Research on HPC mixture proportion design

HPC, known for its excellent mechanical properties and durability, has become a widely used material in modern construction engineering. The proper mixture proportion design of concrete is crucial to ensuring the performance of HPC. Traditional mixture proportion design methods often rely on experience and experimental data, leading to inefficiencies and high costs. With the rapid advancement of artificial intelligence and machine learning technologies, data-driven optimization methods have introduced new approaches and tools for the mixture proportion design of HPC. Chen et al. (Reference Chen, Wang, Feng, Liu, Wu, Qin and Xia2023) proposed a MOO hybrid intelligent framework combining RF and non-dominated sorting genetic algorithm version II (NSGA-II) to effectively predict concrete durability and optimize concrete mixture proportions. They applied this proposed method to a practical highway project. Liu et al. (Reference Liu, Zheng, Dong, Xie and Zhang2023) proposed a method for optimizing the mixture proportion of recycled aggregate concrete (RAC) using machine learning and metaheuristic techniques. They developed six machine learning models to predict the CS of RAC. Based on the best predictive model, they employed a multi-objective PSO algorithm with a competition mechanism to optimize three scenarios involving four objective functions of RAC: CS, material cost, carbon footprint, and energy intensity. Tipu et al. (Reference Tipu, Panchal and Pandya2023) aimed to maximize CS while minimizing costs and carbon dioxide emissions. They proposed using the XGBoost model and NSGA-II to achieve these objectives. The XGBoost model was employed to predict the CS of concrete, and NSGA-II was utilized for three-objective optimization. Zhang et al. (Reference Zhang, Huang, Wang and Ma2020) proposed a MOO method combining machine learning and metaheuristic algorithms to optimize concrete mixture proportions. They utilized a multi-objective PSO algorithm to refine the mixing ratios and achieve optimal goals. Their experiments revealed that BPNNs perform better on continuous data (such as strength), while RF algorithms yield higher prediction accuracy for more discrete data (such as slump). Using their MOO model, they successfully obtained the PF for the MOO of both HPC and plastic concrete mixtures. Zhang et al. (Reference Zhang, Han, Wang and Wang2025) proposed an innovative method for mix proportion design and optimization. It pioneers the application of response surface methodology in the multi-parameter mix design and optimization of manufactured sand concrete, significantly enhancing the efficiency of the design process and providing more accurate solutions for the diversified production of manufactured sand concrete. Mandal and Rajput (Reference Mandal and Rajput2025) explored the application of machine learning techniques in optimizing ceramic waste concrete. They discussed a range of computational paradigms, including decision trees, RFs, XGBoost, ANNs, bagging, adaptive boosting (AdaBoost), gradient boosting, regression models, and SVMs. Zhang et al. (Reference Zhang, Pan, Chang, Wang, Liu, Jie, Ma, Shi, Guo, Xue and Wang2025) developed an intelligent predictive model for the CS of RAC using machine learning techniques, based on 1,255 mix design datasets compiled from published literature and laboratory experiments. Furthermore, they established an intelligent mix proportioning model capable of generating optimal mixtures to meet specified CS targets. Wang et al. (Reference Wang, Ji, Xu, Lu and Dong2025) proposed a method based on the characterization of aggregate physical properties to address the varying requirements for CS, flexural strength, and cost-effectiveness in different types of RAC. Neural networks were employed to capture the complex nonlinear relationships between input parameters and target performance indicators. The algorithm with the highest fitting accuracy was selected as the objective function and integrated with NSGA-II to obtain the PF. Finally, the ideal point method was applied to identify Pareto-optimal solutions, enabling the selection of optimal mix proportion schemes tailored to different performance preferences. Concrete CS is a critical objective in the concrete mixture design process, and it is a primary focus of our attention. Asteris and Kolovos (Reference Asteris and Kolovos2019) researched the prediction of CS in self-compacting concrete using ANNs. Kumar and Pratap (Reference Kumar and Pratap2024) investigated the use of machine learning methods to accurately predict the CS of high-strength concrete. Feng et al. (Reference Feng, Liu, Wang, Chen, Chang, Wei and Jiang2020) proposed an intelligent prediction method for concrete CS based on machine learning techniques. This method utilizes the AdaBoost algorithm. Naderpour et al. (Reference Naderpour, Rafiean and Fakharian2018) utilized ANNs to predict the CS of RAC. Hariri-Ardebili et al. (Reference Hariri-Ardebili, Mahdavi and Pourkamali-Anaraki2024) applied the principle of AutoML solutions to predict the most important mechanical property of various concrete datasets, namely CS, for benchmark testing.

Research on dung beetle optimizer

Since its proposal at the end of 2022, DBO has attracted widespread attention from many scholars due to its fast convergence speed, high solving accuracy, and strong optimization ability. So far, there have been some improvements and applications. Li et al. (Reference Li, Sun, Yao and Wang2024) proposed an improved DBO to optimize the parameters of the BiLSTM model, thereby improving the predictive performance of the dual-optimization wind speed prediction model. Zhu et al. (Reference Zhu, Li, Tang, Li, Lv and Wang2024) introduced a quantum-inspired hybrid dung beetle optimizer (QHDBO), which combines quantum computing techniques with multiple strategy integration, effectively applying it to address engineering challenges. (He et al. (Reference Wang, Li and Wang2024) developed an improved dung beetle optimizer (IDBO) and utilized it to optimize the parameters of variation mode decomposition. Jiachen and Li-hui (Reference Jiachen and Li-hui2024) proposed an IDBO combined with the dynamic window approach (DWA) for path planning problems in both static and dynamic environments. (Cai et al. (Reference Cai, Sun, Wang, Zheng, Zhou, Li, Huang and Zong2024) combined an IDBO with two deep learning models for predicting groundwater depth. To enhance the accuracy of specific dynamic subject recognition, Li et al. (Reference Li, Zhao, Gu and Duan2024) proposed a model that integrates an IDBO to optimize a long short-term memory (LSTM) network for specific dynamic subject identification. Wang et al. (Reference Wang, Huang, Yang, Li, He and Chan2023) introduced a quasi-adversarial learning-based DBO (QOLDBO) that integrates Q-learning, applying it effectively to address classical engineering problems. Mai et al. (Reference Mai, Zhang, Chao, Hu, Wei and Li2024) introduced a novel maximum power point tracking (MPPT) technology for photovoltaic systems, utilizing the DBO to maximize output power across diverse weather conditions. Tu et al. (Reference Tu, Fan, Pang, Yan, Wang, Liu and Yang2024) designed a Q-learning-based multi-swarm beetle optimizer (MODBO-QL) to solve the optimal scheduling model for Hybrid Integrated Energy System (HIES). During the machining process, parts are prone to generating higher surface residual stresses in the cutting direction (CD), thereby reducing production costs. To address this issue, Xue et al. (Reference Xue, Li, Li, Zhang, Huang, Li, Yang and Zhang2023) employed a MOO method that combines DBO–BPNN and improved particle swarm optimization (IPSO) algorithm. Wen-Chao et al. (Reference Wen-Chao, Liang-Duo, Liang and Chu-Tian2023) proposed a method combining variation mode decomposition (VMD) and gated recursive unit (GRU) in DBO, where DBO is used to optimize the parameters of GRU. Zhang et al. (Reference Zhang, Zhang, Liu, Sun, Li, Faiz, Li, Cui and Khan2024) improved the hybrid kernel extreme learning machine model on the basis of the DBO to measure and analyze the Water-energy-food nexus (WEFN) recovery ability of China’s Beidahuang Group. Table 1 lists additional studies on DBO covering various improvements and widespread applications, demonstrating the significant potential of DBO.

Table 1. Literature survey on the DBO

CNN multi-input regression prediction

CNNs not only excel in image processing, but also demonstrate powerful capabilities in multi-input regression prediction tasks. Multi-input regression prediction involves gathering data from multiple input sources to predict one or more continuous output variables. This approach is particularly suitable for modeling complex systems, such as forecasting the CS of concrete, as it can manage the nonlinear relationships between multiple input variables.

By employing a multi-input CNN model, various types of input data (such as material composition, environmental conditions, etc.) can be integrated and analyzed. Convolution layers automatically extract features from these inputs, while pooling layers reduce dimension and prevent overfitting. Finally, the extracted features are transformed into the final predicted values through fully connected layers. This method effectively captures the complex characteristics of the input data, thereby enhancing the accuracy and robustness of the predictions.

Related research has demonstrated that using CNNs for multi-input regression prediction has broad application prospects in the fields of engineering and science. For instance, Pourdaryaei et al. (Reference Pourdaryaei, Mohammadi, Mubarak, Abdellatif, Karimi, Gryazina and Terzija2024) proposed a suitable technique for forecasting electricity prices using a multi-head self-attention and CNN-based approach. Similarly, Buchanan and Crawford (Reference Buchanan and Crawford2024) proposed a model that combines feature recognition and many-to-one mapping capabilities of classical CNN with the probability characteristics of Gaussian process regression (GPR) for predicting battery health status. Wan et al. (Reference Wan, Dong, Sun, Zheng, Cheng, Qiao and Jia2024) explored the application of CNN in predicting the partitioned homogeneous properties (PHPs) of electronic product wiring structures and found that CNN models can accurately predict the performance of previously unpaired wiring structures, which can be directly applied to product-level reliability finite element analysis (FEA) and improve the efficiency of reliability assessment. Bhadra et al. (Reference Bhadra, Sagan, Skobalski, Grignola, Sarkar and Vilbig2024) developed an end-to-end 3D CNN model for predicting soybean yield using RGB images based on multi-temporal drones and approximately 30,000 sample plots.

Dung beetle optimizer

Based on the distinct social roles among dung beetles, the DBO categorizes the population into four groups: ball-rolling dung beetles, spawning dung beetles, small dung beetles, and thief dung beetles. The dung beetles’ location is represented as follows:

(1) $$ X=\left\{{X}_i|i=1,2,\dots, {N}_{all}\right\} $$

For different purposes, the positions of the four dung beetle subpopulations are represented by distinct symbols as follows:

(2) $$ R=\left\{{R}_e|e=1,2,\dots, {N}_{roll}\right\} $$

(3) $$ S=\left\{{S}_m|m=1,2,\dots, {N}_{spawn}\right\} $$

(4) $$ L=\left\{{L}_h|h=1,2,\dots, {N}_{little}\right\} $$

(5) $$ T=\left\{{T}_z|z=1,2,\dots, {N}_{thief}\right\} $$

(6) $$ X=R\cup S\cup L\cup T $$

(7) $$ {N}_{roll}+{N}_{spawn}+{N}_{little}+{N}_{thief}={N}_{all} $$

If the dimension of the optimization problem is $ D $ and the corresponding fitness function is $ f $ , then the position of each beetle is represented as a solution to the problem as follows:

(8) $$ {X}_i=\left\{{x}_{i,1},{x}_{i,2},\dots, {x}_{i,D}\right\} $$

The individual fitness value is represented as $ f\left({X}_i\right) $ . The representations of $ {\mathrm{R}}_{\mathrm{e}} $ , $ {S}_m $ , $ {L}_h $ , $ {T}_z $ , and their corresponding fitness values are similar to $ {X}_i $ . The algorithm updates the positions of dung beetles according to different strategies, evaluates their survival ability based on fitness, and assumes that a smaller fitness value indicates better optimization results. The mathematical expressions for the optimal and worst positions are as follows:

(9) $$ \left\{\begin{array}{l}{X}^b=\left\{{X}_i\in X,i=1,2,\dots, {N}_{all}|\forall {X}_j,f\left({X}_i\right)\le f\left({X}_j\right)\right\}\\ {}{X}^w=\left\{{X}_i\in X,i=1,2,..,{N}_{all}|\forall {X}_j,f\left({X}_j\right)\le f\left({X}_i\right)\right\}\end{array}\right. $$

The final optimal position $ {X}^b $ represents the best solution to the problem.

1. (1) Update the position of the roller-ball dung beetles

The position update for roller-ball dung beetles is divided into two scenarios: obstacle-free and obstacle-present.

When $ \delta <\gamma $ , roller-ball dung beetles are in an obstacle-free state, where $ \delta $ is a random number and $ \delta \in \left[0,1\right] $ , $ \gamma =0.9 $ . The position update formula is as follows:

(10) $$ {R}_{new,e}^{t+1}={R}_e^t+\alpha \times k\times {R}_e^{t-1}+u\times \Delta x $$

(11) $$ \Delta x=\left|{R}_e^t-{X}^w\right| $$

Here, $ t $ represents the current iteration number, and $ {R}_e^t $ is the e-th dung beetle’s position after the t-th iteration. $ k $ is a constant and $ k\in \left(0,0.2\right] $ represents the positional deviation coefficient. $ \alpha $ is a natural coefficient used to model the impact of environmental factors on the direction of movement. When $ \lambda <\eta $ , $ \alpha $ is 1, otherwise $ \alpha $ is −1, where $ \lambda $ is a random number and $ \lambda \in \left[0,1\right] $ , $ \eta =0.1 $ , $ u $ is a constant, and $ u\in \left[0,1\right] $ . $ {X}^W $ represent the worst global position, while $ \Delta x $ is used to simulate changes in lighting intensity.

When $ \delta \ge \gamma $ is in an obstacle state, the position is updated as follows:

(12) $$ {R}_{new,e}^{t+1}={R}_e^t+\tan \left(\theta \right)\left|{R}_e^t-{R}_e^{t-1}\right| $$

where $ \theta $ is the deflection angle represented by radians, $ \theta $ is a random number, and $ \theta \in \left[0,\pi \right] $ . If $ \theta $ is $ 0,\pi /2,\pi $ , the position is not updated. In the above position update formula, $ {R}_{new,e}^{t+1} $ is the candidate position obtained from the individual’s $ t+1 $ iteration, which will be compared with the previously best-known optimal position and retained; that is, if $ f\left({R}_{new,e}^{t+1}\right)>f\left({R}_e^t\right) $ , $ {R}_e^{t+1}={R}_e^t $ , otherwise $ {R}_e^{t+1}={R}_{new,e}^{t+1} $ .

1. (2) Update the position of the spawning dung beetles

Some of the dung balls collected by the beetles are used as food, while the rest are transported to a secure location for spawning, where they serve as brooding balls to nurture the next generation. The boundaries of the area where the brooding balls are located are strictly limited in Eq. (13):

(13) $$ {\displaystyle \begin{array}{l}{Lb}^{\ast }=\mathit{\operatorname{Max}}\left({X}^{b\ast}\times \left(1-Q\right), Lb\right)\\ {}{Ub}^{\ast }=\mathit{\operatorname{Min}}\left({X}^{b\ast}\times \left(1-Q\right), Ub\right)\end{array}} $$

where $ {Ub}^{\ast } $ and $ {Lb}^{\ast } $ represent the upper and lower boundaries of the spawning area, respectively. $ {X}^{b\ast } $ represents the current best position of all beetles in the population, $ Q=1-t/T $ , $ T $ is the maximum iteration number, and $ Ub $ and $ Lb $ are the upper and lower boundaries, respectively.

After transporting the brooding ball to the designated spawning area, spawning dung beetles will lay one egg inside it during each iteration. The brooding ball’s position is then updated as follows:

(14) $$ {B}_{new,m}^{t+1}={X}^{b\ast }+{a}_1\times \left({B}_m^t-{Lb}^{\ast}\right)+{a}_2\times \left({B}_m^t-{Ub}^{\ast}\right) $$

where $ {B}_m^t $ represents the m-th brooding ball’s position after the t-th iteration, $ {a}_1 $ and $ {a}_2 $ are two random vectors of size $ 1\times D $ , and $ D $ is the dimension.

1. (3) Update the position of the little dung beetles

Once the juvenile dung beetles inside the brooding ball reach maturity, they emerge from it to forage for food. Therefore, to achieve the goal of space exploration, it is essential to define an optimal foraging area to guide their search. The boundary of this optimal foraging area is defined as follows:

(15) $$ {\displaystyle \begin{array}{l} Lb^{\prime }=\mathit{\operatorname{Max}}\left({X}^b\times \left(1-Q\right), Lb\right)\\ {} Ub^{\prime }=\mathit{\operatorname{Min}}\left({X}^b\times \left(1+Q\right), Ub\right)\end{array}} $$

where $ {X}^b $ represents the global best position and $ Ub^{\prime } $ and $ Lb^{\prime } $ represent the upper and lower bounds of the optimal foraging area, respectively. The definitions of $ Q $ , $ Lb $ , and $ Ub $ are the same as in Eq. (13). The formula for updating the little dung beetles’ position is as follows:

(16) $$ {L}_{new,h}^{t+1}={L}_h^t+{C}_1\times \left({L}_h^t- Lb^{\prime}\right)+{C}_2\times \left({L}_h^t- Ub^{\prime}\right) $$

where $ {L}_h^t $ represents the h-th little beetle’s position after the t-th iteration, $ {C}_1 $ is a random number that follows a normal distribution, $ {C}_1\in \left[0,1\right] $ , and $ {C}_2 $ is a random vector, with each component ranging between 0 and 1.

1. (4) Update the position of the thief dung beetles

Not all dung beetles will actively push the ball; some will instead steal the dung balls collected by others. The formula for updating the thief dung beetles’ position is as follows:

(17) $$ {T}_{new,z}^{t+1}={X}^b+S\times g\times \left(\left|{T}_z^t-{X}^{b\ast}\right|+\left|{T}_z^t-{X}^b\right|\right) $$

where $ {X}^b $ represents the global best position and $ {X}^{b\ast } $ holds the same meaning as defined in Eq. (13). $ {T}_z^t $ denotes the z-th thief dung beetle’s position after the t-th iteration, while $ S $ is a constant set to 0.5. Finally, g is a $ D $ dimensional random row vector, with each component between 0 and 1.

The flowchart of DBO is shown in Figure 1, and the pseudo-code of DBO is shown in Algorithm 1.

Figure 1. Flowchart of DBO.

Development of a CNN-based regression prediction model for CS in HPC

Data processing

Based on practical engineering and prior studies, the mixture proportion parameters for HPC include the amounts of cement (C), mineral powder (MP), fly ash (FLA), water (W), superplasticizer (SP), coarse aggregate (CA), and fine aggregate (FA). These components significantly impact the performance of concrete. Therefore, these key components are used as input variables for the CNN model. Running a CNN model requires setting crucial hyperparameters, specifically the learning rate and the number of convolutional kernels. It is essential to recognize that the concrete mixture proportion factors vary in dimensions and value ranges. To prevent any input quantity from being disproportionately large and skewing the output results, the samples are scaled to fit within [0,1]. This ensures that each parameter contributes equally. The formula for normalizing the input variables is as follows:

(18) $$ {X}_{norm}=\frac{X-{X}_{\mathrm{min}}}{X_{\mathrm{max}}-{X}_{\mathrm{min}}} $$

where $ {X}_{norm} $ is the normalized data value and $ X $ is the initial value, with $ {X}_{\mathrm{max}} $ and $ {X}_{\mathrm{min}} $ being the initial maximum and minimum values, respectively.

To assess the forecasting capability of CNN, the dataset is typically split into a training set and a test set. CNN is then employed to train a model using the training set, which is subsequently used to make predictions on the testing set. The disparity between the forecast outcomes and the true data is computed to assess the predictive efficacy of the model.

Validation metrics for regression predictive model

The CNN regression prediction model is an algorithm based on CNN designed for predictive tasks. The primary concept of the CNN model is to extract features from input data using convolution operations and then progressively process these features through pooling layers and fully connected layers to achieve prediction objectives. In this study, the CNN model was utilized to predict the CS of HPC. To validate the efficacy of the CNN regression prediction model, four widely employed statistical metrics were employed to evaluate its prediction accuracy. For detailed information on these indicators, please refer to Table 2. These indicators allow us to comprehensively evaluate the accuracy and effectiveness of the CNN model in predicting the CS of HPC.

Table 2. The four performance metrics adopted in the experiment section

MOO of the HPC mixture proportion

Definition of the objective function

1. (1) Objective function of concrete CS based on the DBO–CNN model

To enhance both the calculation rate and accuracy of the optimization procedure, the regression model obtained from the optimized CNN and the mixture proportion is utilized as one of the fitness functions of NSDBO. The regression function based on the CNN is as follows:

(19) $$ CS= CNNR\mathrm{egress}\left(\mathrm{ConcreteCompos}\right) $$

where CS represents the CS of concrete, while $ ConcreteCompos $ refers to the various components that constitute the concrete mixture.

1. (2) Cost function of concrete

The raw material prices in this article are based on the prices in the Pearl River Delta region. There are 7 types of raw materials used, including C price (P_C), FLA price (P_FLA), MP price (P_MP), FA price (P_FA), CA price (P_CA), W price (P_W), and SP price (P_SP). The average prices of various raw materials obtained through market research are shown in Table 3.

Table 3. Unit price of each raw material

The unit cost of concrete in the objective function is calculated using Eq. (20).

(20) $$ Cost={C}_C{P}_C+{C}_{FLA}{P}_{FLA}+{C}_{MP}{P}_{MP}+{C}_{FA}{P}_{FA}+{C}_{CA}{P}_{CA}+{C}_W{P}_W+{C}_{SP}{P}_{SP} $$

In Eq. (20), C_C, C_FLA, C_MP, C_FA, C_CA, C_W, and C_SP, respectively, represent the amount of C, FLA, MP, FA, CA, W, and SP per cubic meter of concrete (kg/m3), and Cost represents the unit cost of concrete (Yuan/m³).

1. (3) Carbon dioxide emissions function of concrete

Cement production is one of the main sources of carbon dioxide emissions. In the process of cement production, limestone (calcium carbonate) is heated to high temperature to form lime (calcium oxide), which releases carbon dioxide. Therefore, the cement production process directly leads to a large amount of carbon dioxide emissions. In order to reduce carbon dioxide emissions, cement should be used less. Therefore, it is necessary to establish another objective function (related to carbon dioxide emissions), and Eq. (21) has been used.

(21) $$ {CO}_2=0.9\times {C}_C $$

Variable constraints

1. (1) Constraints on ingredient ranges

The research integrates upper and lower bounds for concrete ingredients, delineated by Eq. (22). These constraints define the acceptable search space for all variables, ensuring compliance with predefined limits. Typically, the upper and lower bounds for ingredients are determined by the regulatory standards governing concrete mixture proportion design in the respective country.

(22) $$ {A}_{i,\min}\le {A}_i\le {A}_{i,\max } $$

where $ {A}_i $ is the $ ith $ variables value and $ {A}_{i,\max } $ and $ {A}_{i,\min } $ are the upper and lower bounds of the $ ith $ variable, respectively.

Table 4. Variable weights and range restrictions

Table 4 presents the range considered in this investigation, detailing the permissible values for each ingredient. Adhering to these constraints ensures that the mix design remains within acceptable bounds and complies with regulatory standards.

1. (2) Constraints on ingredient volumes

The overall volume of concrete is fixed at 1 m³, calculated according to Eq. (23).

(23) $$ {V}_{total}=1{m}^3=\frac{C_C}{\rho_C}+\frac{C_{MP}}{\rho_{MP}}+\frac{C_{FLA}}{\rho_{FLA}}+\frac{C_W}{\rho_W}+\frac{C_{SP}}{\rho_{SP}}+\frac{C_{CA}}{\rho_{CA}}+\frac{C_{FA}}{\rho_{FA}} $$

where $ {\rho}_C,{\rho}_{FLA},{\rho}_{MP},{\rho}_{FA},{\rho}_{CA},{\rho}_W,\mathrm{and}{\rho}_{SP} $ are density in kg/m³, whereas $ {C}_C,{C}_{FLA},{C}_{MP},{C}_{FA},{C}_{CA},{C}_W,{\mathrm{and}\ \mathrm{C}}_{SP} $ are quality in kg of C, FLA, MP, FA, CA, W, and SP, respectively.

1. (3) Constraints on ingredient proportion

In the proportion constraint, relationships among variables are defined by setting specific ratios. Ratio constraints SCM/C, W/C, C/CA, C/(FA + CA), and CA/(FA + CA) are considered. Eq. (24) defines the boundaries for the ratio constraint values in Table 5.

(24) $$ {B}_{i,\min}\le {B}_i\le {B}_{i,\max } $$

where $ {B}_i $ is ratio constraints and $ {B}_{i,\min } $ and $ {B}_{i,\max } $ are the minimum and maximum values of the $ ith $ ratio constraint.

Table 5. Constraints on the ratios between variables

Non-dominated sorting dung beetle optimizer

Once the fitness function and constraints are defined, NSDBO (Zhu et al., Reference Zhu, Ni, Chen and Guo2023) is employed to perform MOO of HPC mixture proportion. This aims to get a Pareto solution set that achieves the optimal mixture proportion with the lowest economic cost and minimal carbon dioxide emissions while maximizing the CS of the concrete. Non-dominated sorting is an efficient and powerful method commonly used in MOO algorithms. It ranks solutions based on the degree of their Pareto optimal. Solutions that are not dominated by any other solutions are given rank 1. Those dominated by only one solution are given rank 2, those dominated by two others are given rank 3, and this pattern continues accordingly. The solutions are then selected based on their ranks to improve the quality of the population. The main steps required to obtain PF through NSDBO are as follows:

1. (1) First, define the population size, dimension, maximum iterations, upper and lower limits of decision variables, and Pareto archive size, randomly initialize the dung beetles population, and store them into matrices. Calculate the fitness value for each individual in the population, identify the non-dominated solutions in the initial population, and store them in the Pareto archive. Then, compute the crowding distance for each member of the Pareto archive.

2. (2) Then, update the positions of the roller-ball dung beetles, spawning dung beetles, little dung beetles, and thief dung beetles. Therefore, a new generation of offspring has been obtained.

3. (3) Update population fitness. The parent and offspring populations are merged together. Perform the non-dominated sorting, and calculate the crowding distance. Start adding each front based on rank and crowing distance until the whole Pareto archive is filled.

This process continues until the algorithm reaches its maximum number of iterations or another stopping criterion is satisfied. At that point, the Pareto-optimal solution set is produced. The flowchart of NSDBO is shown in Figure 2.

Figure 2. Flowchart of NSDBO.

Intelligent decision-making through EWTOPSIS method

Employing the EWTOPSIS method, we conducted a thorough evaluation and decision analysis for 60 individuals within the PF, representing 60 mixture proportion optimization schemes, to pinpoint the optimal solution within this frontier. This method, using a normalized decision matrix, is a MOO analysis technique. It incorporates the principle of information entropy to ascertain the objective weights of the indicators, computes the Euclidean distance between each assessed object and the ideal target, ranks the schemes according to their relative proximity, and identifies the optimal solution (Lv et al., Reference Lv, Liu, Li, Meng, Fu, Ji and Hou2023). This approach accurately captures the significance of the evaluated indicators, thereby enhancing the objectivity of the evaluation results. The calculation method is outlined as follows:

1. (1) Establish the initial assessment matrix

Define the initial assessment matrix as the set of assessment criteria for the 60 mixture proportion optimization schemes in the PF. It can be represented as follows:

(25) $$ A=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1j}\\ {}{a}_{21}& {a}_{22}& \cdots & {a}_{2j}\\ {}\dots & \dots & \cdots & \dots \\ {}{a}_{i1}& {a}_{i2}& \cdots & {a}_{ij}\end{array}\right) $$

where $ A $ is the initial assessment matrix, $ {a}_{ij} $ represents the $ j $ -th evaluation index value in the $ i $ -th mix proportion optimization scheme, $ i $ =1,2,…, m, $ j $ =1,2,…, n, m is the number of optimized mix proportion schemes, and n is the number of assessment indicators.

1. (2) Build a standardized decision matrix

Perform dimensionless processing on matrix A and normalize all indicators. The dimensionless formulas corresponding to each indicator type are shown in Eqs. (26) and (27).

Positive indicators:

(26) $$ {b}_{ij}=\frac{a_{ij}-\min \left({a}_{ij}\right)}{\max \left({a}_{ij}\right)-\min \left({a}_{ij}\right)} $$

Reverse indicators:

(27) $$ {b}_{ij}=\frac{\max \left({a}_{ij}\right)-{a}_{ij}}{\max \left({a}_{ij}\right)-\min \left({a}_{ij}\right)} $$

where $ {b}_{ij} $ represents the elements in the normalized decision matrix.

1. (3) Calculate the entropy weight of evaluation indicators

The formula for calculating entropy weight is shown in Eq. (28).

(28) $$ {w}_j=\frac{1-{e}_j}{\sum \limits_{j=1}^n\left(1-{e}_j\right)} $$

where $ w{}_j $ is the entropy weight of the j-th assessment indicator; $ {e}_j=-k\left[\sum \limits_{i=1}^m{f}_{ij}\ln {f}_{ij}\right] $ , representing the information entropy of the j-th assessment indicator; $ k=1/\ln m $ ; $ {f}_{ij}=\left(1+{b}_{ij}\right)/\sum \limits_{i=1}^m\left(1+{b}_{ij}\right) $ .

1. (4) Construct weighted decision matrix

(29) $$ {c}_{ij}={w}_j\times {b}_{ij} $$

where $ {c}_{ij} $ is the element in the weighted decision matrix.

1. (5) Determine positive and negative ideal solutions

(30) $$ \left\{\begin{array}{l}{C}^{+}=\left\{{c}_1^{+},{c}_2^{+},\dots, {c}_n^{+}\right\}\\ {}{C}^{-}=\left\{{c}_1^{-},{c}_2^{-},\dots, {c}_n^{-}\right\}\end{array}\right. $$

where $ {C}^{+} $ and $ {C}^{-} $ represent the positive ideal solution set and the negative ideal solution set, respectively; $ {c}_n^{+} $ and $ {c}_n^{-} $ represent positive ideal solutions and negative ideal solutions, respectively.

1. (6) Determine the Euclidean distance between the optimization scheme of each mix proportion and the positive and negative ideal solutions

(31) $$ \left\{\begin{array}{l}{D}_i^{+}=\sqrt{\sum \limits_{j=1}^n{\left({c}_{ij}^{+}-{c}_{ij}\right)}^2}\\ {}{D}_i^{-}=\sqrt{\sum \limits_{j=1}^n{\left({c}_{ij}^{-}-{c}_{ij}\right)}^2}\end{array}\right. $$

where $ {D}_i^{+} $ and $ {D}_i^{-} $ , respectively, represent the distance between the i-th mix ratio optimization scheme and the positive and negative ideal solutions.

1. (7) Determine the relative closeness of each mix proportion optimization scheme to the ideal solution

(32) $$ {O}_i=\frac{D_i^{-}}{D_i^{-}+{D}_i^{+}} $$

where $ {O}_i $ represents the relative closeness of the i-th optimized mix proportion scheme.

Implementation of NSDBO in three-objective HPC mixture proportion optimization

Three-objective optimization of concrete mixture proportions to identify optimal ingredient proportions is conducted through the integration of the CNN model and NSDBO algorithm, following the steps outlined below. The workflow of this process is depicted in Figure 3.

Figure 3. Flowchart of CNN–NSDBO–EWTOPSIS for high-strength concrete mix proportion optimization.

Step 1: The CS is represented using a CNN, while the DBO algorithm is employed to explore and determine optimal hyperparameter values.

Step 2: Determine the cost and carbon dioxide (CO₂) emission by applying Eqs. (20) and (21), respectively.

Step 3: Construct a three-objective optimization task by combining all objectives using Eqs. (33) and (34).

(33) $$ Objective\ function:\left\{\begin{array}{l}\max f(CS)\\ {}\min f(Cost)\\ {}\min f\left({CO}_2\right)\end{array}\right. $$

(34) $$ Subjected\ to:\left\{\begin{array}{l} Range\ Constra\operatorname{int}s\\ {} Volume\ Constra\operatorname{int}s\\ {} Ratio\ Constra\operatorname{int}s\end{array}\right. $$

Step 4: Utilize the NSDBO algorithm to search for optimal solutions to the formulated MOO problems.

Step 5: Obtain non-dominated PF solutions.

Step 6: Choose a solution from the PF based on Eq. (32).

High-strength concrete compressive strength prediction based on CNN

Data collection and processing

The experimental dataset on concrete CS is sourced from the UCI public databases, initially curated by Yeh (Yeh, Reference Yeh1998). It consists of 1133 samples. Initially, the data sample had a total of eight input variables, namely C, MP, FLA, W, SP, CA, FA, and Age. The output, representing concrete properties, is the uniaxial CS of the concrete. Furthermore, the statistical distributions of the relevant parameters are presented in Figure 4, enabling direct observation. The figure clearly shows the range and frequency of each parameter’s distribution. In addition, five new features were introduced in the dataset (SCM/C (MP + FLA = SCM), W/C, C/CA, C/A, and CA/A). These features exhibit a high correlation with CS, thereby enhancing the model’s prediction accuracy. The sensitivity analysis is shown Figure 5. Table 6 presents detailed static data samples. The correlation coefficients between variables are illustrated using the Pearson correlation coefficient (Benesty et al., Reference Benesty, Chen, Huang and Cohen2009) in Figure 6. The dataset contains no missing samples, and the variable ranges are evenly distributed.

Figure 4. Statistical distributions of the input/output variables.

Figure 5. The importance of input variables in determining the CS of concrete.

Table 6. Statics of the dataset

Figure 6. Pearson correlation coefficient between variables.

CNN hyperparameter selection

In this section, the experiment involves optimizing the hyperparameters of CNN, using the DBO. Hyperparameter tuning aims to find the optimal hyperparameter combination for a CNN regression prediction model to achieve the best predictive performance. We choose to optimize the learning rate and the number of convolution kernels. This process can be viewed as an optimization problem. Specifically, the combination of hyperparameters is considered a vector of decision variables, analogous to dung beetles’ position. The goal is to minimize the root mean square error, which is treated as the objective to be optimized. The optimal hyperparameters of the model are identified as the global best position found in the final iteration of the DBO algorithm. The detailed steps for using DBO in hyperparameter tuning are illustrated in Figure 7.

Figure 7. DBO–CNN model.

In the optimization process using the DBO, fitness serves as the primary metric for assessing individual performance, driving the optimization process, and guiding the selection of individuals. To enhance both the speed and accuracy of this optimization process, we employ the root mean square error between the concrete CS predicted by a trained CNN and the actual values as the fitness function for the DBO. This approach effectively captures the complex nonlinear relationship between the input variables and the output objectives. Thus, the fitness function based on the CNN can be expressed as follows:

(35) $$ RMSE= FitFun\left(X,\mathrm{TraIn},\mathrm{TraOut},\mathrm{TraNum},\mathrm{DR},\mathrm{NumF},\mathrm{NumR},\mathrm{FS}\right) $$

where $ X $ represents the variables to be optimized, specifically the learning rate and the number of convolution kernels. $ TraIn $ denotes the input variables of the training set, while $ TraOut $ signifies the output of the training set. $ TraNum $ stands for the maximum number of training iterations, and $ DR $ represents the forgetting rate. Additionally, $ NumF $ and $ NumR $ indicate the number of input features and output features, respectively. Finally, $ FS $ denotes the size of the convolution kernel.

The parameter settings for DBO-optimized CNN are shown in Table 7. The hierarchical structure diagram of the constructed one-dimensional CNN is shown in Figure 8. The iterative convergence curve of the DBO-optimized CNN is shown in Figure 9.

Table 7. Parameters setting of DBO–CNN

Figure 8. Hierarchical structure of one-dimensional CNN.

Figure 9. The iterative convergence curve of DBO-optimized CNN.

Model validation

The training set is utilized to learn and establish a predictive model for concrete CS based on DBO–CNN. Subsequently, the established model is validated using the test set to assess its predictive performance. The prediction results and error diagrams for concrete CS obtained from the training set are depicted in Figure 10, with corresponding results and error diagrams for the test set shown in Figure 11. Additionally, Figure 12(a) illustrates the scatter plot of concrete CS obtained from the training set, while Figure 12(a) presents the corresponding scatter plot for the test set. According to the training outcomes of the DBO–CNN model, the following conclusions can be drawn: the DBO–CNN model exhibits outstanding regression learning capabilities and can serve as an effective alternative for capturing the relationship between concrete mix proportions and CS, thereby enhancing the efficiency of concrete design.

Figure 10. Train output results and error output results of prediction models based on DBO–CNN.

Figure 11. Test output results and error output results of prediction models based on DBO–CNN.

Figure 12. Prediction results for the concrete CS on training set and test set.

Furthermore, to demonstrate the effectiveness of DBO–CNN, we trained and tested CNN, BP, GA–BP, PSO–BP, ELM, RBF, and DBO–CNN separately on the training and testing sets. We evaluated our models using the four evaluation metrics mentioned earlier. The results are presented in Table 8, where it can be observed that DBO–CNN achieved the best performance. We also estimated the confidence interval based on the predicted values using statistical methods. Assuming the prediction errors follow a normal distribution, the confidence interval at the 95% confidence level was calculated using the mean and standard deviation. This calculation quantifies the uncertainty of the model’s predictions, enhancing the reliability of decision-making. The histogram of error distribution and the fitting curve based on normal distribution are shown in Figure 13. Taking the first sample’s predicted value as an example, its corresponding 95% confidence interval is [17.74, 34.83].

Table 8. Training and testing results of prediction models

Figure 13. Error distribution histogram and fitting curve based on normal distribution.

MOO of the HPC mixture proportion design based on CNN–NSDBO–EWTOPSIS

Definition of the objective function

1. (1) Objective function of concrete CS based on the DBO–CNN model

Previously, we optimized the hyperparameters of the CNN using DBO, resulting in the DBO–CNN regression prediction model. This model is used as the first objective function to be solved for the design problem of HPC mixture proportion. The objective function for maximizing the CS of concrete, $ \max {f}_1 $ , can be formulated as follows:

(36) $$ {f}_1=\max \left\{ CNN\operatorname{Re} gression\left(C, MP, FLA,W, SP, CA, FA, Age\right)\right\} $$

1. (2) Concrete Cost Function

In this context, the real price of concrete primary materials depends on the unit prices of C, MP, FLA, W, SP, CA, and FA, which are equivalent to 0.5, 0.4, 0.28, 0.005, 0.16, 0.11, and 0.125 yuan/kg, respectively. The objective function for minimizing the concrete cost, $ \min {f}_2 $ , can be formulated as follows:

(37) $$ \min {f}_2=0.5C+0.4 MP+0.28 FLA+0.005W+0.16 SP+0.11 CA+0.125 FA $$

1. (3) Carbon dioxide emissions function of concrete

Typically, the amount of carbon dioxide produced by manufacturing 1 m³ of concrete primarily depends on the quantity of cement used. When preparing concrete, using 1 kg of cement will produce approximately 0.9 kg of carbon dioxide emissions. The objective function for minimizing the carbon dioxide emissions of concrete, $ \min {f}_3 $ , can be formulated as follows:

(38) $$ \min {f}_3=0.9C $$

Variable constraints

For the preparation of HPC, C and FLA are commonly used as binding materials, with super-plasticizes added to enhance workability. According to HPC design specifications and engineering practices, the dosage of each component in the HPC mixture should fall within an acceptable range and comply with the corresponding proportion constraints. The constraint conditions for optimizing the HPC mixture proportion are determined according to Eq. (39).

(39) $$ {\displaystyle \begin{array}{l}102\le C\le 540\\ {}0\le MP\le 360\\ {}0\le FLA\le 260\\ {}121\le W\le 247\\ {}0\le SP\le 33\\ {}708\le CA\le 1145\\ {}594\le FA\le 993\\ {}1\le Age\le 365\\ {}0\le \left( MP+ FLA\right)/C\le 0.35\\ {}0.3\le W/C\le 0.6\\ {}0.2\le C/ CA\le 0.6\\ {}0.05\le C/\left( CA+ FA\right)\le 0.33\\ {}0.4\le CA/\left( CA+ FA\right)\le 0.65\\ {}1=\frac{C}{1440}+\frac{MP}{2800}+\frac{FLA}{2500}+\frac{W}{1000}+\frac{SP}{1200}+\frac{CA}{2700}+\frac{FA}{2600}\end{array}} $$

Three-objective optimization results and discussion

During the optimization process, NSDBO performs four different operations. Therefore, several parameters must be set before the iterative optimization process, including the population size, the maximum number of iterations, and the parameters $ k $ , $ b $ , and $ s $ . To balance optimization effectiveness and convergence speed, the population size is set to 60, and the maximum number of iterations (serving as the termination criterion) is set to 100. Additionally, $ k $ is set to 0.1, $ b $ to 0.3, and $ s $ to 0.5. To demonstrate the effectiveness of NSDBO, we selected NSGA-II and Multi-Objective Particle Swarm Optimization (MOPSO) as comparative algorithms, as they have been widely used in the literature to solve similar problems. The algorithm parameter settings are detailed in Table 9.

Table 9. Algorithm parameter settings

With the objective function, variable constraints, and model parameters set as described, 60 Pareto-optimal concrete mixtures were generated using 100 iterations of NSDBO, as illustrated in Figure 14. In the figure, the proportions of the ingredients in the Pareto solution are shown, where the y-axis represents the CS, and the x-axis represents all the components of the concrete mix. Additionally, we successfully obtained the PF, and the EWTOPSIS method was used for decision-making. Figure 15 shows the EWTOPSIS scores corresponding to the optimal front of the three-objective optimization problem. The solution with the highest score was chosen for practical implementation. In the figure, the color of the solutions corresponds to their assigned EWTOPSIS score. The obtained Pareto solution set and PF are presented in Table 10. To demonstrate the superiority of NSDBO in solving this problem, we also employed NSGA-II and MOPSO, which have been used to solve similar problems. The chart comparing the experimental results is shown in Figure 16. When solving this problem, only the spacing metric was used for evaluation because the true PF was unknown. A smaller spacing value indicates a more evenly distributed solution set. The values of the spacing metric obtained from the experiment are presented in Table 11. From Table 11 and Figure 16, we can observe that the distribution of the Pareto solution set for NSDBO is more uniform.

Figure 14. Effect of ingredient proportions on CS and cost in the obtained PF optimal solutions.

Figure 15. Three objective PF with EWTOPSIS evaluation.

Table 10. Obtained Pareto solution set and Pareto front

Figure 16. Algorithm comparison chart.

Table 11. Spacing metric