_{c}), and the mutation rate (p_{m}), and these settings directly affect the algorithm’s performance and search capabilities^{60,61,62}, as detailed in Table 1.

Although the significance of genetic algorithm parameters has been clearly outlined, there is no unified theoretical basis to guide the selection of parameters, and most parameters are obtained through repeated experiments to determine the optimal values^{63}. The steps mainly involve determining the population size N, setting initial values for the crossover rate (p_{c}) and mutation rate (p_{m}) based on the complexity of the research problem to obtain better solutions. Parameters can be fine-tuned within a small range to improve the computational results. Repeat these steps until the most scientific parameters are determined. The number of individuals in the initial population is set as N.

Step 3: Make the first evolutionary generation of the population that is generated Gen = 1, calculate the fitness level of the first N individuals in accordance with the objective function, perform fast non-dominated sorting, congestion calculation, and determine whether the fitness level meets the termination criterion or not. If it does, the generation is terminated; if not, it moves on to Step 4 for execution.

Step 4: The population produced by Step 3 is put through a tournament selection process, which chooses people with high adaptability and weeds out people with low adaptation.

Step 5: Crossover, mutation to create a population of the next generation, objective function calculation for each population member, elite retention strategy, sequential fast non-dominated sorting, congestion calculation to create the population, and the population with the best members as the new population;

Step 6: Determine whether the iteration number Gen is equal to the maximum iteration number. If it is less than the maximum iteration number, then Gen = Gen + 1 and move on to Step 3. Return the execution result as the new generation population to run Step 3. If not, the algorithm stops the operation;

### Cost objective model

In this paper, the cost of components is mainly divided into six parts: labor cost, material cost, machinery cost, management fee, profit and tax, of which the proportion of management fee, profit and tax is selected with reference to the comprehensive quota of the project location. Therefore, the cost accounting for different types of components is shown in the following Eqs. (1), (2).

$$ C = \sum\limits_{i = 1}^{n} {C_{i1} X_{i} + C_{i2} (1 – X_{i} )} $$

(1)

$$ C_{ij} = \sum\limits_{k \in K} {p_{ij}^{k} c_{ij}^{k} + m_{ij}^{k} c_{ij}^{k} + s_{ij}^{k} c_{ij}^{k} + M_{ij} + O_{ij} + T_{ij} } $$

(2)

where C is the total cost of the project; n is the total number of types of components; i denotes the i-th component; j denotes the j-th construction techniques, j = 1,2, j = 1 denotes the cast-in-place technique, j = 2 denotes the prefabricated technique; C_{ij} is the cost of the j-th construction techniques of the i-th component; X_{i} = 0 or 1, X_{i} = 0 denotes the prefabricated technique of the i-th component, X_{i} = 1 denotes the cast-in-place technique of the i-th component; \(p_{ij}^{k} ,\;m_{ij}^{k} ,\;s_{ij}^{k}\) is the consumption of man, material and machine in the k-th under the j-th construction techniques of the i-th kind of component; \(c_{ij}^{k}\) is the unit cost of man, material and machine in the k-th under the j-th construction techniques of the i-th kind of component; M_{ij}, O_{ij}, T_{ij} are the overhead, profit and tax under the j-th construction techniques of the i-th kind of component, respectively.

### Construction of duration target model

By analyzing the resources that can be allocated to each process of the project, as well as the external environment, policies and other relevant elements of the construction conditions, the construction duration of each process can be determined^{64}. In this paper, based on the enterprise time quota of an enterprise in Shenzhen, the duration of the processes included in the construction of the component is calculated, so as to summarize the duration required for the component, which is calculated as shown in Eqs. (3), (4).

$$ T = \sum\limits_{i = 1}^{n} {T_{i1} X_{i} + T_{i2} (1 – X_{i} )} $$

(3)

$$ T_{ij} = \sum\limits_{l \in L} {\frac{{n_{ij}^{l} t_{ij}^{l} }}{{a_{ij}^{l} b_{ij}^{l} }}} $$

(4)

where T is the total duration of the project; n is the type of component; i represents the i-th component; j represents the j-th construction techniques, j = 1,2, j = 1 represents the cast-in-place technique, j = 2 represents the prefabricated technique; T_{ij} is the duration of the j-th construction techniques for the i-th component; X_{i} = 0 or 1, X_{i} = 0 indicates that the i-th component adopts the prefabricated technique, and X_{i} = 1 indicates that the i-th component adopts the cast-in-place technique; \(n_{ij}^{l}\) is the consumption required for the l-th construction technique under the j-th construction techniques of the i-th component; \(t_{ij}^{l}\) is the time quota of the l-th construction techniques under the j-th construction techniques of the i-th component; \(a_{ij}^{l}\), \(b_{ij}^{l}\) is the number of shifts arranged for the l-th construction technique under the j-th construction techniques of the i-th component and the number of people in each shift.

### Carbon emission target modeling

In this paper, the carbon emission of prefabricated components is calculated mainly based on the “Building Carbon Emission Calculation Standard” (GB/T 51366-2019)^{65}, and the basic equation of carbon accounting is: carbon emission of the building = energy and material consumption × Carbon Emission Factor, and the carbon emission of the components should be calculated according to different stages, and the results of the segmented calculation should be accumulated to get the carbon emission of its whole life cycle. The physical stage of the building is mainly divided into three stages: the production of building materials, the transportation of building materials and the construction of the building, so the calculation of the carbon emission of the components is shown in Eqs. (5)–(9):

$$ E = \sum\limits_{i = 1}^{n} {E_{ij} X_{i} + E_{ij} (1 – X_{i} )} $$

(5)

$$ E_{ij} = E_{scij} + E_{ysij} + E_{jzij} $$

(6)

$$ E_{{{\text{scij}}}} = \sum\limits_{k = 1}^{n} {M_{ijk} F_{ijk} } $$

(7)

$$ E_{{{\text{ysij}}}} = \sum\limits_{k = 1}^{n} {M_{ijk} D_{ijk} T_{ijk} } $$

(8)

$$ E_{{{\text{jzij}}}} = \sum\limits_{k = 1}^{n} {N_{ijk} NF_{ijk} } $$

(9)

where E is the total carbon emission of the project; n is the type of component; i denotes the i-th component; j denotes the j-th construction techniques, j = 1,2, j = 1 denotes cast-in-place technique, j = 2 denotes prefabrication technique; E_{ij} is the carbon emission under the j-th construction techniques of the i-th component; X_{i} = 0 or 1, X_{i} = 0 denotes that the prefabrication process is adopted in the i-th component, X_{i} = 1 denotes that the cast-in-place technique is adopted in the i-th component. E_{sij}, E_{ysij}, and E_{jzij} are the carbon emissions of the production, transportation, and construction phases of building materials under the j-th construction techniques of the i-th component, respectively; M_{ijk}, F_{ijk}, D_{ijk}, and T_{ijk} are the consumption of the k-th major building materials under the j-th construction techniques of the i-th component, the carbon emission factor, the average transportation distance, and the carbon emission factor of the transportation distance per unit weight, respectively; N_{ijk}, NF_{ijk} are the total consumption of the k-th energy source for the j-th construction techniques of the i-th component and the carbon emission factor of the energy source.

### Constraint construction of precast rate

The case used in this paper is located in Shenzhen, Guangdong Province, so the calculation of precast rate mainly adopts the local standard of Shenzhen. The calculation method of precast rate in the Calculation Rules for Precast Rate and Assembly Rate of Residential Industrialization Projects in Shenzhen issued by Shenzhen is shown in Eq. (10).

$$ {\text{V}}_{{\text{Prefabrication rate}}} = \frac{{{\text{V}}_{{\text{Standard floor prefabricated concrete components}}} }}{{{\text{V}}_{{\text{Standard layer full volume}}} }} \times 100\% $$

(10)

From the above formula precast rate calculation we can see that the overall precast rate is the sum of the precast rate of each component, so the precast rate of each prefabricated component is the ratio of the volume of the different components to the overall volume of the component, so for the calculation of the precast rate of the project is as shown in Eqs. (11), (12).

$$ A = \sum\limits_{i = 1}^{n} {A_{i1} X_{i} + A_{i1} (1 – X_{i} )} $$

(11)

$$ A_{{{\text{ij}}}} = \frac{{V_{{\text{Prefabricated components}}} }}{{V_{{\text{Total volume}}} }} $$

(12)

where A is the total prefabricated rate of the project; n is the type of components; i indicates the i-th component; j indicates the j-th construction techniques, j = 1,2, j = 1 indicates the cast-in-place technique, and j = 2 indicates the prefabricated technique; A_{ij} is the prefabricated rate under the j-th construction techniques for the i-th type of component; X_{i} = 0 or 1, X_{i} = 0 indicates that the i-th type of component adopts prefabricated technique, and X_{i} = 1 indicates that the i-th type of component adopts cast-in-place technique; V_{Prefabricated components} is the volume of the i-th type of component adopting the prefabricated technique; V_{total volume} is the total volume of the component.

### Constraint construction for quality optimization contribution rate

Quality Function Deployment (QFD) is a systematic quality management tool used to translate customer requirements into specific design requirements for products or services, and to convert these design requirements into specific operational guidelines in the actual production process^{66,67}. The key to implementing QFD is a waterfall-like decomposition approach, which, through cascading layers, explores the importance of different components in relation to quality optimization compared to traditional construction methods^{68}. Additionally, QFD allows for quantifiable assessment, using a quantitative scoring system to evaluate the quality optimization contribution rate of prefabricated components compared to cast-in-place components^{69}. This enables the exploration of the degree of quality optimization relative to traditional construction methods under different component combinations and prefabrication rates. Therefore, this paper introduces Quality Function Deployment (QFD) to quantify the quality optimization contribution rate indicator^{70}, which can effectively investigate the importance of different components in relation to quality optimization compared to traditional construction methods.

The House of Quality is a fundamental tool in QFD, and by establishing the House of Quality, the transformation of each level of elements can be completed, converting the core processes into tangible expressions^{71}. The QFD House of Quality consists of two attributes: customer requirements and engineering measures^{72}. The first level of the quality house is built as illustrated in Fig. 4. QFD is really the level of correlation between each process and each quality indicator.

As seen in Fig. 4, the quality house is made up of six parts: the left wall, ceiling, room, roof, right wall, and basement. Each part has a unique meaning, which is depicted in the image. Each value Z_{nm} in the quality house represents the degree of connection between the n-th process and the m-th quality index. Z_{nm} is assessed using the expert scoring system, and the score range of 1 to 5 represents how well a process has been optimized for quality in comparison to more conventional building methods. Through the relationship matrix can be calculated for each construction techniques accounted for the proportion of the quality optimization of the whole project, the calculation formula is:

$$ v_{{{\text{nm}}}} = \sum\limits_{{{\text{m}} = 1}}^{{\text{m}}} {w_{{\text{m}}} z_{{{\text{nm}}}} } $$

(13)

where \(w_{{{\text{nm}}}}\) is the weighting factor of each quality indicator.

We use Structural Equation Modeling (SEM) to quantify the weight of quality indicators w_{nm} in prefabricated construction. Prefabricated construction, as a complex field involving multiple factors, is influenced by numerous factors. These factors vary in their sources, timing, and degree of impact, and they intertwine and interact with each other^{70}. Structural Equation Modeling (SEM) is a statistical analysis method used to explore complex relationships between variables and predict causal relationships between variables^{73}. SEM can consider relationships between multiple variables simultaneously, and the size and direction of path coefficients can reflect the strength and direction of relationships between different variables, visually displaying the importance of different factors and providing a scientific basis for decision-making^{74,75}. Therefore, path coefficients have a certain theoretical basis and reliability in determining the importance of factors.

The second layer of the quality home primarily creates a connection between each building process p_{n} and the construction type X_{h}. The second layer’s quality house matrix is a 0–1 matrix, scoring 1 if a process is used on the precast component and 0 otherwise. It is feasible to summarize the important scores of the various prefabricated components in terms of the value degree S_{h} by building the second layer of the quality house, which is calculated using the following formula:

$$ S_{{\text{h}}} = \sum\limits_{{{\text{n}} = 1}}^{n} {{\text{g}}_{hn} v_{hn} } $$

(14)

where \(v_{hn}\) is the importance weight obtained for the first level of the mass house, h is the number of components, and g_{hn} is the value of each term of the 0–1 matrix for the second level of the mass house.

The quality optimization rate of prefabricated components can be obtained by carrying out the calculation of the proportion of different components through the importance score values of different components, the numerator is the importance score situation of each component, and the denominator is the importance score situation of the component assuming that the optimization degree is all full scores. On this basis, by calculating the proportion of the component quality optimization rate to the sum of the total component quality optimization rate, the quality optimization contribution rate of prefabricated components Q can be obtained, so the quality optimization contribution rate of prefabricated component combination is calculated as follows:

$$ Q = \sum\limits_{i = 1}^{n} {Q_{ij} X_{i} + Q_{ij} (1 – X_{i} )} $$

(15)

where Q is the quality optimization contribution rate of the project; n is the type of components; i denotes the i-th component; j denotes the j-th construction techniques, j = 1, 2, j = 1 denotes the cast-in-place technique, j = 2 denotes the prefabrication process; Q_{ij} is the quality optimization contribution rate of the i-th component under the j-th construction techniques; X_{i} = 0 or 1, X_{i} = 0 denotes that the i-th component adopts prefabrication process, X_{i} = 1 denotes that the i-th component adopts the cast-in-place technique; m_{i} denotes that there are m_{i} types of construction techniques for the i-th component.

Simulated annealing algorithm to optimize the projection pursuit evaluation model. In order to get the optimum projection that can accurately reflect the original data, the objective function created by the projection pursuit is optimized in this study using the simulated annealing approach. The projection pursuit model flow chart Fig. 5 is optimized using the simulated annealing approach, and the calculation procedure is as follows:

Step 1: Data preparation. In the prefabricated component combination solution, normalize the cost, time, and carbon emission optimization objectives.

Step 2: Create the projection’s index function. In order to convert the multi-objective optimization problem into a one-dimensional projected eigenvalue sequence, convert the cost, duration, and carbon emission high-dimensional data into low-dimensional data. The projection value of sample i in the projection direction is set to the following value, assuming that a_{j} is the projection direction vector:

$$ z_{i} = \sum\limits_{j = 1}^{n} {a_{j} x_{ij}^{*} ,\quad i = 1,\;2,\; \ldots ,\;n} $$

(16)

In selecting the optimal projection direction, the indicator is considered to be optimal when it is infinitely close to the maximum value, and the projection indicator function Q(a) is expressed as:

$$ Q(a) = S(a) \times D(a) $$

(17)

$$ S(a) = \sqrt {\frac{{\sum\limits_{i = 1}^{n} {\left( {z_{i} – \overline{z}} \right)}^{2} }}{(n – 1)}} $$

(18)

$$ D(a) = \sum\limits_{i = 1}^{n} {\sum\limits_{k = 1}^{n} {(R – r_{ik} )f(R – r_{ik} )} } $$

(19)

$$ r_{ik} = \left| {z_{i} – z_{k} } \right| $$

(20)

where S(a) is the inter-class dispersion, D(a) is the local density of the projected values, is the mean value of the projection, R is the window radius of the local density, and the function f is the unit step function, which takes the value of 1 when R ≥ r_{ik}, and 0 otherwise.

Step 3: Generate the prefabricated component combination solution randomly, and regard it as the optimal solution to calculate the objective function.

Step 4: Set the initial temperature, and set the initial number of iterations: t = 1;

Step 5: Randomly vary the current optimal prefabricated component solution and generate a new prefabricated component solution and calculate its objective function value and increment \(\Delta Q\);

Step 6: Accept the solution as the current optimal best solution when \(\Delta Q\) < 0; when \(\Delta Q\) ≥ 0, select the new solution as the optimal solution through \(P = \exp \left( { – \frac{\Delta Q}{T}} \right)\);

Step 7: If t< maximum iteration number, then t = t + 1, return to Step 3:

Step 8: When there is no complete cooling, \({\text{T = }}T\left( {\text{t}} \right)\),enter the process Step 2; when for the cooling state, directly output the current results.

Step 9: According to the best projection a*, in accordance with Eq. (21), calculate the best projection value of each prefabricated component solution \(z_{i}^{*}\), sorted by the value of \(z_{i}^{*}\) from the largest to the smallest.

$$ z(i) = \sum\limits_{j = 1}^{p} {a(j)x(i,\;j),\quad i = 1,\;2,\; \ldots ,\;n} $$

(21)