Multi-objective optimization of ultra-high performance concrete based on life-cycle assessment and machine learning methods

Abstract

Ultra-high performance concrete (UHPC) has gained a lot of attention lately because of its remarkable properties, even if its high cost and high carbon emissions run counter to the current development trend. To lower the cost and carbon emissions of UHPC, this study develops a multi-objective optimization framework that combines the non-dominated sorting genetic algorithm and 6 different machine learning methods to handle this issue. The key features of UHPC are filtered using the recursive feature elimination approach, and Bayesian optimization and random grid search are employed to optimize the hyperparameters of the machine learning prediction model. The optimal mix ratios of UHPC are found by applying the multi-objective algorithm non-dominated sorting genetic algorithm-III and multi-objective evolutionary algorithm based on adaptive geometric estimation. The results are evaluated by technique for order preference by similarity to ideal solution and validated by experiments. The outcomes demonstrate that the compressive strength and slump flow of UHPC are correctly predicted by the machine learning models. The multi-objective optimization produces Pareto fronts, which illustrate the trade-off between the mix’s compressive strength, slump flow, cost, and environmental sustainability as well as the wide variety of possible solutions. The research contributes to the development of cost-effective and environmentally sustainable UHPC, and aids in robust, intelligent, and sustainable building practices.

Graphical abstract

Keywords

ultra-high performance concrete / machine learning / multi-objective optimization / life-cycle assessment

1 Introduction

Recent years have seen a significant increase in the use of ultra-high performance concrete (UHPC), a highly inventive cementitious engineering material [1]. UHPC has superior mechanical, rheological, and durability properties than conventional concrete. UHPC employs a variety of supplementary cementitious materials (SCM), appropriate sand gradation, low water/cement ratio, fiber reinforcing, and high-efficiency water-reduction agents to produce high fluidity, high mechanical performance, and high durability. Compared to conventional concrete, UHPC is more durable due to its discontinuous pore structure. Brittle failure can become ductile failure, and the interior fibers can give UHPC tensile strain hardening capabilities. The stress between fibers can be transferred to the uncracked UHPC matrix by means of fibers. Because fine blend materials often have a high amorphous SiO₂ content and may enhance matrix packing density, it is a significant factor in impacting the system’s overall performance when used. It can assist in releasing water that has been trapped between particles and create the rolling motion like a ball. As a result, the mixture’s viscosity can be decreased, resulting in improved dispersion, less porosity, and a denser microstructure. Furthermore, it can enhance the transition zone at the interface between the matrix and fibers, therefore removing calcium hydroxide crystals and eventually enhancing uniformity. As a result, UHPC is a superior cement-based material that may provide civil structures with more strength and longevity [2–5]. However, because UHPC has a very high cement content and steel fiber addition, its initial material cost is often greater than that of conventional concrete. Meanwhile, the high carbon emissions of UHPC hinder its widespread application [6]. Trial mixing experiments are the mainstay of the conventional UHPC design. When utilizing different raw materials, the method’s adaptability and dependability are weak due to its reliance on trial and error and experience [6]. It is challenging to achieve precise dynamic mixing ratio optimization for multi-objectives in real-time while simultaneously taking UHPC’s environmental protection indicators and economic advantages into account.

Machine learning (ML) is an artificial intelligence technique that processes and analyzes data using various algorithms and calculation models [7–9]. It can determine how concrete components affect its performance by analyzing and modeling a sizable amount of data. The ML prediction model is more effective, accurate, and widely applicable than the conventional approach. ML has been widely used in civil engineering [10,11]. Many types of cement-based materials have been optimized using ML methods, like graphene-modified cementitious composites [12,13], rubber-modified recycled aggregate concrete [14], reinforced concrete [15], sustainable high-strength concrete [16], and waste glass reinforced cement [17]. Sun et al. [13] established a multi-objective optimization model to identify the optimal design scheme and the important variables that have the most influence on the properties of graphite-slag conductive composite. Mahjoubi et al. [18] considered the life cycle cost and carbon footprint with three models to forecast the compressive strength, tensile strength, and ductility of cementitious composites. With multi-objective optimization, Chen et al. [19] utilized the hybrid intelligent framework of the random forest algorithm, support vector machine, and non-dominated sorting genetic algorithm to reach multiple optimal solutions on the cost and performance of high-performance concrete. A multi-objective optimization framework was developed by Shamsabadi et al. [20] to minimize manufacturing costs and environmental issues associated with green concrete while preserving the optimal compressive strength. A multi-objective optimization framework was proposed by Zheng et al. [21] to optimize the mix proportion of concrete. The framework built suitable ML models with high prediction accuracy and the capacity to resolve multi-objective hybrid optimization problems using K-fold cross validation, Bayesian hyperparameter optimization, regression feature elimination, and the constrained two-archive evolutionary algorithm (C-TAEA).

While numerous strategies are being developed to lessen the carbon footprint of the construction industry, the most workable and realistic ones entail reducing the quantity of Portland cement in concrete by substituting it entirely or partially with SCMs like ground granulated blast-furnace slag, fly ash (FA), and silica fume [22–24]. Furthermore, recycled concrete aggregates, recycled glass, and recycled plastic aggregates can be used in place of coarse aggregates, either entirely or partially, to maximize the quantity of recycled material in concrete [25,26]. These green techniques have demonstrated a great deal of promise in terms of embodied carbon reduction. Miller [27] stated that another factor influencing the likelihood of global warming is the mix ratio of the concrete. Alternative cement materials should have their environmental impact studied in order to reduce CO₂ emissions. The life-cycle assessment of concrete includes the impact of each stage on the environment and cost from the production, use to scrapping of concrete. The evaluation method includes not only the production process of concrete, but also the stages of raw material acquisition, transportation, construction, use, waste and recycling of concrete [28]. At present, concrete life-cycle assessment mainly focuses on the economic benefits and environmental impact of concrete. However, previous research on UHPC has rarely considered the cost and carbon emissions throughout its entire life cycle optimized by ML methods.

The research predicts the compressive strength and slump flow of UHPC through ML algorithms, evaluates the carbon emission and cost of UHPC in the life-cycle, and carries out multi-objective optimization of mix proportion. The multi-criteria decision-making methodologies are applied to choose the optimal solution. The flowchart of the entire framework is exhibited in Fig.1. The optimization framework takes into account the mechanical properties, working performance, economic benefits and carbon emissions, effectively reduces the consumption of manpower and time in the traditional trial process, saves the cost of UHPC, reduces the carbon emissions of it, and realizes the precise control of various properties of the UHPC. Researchers and engineers can easily build and tune UHPC to satisfy various on-site needs in real-world projects due to the Pareto frontier that the optimization model yields.

2 Machine learning methods

2.1 Prediction model based on machine learning

This study compares the prediction accuracy of six ML algorithms for the compressive strength and slump flow of UHPC, including: 1) Random forest regression (RFR) [29–31]; 2) Support vector machine regression (SVR) [32,33]; 3) Decision tree regression (DTR) [34]; 4) Extreme tree regression (ETR) [35]; 5) Multi-layer perceptron regression (MLPR) [36]; 6) Extreme lifting tree regression (XGBR) [37].

The performance of the prediction model of the ML algorithm is evaluated and the hyperparameter is optimized. The following four indicators, mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE) and determination coefficient (R²), are defined as follows:

(1)MAE=\frac{1}{n}\sum _{i=1}^n\left|y_i-{\hat{y}}_i\right|,

(2)MSE=\frac{1}{n}\sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2,

(3)RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2},

(4)R^2=1-\frac{{\displaystyle \sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(y_i-{\overline{y}}_i\right)}^2}},

where y_i means the real value, {\hat{y}}_i represents the predicted value, {\overline{y}}_i is the average value of the real value of the sample, and n refers to the number of samples. Because the absolute value of error is used, MAE is not affected by the positive and negative directions. The smaller MAE indicates that the fitting degree of the model is better. MSE increases the error of the prediction model by amplifying the influence of the error fitting degree through the error square, but it needs to deal with the influence of abnormal errors. MSE represents the average value of the square error between the real value and the predicted value of the model, and the smaller MSE represents the better fitting degree of the model. RMSE represents the value of the square root of the error square between the actual value and the predicted value of the model, and the smaller represents the better fitting degree of the model. R² indicates the goodness of the fit of the regression model, and a higher R² demonstrates a better fit of the model.

2.2 Random grid search method

The ML model’s parameters can be optimized using the random grid search approach. It is an enhanced form of the grid search technique, which increases some degree of unpredictability in the parameter space search [38]. The random grid search principle is illustrated in Fig.2. By randomly selecting the preselected parameter sets, the random grid search method can more effectively search for hyperparameter combinations. Random sampling can save time and computational resources by finding an optimal parameter combination in fewer repetitions. The random grid search technique may explore the parameter space more freely, especially when the parameter space is big or continuous, since it permits random sampling in the defined parameter space instead of requiring a list of parameters as grid search. The majority of ML models, including but not limited to decision trees, support vector machines, random forests, etc., and their parameter optimization may be used with the random grid search approach [39].

Cyclic iteration is employed in the random grid search. Randomly chosen parameters are modeled in a single iteration. For modeling, parameters are chosen at random in the following iterations. The size of the parameter subspace may be manipulated by varying the number of random grid search iterations. The random grid search is stopped after the specified computation time is reached [38].

2.3 Bayesian optimization

Based on probability models, the Bayesian approach is an optimization methodology for hyperparameters. Bayesian inference is used to continuously update this distribution in order to find the hyperparameter that is most likely to achieve the optimal value of the objective function. This is done by building a probability model of the objective function and using Gaussian processes or tree structures to establish a probability distribution of the objective function in the hyperparameter space [40]. The fundamental concept involves creating a probability model for the goal function, updating the estimate of the objective function on a regular basis based on observational data and past knowledge, and directing the subsequent hyperparameter selection process [41]. By using past observational data to dynamically modify the upcoming exploration direction, the Bayesian technique more effectively searches the parameter space. Throughout the search process, the position and quantity of sampling points can be adaptively changed using the Bayesian approach. It handles complicated and high-dimensional hyperparameter spaces well. The present probability model allows for the simultaneous evaluation of numerous hyperparameter combinations in each iteration, hence increasing search efficiency. By completely utilizing observation findings and previous knowledge, the Bayesian technique continually updates the objective function estimation, leading to faster convergence to the optimal solution [42].

The posterior distribution of the objective function is updated until it almost resembles the real distribution by giving an optimized objective function and iteratively adding sample points. This process is also referred to as a Gaussian process [41]. The Bayesian technique is quicker and requires fewer iterations. For non-convex problems, global optimization can be accomplished.

2.4 K-fold cross validation

One popular technique for evaluating models is K-fold cross validation. One of the k subsets of the original data set is kept as the validation set in K-fold cross validation, while the remaining k – 1 subsets are utilized to train the model. Every subset is utilized as a validation set, and this procedure is repeated k times. The average of k validation findings is the final result [21]. Reusing these subsets to train and verify the model can help eliminate bias and variance concerns caused by poor data set segmentation, as well as offer a more accurate assessment of the model’s generalization capacity. This strategy fully utilizes all available data for training and validation. Because every sample serves as a validation set, the likelihood of overfitting is decreased because the model cannot depend too much on a particular segmentation of the training validation [43,44].

2.5 Recursive feature elimination

By identifying and eliminating features that have minimal impact on the model’s performance, recursive feature elimination (RFE) increases the model’s resistance to overfitting and qualifies the model for data from larger training sets. The wrapper technique, which uses recursive training to eliminate the least significant features from the model in an effort to enhance performance, is the basis of the feature selection concept of recursive feature removal [45]. To identify the feature subset that has the least influence on model performance and lower the prediction model’s complexity, the basic strategy is to identify the important features and minimize the size of the feature set that contains weak features [46]. RFE makes the prediction model appropriate for a greater training set by finding and eliminating characteristics that have little effect on model performance, hence improving the model’s resistance to overfitting. RFE may combine several ML methods to increase the prediction performance of models and lower computing costs because it is not dependent on any particular type of model.

2.6 Multi-objective optimization algorithms

2.6.1 Multi-objective evolutionary algorithm based on adaptive geometric estimation

To better adapt to the features of the present optimization issue, the multi-objective evolutionary algorithm based on adaptive geometric estimation (AGE-MOEA) modifies the geometric structure throughout the search phase [47]. To more successfully investigate the global features of the Pareto front, AGE-MOEA calculates the geometric form of the Pareto front and adaptively modifies algorithm parameters and operations [48]. While keeping the general layout of non-dominated sorting genetic algorithm-II (NSGA-II), AGE-MOEA adjusts the crowding distance. Sort the non-dominant frontiers using a non-dominant sorting tool. Then, use the initial frontiers to normalize the target space and estimate Pareto frontiers. Because of its flexibility and geometric estimating methodologies, AGE-MOEA can identify more high-quality solution sets and converge to the Pareto front more quickly than typical multi-objective evolutionary algorithms [49].

2.6.2 Non-dominated sorting genetic algorithm-III

The primary goal of Non-dominated Sorting Genetic Algorithm-III (NSGA-III) is to optimize the distribution and convergence of Pareto optimal solution sets in multi-objective optimization problems. It is an advancement over NSGA-II and genetic algorithm (GA) [50,51]. NSGA-III chooses more representative populations and uses crowding calculations to guarantee population variety.

By using a hierarchical optimization approach, NSGA-III separates the target space into many layers, each of which represents a distinct range of target values. A more thorough exploration of the multidimensional target space is made possible by locating Pareto frontier solutions at various levels. The NSGA-III concepts are displayed in Fig.3. NSGA-III presents the idea of reference points for search based on a GA. Pre-selected goal values serve as reference points. The algorithm is led to search for the Pareto front and adaptively modify the search direction based on the reference points’ positions in the target space. To preserve solutions with variety and excellent distribution, NSGA-III picks solutions at each level depending on the separation between individual and reference points as well as the level of individuals in non-dominated sorting.

2.6.3 Technique for order preference by similarity to ideal solution

Technique for order preference by similarity to ideal solution (TOPSIS) is an analytical method suitable for comparing and selecting multiple options based on multiple indicators. The main idea of this method is to first determine the positive ideal solution (optimal solution) and negative ideal solution (worst solution) for each indicator. The so-called positive ideal solution is the best value (solution) of one combination, and its various attribute values reach the best value among the candidate solutions, while the negative ideal solution is the worst value (solution) of another combination. Then, the Euclidean distance between each solution and the positive ideal solution and the negative ideal solution is calculated, and the degree of closeness between each solution and the optimal solution is obtained as the criterion for evaluating the quality of the solution.

3 Life-cycle assessment of ultra-high performance concrete

3.1 Life-cycle cost assessment of ultra-high performance concrete

The design philosophy of construction engineering is dedicated to lowering total costs throughout the whole life cycle, starting from the first stage of UHPC’s mix design. This design idea properly evaluates the economic benefits of UHPC under various mix ratios, taking into account the expenses associated with engineering projects at different stages, such as planning, building, and scrapping, in order to select the best design scheme. In this procedure, it is important to thoroughly analyze not just the original building cost but also the end processing cost. By using this life-cycle cost analysis approach, it is possible to make sure that UHPC not only satisfies engineering specifications but also maximizes economic advantages over the entire life cycle. The life cycle of UHPC is shown in Fig.4.

The full life-cycle cost mainly includes raw material cost, transportation cost, mixing cost, and scrap cost [52,53]. The detailed parts within each major cost can be refined through a list to assess specific situations. For the convenience of evaluation, the full life-cycle cost of UHPC (C) is divided into raw material cost (C₁), transportation cost (C₂), mixing cost (C₃), and scrap cost (C₄), as illustrated in Eq. (5).

(5)C=C_1+C_2+C_3+C_4.

3.1.1 Raw material costs

The material cost comes from the production and supply of raw materials, including cement, water, aggregates, fly ash (FA), silica fume (SF), granulated ground blast furnace slag (GGBS), superplasticizer agents, etc. To evaluate the impact of mix ratios on the entire life-cycle cost, the selected raw materials and their unit costs should be kept the same under different mix ratios. The raw material cost (C₁) is evaluated by

(6)C_1=\sum M_i\times C_i,

where M_i is the quality of raw materials required for producing UHPC, and c_i is the price per ton of material. Prices of raw materials are obtained from Tab.1. The unit of cost is China Yuan in the study.

3.1.2 Transportation costs

During the stage of transporting raw materials to the concrete mixing site, it is inevitable to have transportation costs. These costs involve the transportation costs of raw materials from the collection site to the mixing site. The transportation costs of transporting fresh concrete to the construction site, and the transportation costs of concrete waste after demolition and scrapping should also be included in the transportation costs. The unit transportation cost, construction waste landfill disposal cost, and UHPC mixing cost are listed in Tab.2. According to previous R [13,52], it is assumed that the general distance for transporting goods is illustrated in Tab.3.

According to the criteria proposed by Thomas and Griffin [57], transportation costs are determined by the quantity of goods, transportation distance, and unit transportation cost. The transportation cost (C₂) can be calculated by

(7)C_2=\sum M_i\times X_{it}\times D_{it},

where X_it is the unit transportation cost of material i transported to destination t, as displayed in Tab.2. D_it refers to the distance of material i to the destination t, as listed in Tab.3. M_r is the mass of 1 m³ of UHPC.

3.1.3 Mixing costs

The mixing cost in the production process of UHPC includes multiple aspects, such as equipment cost, fuel cost, and labor cost. The cost of equipment involves expenses related to the purchase, maintenance, and depreciation of concrete mixing equipment. The fuel cost mainly refers to the consumption of energy required for the concrete mixing process, including the cost of energy such as fuel or electricity. Labor costs refer to the labor costs required during the concrete mixing production process, including expenses for operator salaries, benefits, and training. The mixing cost (C₃) can be obtained by

(8)C_3=M_{\text{r}}\times c_{\text{r}},

where c_r is the mixing cost of UHPC, as exhibited in Tab.2.

3.1.4 Discard costs

At the end of the life cycle, the old UHPC needs to be removed, treated and disposed of. This stage involves the demolition of concrete structures, waste disposal and subsequent landfill or recycling processes. The whole process needs to comprehensively consider factors such as process, cost and environment. The scrap cost (C₄) can be calculated by

(9)C_4=M_{\text{r}}\times c_{\text{t}},

where c_t is the cost required for the landfill of each ton of UHPC waste, as provided by Tab.2.

3.2 Carbon emission assessment of ultra-high performance concrete during its life-cycle

The carbon emissions produced at each stage are taken into account in the life-cycle assessment. The negative environmental effects of producing UHPC are measured through a life-cycle analysis of the raw materials, energy and technology used, transportation, and waste management. This provides a scientific foundation and recommendations for cutting carbon emissions during the usage of UHPC.

The carbon emissions of UHPC in the whole life-cycle (E) are divided into carbon emissions of raw materials (E₁), carbon emissions of transportation (E₂), carbon emissions of mixing (E₃), and carbon emissions of scrap (E₄).

(10)E=E_1+E_2+E_3+E_4.

3.2.1 Carbon emissions of raw materials

The material carbon emission coefficients are sourced from the previous research [58], as listed in Tab.4. The electricity and diesel consumption data are both sourced from Ref. [59], as listed in Tab.5.

The carbon emission of raw materials mainly comes from the energy consumption in the production of raw materials. At this stage, the use of energy involves the operation of production equipment, including the exploitation, processing, transportation and other links of raw materials, so a comprehensive statistical assessment of carbon emissions of raw materials should be carried out. The carbon emission of raw materials (E₁) can be calculated by

(11)E_1=\mathrm{\Sigma }{M}_i\times e_i,

where e_i is the carbon emission coefficient per ton of material, as displayed in Tab.4.

3.2.2 Carbon emissions during transport

In the transportation stage, the transportation distance is usually used to calculate the carbon emissions in the transportation process. Generally, it refers to the transportation of raw materials to the concrete mixing plant and the transportation of UHPC to the construction site. This method is based on factors such as the transportation distance and mode of goods, and comprehensively considers the energy consumed in the transportation process, so as to evaluate the carbon emissions. The calculation of transportation carbon emissions (E₂) is:

(12)E_2=\sum M_i\times D_{i\text{t}}\times e_j\times K_j,

where e_j refers to j kinds of energy consumption for transporting 1 km per ton of material. In the transportation stage, this study assumes that the material transportation vehicle is a 12 t diesel truck, and the fuel consumption is about 20 L/100 km [6]. Hence, e_j is taken as 0.01667. K_j is the carbon emission coefficient, as is illustrated in Tab.5.

3.2.3 Carbon emissions during mixing

In the production process of mixing raw materials, the carbon emissions mainly come from the large amount of energy consumed by the mixing equipment, including electricity or fuel, which directly affects the carbon emissions. The carbon emission of mixing (E₃) is listed in Tab.4.

3.2.4 Carbon emissions during scrapping

In the scrapping stage of UHPC, it is necessary to classify, clean, crush and screen the demolished concrete debris to reduce the volume of waste and the content of pollutants, and prepare for subsequent landfill or recycling. The carbon emissions of the whole process need to comprehensively consider factors such as process, cost, environment and sustainability. The carbon emissions from scrap (E₄) can be evaluated by

(13)E_4=M_{\text{r}}\times E_{\text{t}},

where E_t is the carbon emission per ton of discarded UHPC, as illustrated in Tab.4.

4 Materials and experiments

4.1 Materials

Sichuan Esheng Company supplies Portland cement (P.I. 52.5R), which has a density of 3190 kg/m³. First-class FA with a density of 2270 kg/m³ and a specific surface area of 415 m²/kg is supplied by the Chongqing Fuhuang Company. The Ningxia Boyu firm produces the mineral powder (GGBS), which has a density of 2750 kg/m³ and a specific surface area of 430 m²/kg. Shanghai Shanying Environmental Protection Technology Co., Ltd. produces silica fume with an average particle size of 13.3 μm and a density of 2310 kg/m³. Tab.6 lists the chemical composition of the silica fume, FA, cement, and GGBS that are employed. Chongqing Construction Industrial Products Co., Ltd. is the company that produces the aggregate. The fine aggregate, which has a density of 2630 kg/m³ and a fineness modulus of 3.34, is produced by a pebble-crushing machine that has good sand particle hardness. The fundamental characteristics of the steel fiber, which is manufactured by Chongqing Fuxiang Metal Fiber Co., Ltd., are displayed in Tab.7. The steel fibers are straight with a smooth surface. Its properties meet the related requirements [6]. Produced by China West Construction Group Co., Ltd., polycarboxylate superplasticizer has a density of 1190 kg/m³, a water reduction rate of 45%, and a solid content of 40%. The laboratory tap water is used as the test water.

4.2 Experiments

After weighing the raw materials, add cement, admixtures and machine-made sand into the mixer and mix for 1 min to obtain a uniform dry mixture. Then add the superplasticizer into the water, gradually add it into the dry mixture, and keep stirring for 3 min. Continue to slowly disperse the steel fiber into the dry mixture, and keep stirring for 4 min. Based on our previous research, the fine blend materials in the paste remain partly agglomerated [60]. Therefore, vibration and ultrasound can be used for the dispersion of UHPC. Experiments have proved that vibration and ultrasound can uniformly disperse fine blend materials in cement-based materials [60–63].

The research refers to GB/T 50080-2016 [64] for slump tests, and follows GB/T 31387-2015 [65] for compressive tests. The research uses the 100 mm × 100 mm × 100 mm test piece specified in Ref. [65].

5 Development of the prediction model using machine learning methods

5.1 Establishment of database

In the research, 442 sets of data are sourced from Ref. [6]. Additionally, the Chongqing Communications Research and Design Institute Co., Ltd. contributes 132 sets of data. Hence, 574 groups of UHPC mix proportion data are obtained. It includes 13 concrete mix design variables: cement content, cement compressive strength, FA content, FA fineness, slag content, slag activity index, silica fume content, fine aggregate content, sand fineness modulus, sand crushing index, steel fiber volume content, the amount of superplasticizer, and water content. The experimental results contain the compressive strength and slump flow.

First, the SVR model is developed based on the data set. The results, MAE = 0.22 and R² = 0.620, are obtained. The prediction accuracy is low. Hence, the RFE is adopted to eliminate the parameters. A predetermined amount of features to be chosen or a certain performance metric that the model meets might serve as the stopping criterion. Using RFE, after recursive elimination, the variables are reduced to 9, which includes cement content, FA content, FA fineness, slag content, silica fume content, fine aggregate content, steel fiber volume content, superplasticizer content, and water content.

Prior to analysis, the input data has to be normalized because the retrieved data set contains varying units and ranges. The gathered information is standardized by applying MinMax to the input variables, as illustrated in Eq. (14) [6].

(14)x_{\text{norm}}=\left(x-x_{\text{min}}\right)/\left(x_{\text{max}}-x_{\text{min}}\right),

where x_min and x_max refer to the minimum and maximum values of a certain class of input values, respectively. x_norm indicates the normalized result.

Three data sets—a training set, a validation set, and a testing set—are randomly selected from the data set. 10% of the data set is used for testing. 90% is applied for training and validation. The model is trained on the training set, tested for accuracy and generalizability on the validation set, and evaluated for final goodness of fit on the testing set.

Considering the low number of cases, ML gives better results if applied to one output. Here, two results (i.e., compressive strength and slump flow) are predicted. Hence, two separate ML models are developed.

5.2 Prediction of compressive strength

5.2.1 Preliminary prediction of compressive strength

According to the ML technology described above, six methods, including RFR, SVR, DTR, ETR, MLPR and XGBR, are used to predict the compressive strength of UHPC. Using a 10-fold cross validation (CV), ten parts are randomly picked from the training set, nine of which are the training set and one of which is the validation set. The process is performed on separate samples for the validation group each time. Averaging the error is used to assess the forecast following the application of a 10-fold CV. The mean values of the MAE, MSE, RMSE, and R² are then examined for comparison. The performance evaluation indexes of the prediction results are shown in Tab.8.

The performance index R² of the test set of six compressive strength ML models without hyperparameter optimization fails to exceed 0.900, indicating that the model’s fitting ability is poor, according to the analysis of the results of multiple performance evaluation indexes of the preliminary fitting model. When compared to the training set, the testing set’s MAE and MSE error indices increase. It demonstrates that there are significant errors and that the model cannot reach a high fitting effect and generalization ability at the early fitting stage without hyperparameter optimization. Hence, the compressive strength model should undergo hyperparameter optimization to increase the model’s fitting precision and capacity for generalization.

5.2.2 Selection of prediction model for compressive strength after hyperparameter optimization

A proper hyperparameter setting may greatly enhance the model’s performance, and adjusting the hyperparameter is a crucial step in the field of ML. A suitable combination of hyperparameters may be found rapidly with random search, and in a minimal number of iterations, Bayesian optimization can obtain the global optimal solution. In actuality, different optimization techniques can be combined, such as employing the random grid search method for preliminary search and Bayesian optimization for further refinement, to increase optimization efficiency. The optimization results of six ML models using the random grid search method are displayed in Tab.9.

The performance assessment index is derived from the compressive strength model produced by RFR, SVR, DTR, ETR, MLPR, and XGBR, after hyperparameter optimization and application. After further investigation, the model with the best generalization effect and the highest fitting effect is chosen. Tab.10 displays the performance evaluation index.

Additionally, the performance indicators of the training and testing data sets are shown in Fig.5(a) and Fig.5(b), respectively, to further demonstrate the efficacy of the constructed model. With the lowest RMSE, MSE, and MAE of 0.013, 0.000225, and 0.015, respectively, and the highest R² value of 98.8%, DTR demonstrates the greatest performance in the analyzed model on the training data set. However, DTR has poor generalization ability on the testing set. The XGBR model yields values of 0.06, 0.0065, 0.081, and 0.912 for MAE, MSE, RMSE, and R² on the testing data set. With the lowest errors (MAE and RMSE) and greatest R², the XGBR model’s performance has greatly improved. This suggests that the XGBR model can handle unseen data more effectively because of its higher generalization capabilities. The findings demonstrate the excellent degree of fitting of the XGBR-based prediction model for UHPC’s compressive strength. As a result, the XGBR approach is recommended for estimating UHPC’s compressive strength.

The XGBR model’s hyperparameter is further optimized to find the optimal hyperparameter using the random grid search method and the Bayesian optimization method in order to better understand the differences in the hyperparameter performance of the model obtained under the two hyperparameter optimization methods. Tab.11 displays the model performance evaluation indexes.

The testing set’s R² of the XGBR prediction model optimized by Bayesian optimization achieves 0.971, and the fitting effect is superior to that of the prediction model optimized by random grid search, with the testing set’s R² being 0.932, according to a comparison of the model’s performance evaluation indexes. In addition, the random grid search in the other three error indexes is less effective at fitting and has worse prediction accuracy than the Bayes-optimized XGBR prediction model. Thus, the XGBR approach model with a superior fitting effect is chosen as an objective function for compressive strength prediction in the subsequent section.

5.3 Prediction of slump flow

5.3.1 Preliminary prediction of slump flow

Six methods—including RFR, SVR, DTR, ETR, MLPR, and XGBR—are also utilized to estimate the slump flow of UHPC in accordance with the prediction technique of compressive strength. Tab.12 shows the model performance assessment indices that are obtained.

The preliminary fitting R² results of six ML models—RFR, SVR, DTR, ETR, MLPR, and XGBR—without hyperparameter tuning do not surpass 0.9 on the testing set, according to Tab.12. The testing set’s performance index sharply declines, despite the training set’s higher performance index, suggesting weak generalization. Large errors result from the model’s inability to attain a high fitting effect during the early fitting step without hyperparameter optimization. The slump flow model’s hyperparameter tuning is done to get a better degree of fitting and generalizability.

5.3.2 Selection of prediction model for slump flow after hyperparameter optimization

The analysis in Subsubsection 5.2.2 indicates that, for the six types of slump flow models, the random grid search method with less computation is preferred to optimize the hyperparameter. This method effectively explores the parameter space in a comparatively short amount of time in order to find the ideal combination of parameters. The hyperparameter search range aligns with the compressive strength range. The ideal outcome is different, though. Tab.13 displays the best outcomes of the identified hyperparameters.

After being trained, the slump flow prediction models developed by RFR, SVR, DTR, ETR, MLPR, and XGBR with hyperparameter optimization are compared and their prediction performance is assessed. Tab.14 lists the performance assessment indices of the derived slump flow forecast models.

The training and testing data sets’ performance metrics are shown in Fig.6(a) and Fig.6(b), respectively. The RFR model yields the following results on the test data set: 0.067, 0.015, 0.124, and 0.916 for MAE, MSE, RMSE, and R², respectively. With the lowest errors (MAE, MSE, and RMSE) and the greatest R², the RFR model performs the best. RFR shows greater R² and lower error in the testing data set, while having a lower R² than DTR in the training data set. This suggests that the RFR model has a better capacity for generalization than the DTR model.

When comparing the prediction performance evaluation indices of several ML algorithms for slump flow, it can be shown that, in contrast to compressive strength prediction, the RFR model performs optimally in slump flow prediction. However, the fitting impact still can be increased. To enhance the RFR model’s fitting performance and achieve more fitting effect and generalization capacity, it is important to enhance the model’s performance index, elevate R², and decrease the MAE, MSE, and RMSE error performance index. Consequently, the RFR slump flow prediction model’s hyperparameters are optimized using the Bayesian optimization approach.

As demonstrated by Tab.15, the performance assessment indices derived from the original random grid search method and the RFR model refined by Bayes are compared.

The RFR slump flow prediction model optimized by Bayes attains an MAE of 0.062 using the testing set, and its fitting impact outperforms that of the prediction model optimized by random grid search. In addition, the random grid search in the other three error indexes is less effective at fitting and has worse prediction accuracy than the Bayes-optimized RFR prediction model. Therefore, the RFR method model with a superior fitting effect is chosen as an objective function for the prediction of UHPC’s slump flow in the subsequent section.

ML models can be used to obtain an approximate result. However, the carbon emission and cost can be calculated directly by Eqs. (5) and (10) after the mix ratio is given. Hence, carbon emission and cost are not outputs from ML models. The ML model is developed for further optimization.

6 Multi-objective optimization of ultra-high performance concrete

6.1 Constraint condition

One essential component of multi-objective optimization is the constraint condition. Decision variables have mathematical functions that define constraints. If the limitations are not fulfilled, the individual becomes unsuitable and must be removed from the population. Four types of restrictions are created for this study: absolute volume constraint, component content constraint, workability constraint, and strength constraint. The compressive strength and slump flow prediction model serves as the foundation for both the strength and workability constraints.

1) Strength constraint

The predicted compressive strength of the UHPC must exceed the required strength. This restriction is a part of the restrictions the designer placed on the objective function. The following is the strength constraint [6]:

(15)f_{\text{c}}>f_{\text{r}},

where f_c is the predicted compressive strength of the UHPC. The required amount for UHPC’s compressive strength, or f_r, must be chosen in accordance with real technical specifications. In this study, it is assumed to be 120 MPa based on UHPC’s fundamental mechanical properties [6].

2) Slump flow constraint

The predicted slump flow of fresh UHPC must be greater than the required slump flow. This restriction is a part of the restrictions the designer placed on the objective function. The following is the slump flow constraint:

(16)Slump>Slum{p}_{\text{r}},

where Slump is the fresh UHPC slump flow predicted value. Slump_r refers to the amount required for the slump flow of fresh UHPC, and it should be chosen based on the project’s specific needs. In this study, 600 mm is chosen in consideration of UHPC’s fundamental functioning properties [6].

3) Component content constraint

There should be a reasonable range for the optimum concrete component content. In this study, the designer’s constraint on decision variables is represented by the data range in the data set, which serves as the constraint range of component content. The following is the component content constraint:

(17)W_{\text{l}}⩽W_{\text{com}}⩽W_{\text{u}},

where W_com represents the content of all the constituents, which include water, high-performance water-reducer, steel fiber, cement, FA, slag, silica fume, and fine aggregate. The bottom and upper bounds of component content are denoted by W_l and W_u, respectively. Tab.16 shows the data set’s statistical properties.

4) Absolute volume constraint

For the purpose of designing mix ratios, the absolute volume approach is recommended by standard GB/T 31387-2015 [65]. Equation (18) computes the absolute volume. The formula indicates that each component’s volume total in 1 m³ of UHPC should equal 1 m³. An extra restriction imposed by the standard as a separate function of the optimization scheme’s decision variable is the absolute volume constraint.

(18)\frac{M_{\text{C}}}{{\rho }_{\text{C}}}+\frac{M_{\text{FL}}}{{\rho }_{\text{FL}}}+\frac{M_{\text{GGBS}}}{{\rho }_{\text{GGBS}}}+\frac{M_{\text{Si}}}{{\rho }_{\text{Si}}}+\frac{M_{\text{FA}}}{{\rho }_{\text{FA}}}+\frac{M_{\text{SF}}}{{\rho }_{\text{SF}}}+\frac{M_{\text{SP}}}{{\rho }_{\text{SP}}}+\frac{M_{\text{W}}}{{\rho }_{\text{W}}}=1,

where M_C, M_FL, M_GGBS, M_Si, M_FA, M_SF, M_SP, and M_W are the mass of cement, FA, slag, silica fume, fine aggregate, steel fiber, water reducer, and water in 1 m³ UHPC, respectively. ρ_C, ρ_FL, ρ_GGBS, ρ_Si, ρ_FA, ρ_SF, ρ_SP, and ρ_W are the density of cement, FA, slag, silica fume, fine aggregate, steel fiber, superplasticizer, and water, respectively.

6.2 Multi-objective optimization of mix ratios

The multi-objective optimization process takes into account not only the compressive strength and slump flow of UHPC, but also its life-cycle cost and carbon emissions. The compressive strength and slump flow of UHPC are modeled using the XGBR and RFR models as the objective function, respectively. UHPC’s life-cycle cost and carbon emissions are decreased while maintaining the required compressive strength and slump flow because of the multi-objective optimization technique. Two multi-objective optimization methods, NSGA-III and AGE-MOEA, are applied within the constraints to find the optimal mix ratio solution and determine the Pareto front.

1) Application of NSGA-III

To optimize the mix ratios to achieve reduced costs and carbon emissions, and improved compressive strength and slump flow, the NSGA-III approach is applied. Finally, five groups of optimum solutions are found on the Pareto front, as Fig.7 illustrates.

According to Ref. [59], the cost of UHPC is between 1048 and 7586 Yuan/m³, and the carbon emissions are between 353 and 2295 kg/m³. The current research results in Fig.7 are generally consistent with the results in the literature.

Fig.7’s Pareto frontier results show how cost and environmental sustainability are traded off. This trade-off emphasizes the significance of balancing competing objectives and is a typical challenge in multi-objective optimization problems. This finding gives vital insights into the optimal ratio of each component material necessary to generate the most ecologically friendly and inexpensive UHPC. The numerous UHPC depicted at the Pareto front in Fig.7 provide a valuable tool for designers and decision-makers to determine the ideal UHPC combination that fulfills the unique objectives of the project. The figure’s outcomes highlight the difficulties in identifying the single best solution in multi-objective optimization issues. Despite these drawbacks, as Subsubsection 2.6.3 explains, TOPSIS technology may be used to find appropriate solutions from the Pareto frontier, offering a more thorough method for optimizing multiple objectives. According to TOPSIS, an alternate solution is deemed superior if it has a higher score since it is more comparable to the ideal answer. The best mix design may be found using TOPSIS technology for certain UHPC projects on the Pareto frontier. Tab.17 displays the mix ratios and outcomes.

In Tab.17, NSGA-III discovers 5 nondominated solutions for cost and carbon emission optimization as illustrated. The ideal solutions are scored using the TOPSIS assessment method. Of these, No. 3 UHPC is the best option with the greatest TOPSIS score; it has a compressive strength of 127 MPa, a carbon emission of 422 kg, a slump flow of 612 mm, and a cost of 1030 Yuan/m³. It is advised as the best option left at the end. In addition to the TOPSIS technique, the engineering requirement may be used to determine the best scheme in a concrete engineering application. For instance, No. 1 UHPC is less expensive but has a greater carbon emission, making it a good choice for situations where cost is a major consideration. The carbon emissions of No. 5 UHPC are 102 kg lower than those of No. 1 UHPC. No. 5 UHPC is more expensive but emits less carbon dioxide, thus it can meet the criterion for green concrete in situations when CO₂ emissions are more crucial. As an alternative, common building projects can employ No. 3, which has a more balanced cost and carbon emission than Nos. 1 and 5.

As can be observed, the concentrated Pareto solution set that is advised costs more than 1001 Yuan. The Pareto solution set with lower prices, however, has a very low score, suggesting that a little cost drop occurs when the mixture’s cost falls below 1001 Yuan, but at the expense of a significant amount of CO₂ emissions. Furthermore, by comparing Pareto solutions with the lowest cost and carbon emissions, it can be found that for Pareto solutions with the lowest cost, expensive components such as steel fibers, water-reducing agents, and silica fume are lower, while cement with higher carbon emissions is more. This is so that costs are lowered due to the decreased amount of pricey components.

The use of a high proportion of by-products is what causes the carbon footprint to be reduced. The most sensible and cost-effective method of lowering CO₂ emissions in the concrete industry is to substitute high-volume SCMs for cement [66]. Mineral admixtures have been shown to lower the CO₂ emissions of concrete [67]. Meanwhile, the carbon emission may be lowered since UHPC has superior durability and requires less maintenance [6]. All things considered, UHPC has the potential to turn into an ecologically friendly material because of its increased durability, environmental concerns, and financial benefits.

2) Application of AGE-MOEA

To optimize the mix proportion and achieve lower costs and carbon emissions, the AGE-MOEA approach is utilized. Finally, six groups of optimal solutions are found on the Pareto front, as shown in Fig.8. Tab.18 provides the mix proportion composition and outcomes.

Six nondominated solutions for cost and carbon emission optimization are found using the AGE-MOEA technique (Fig.8). The TOPSIS assessment method is used to provide scores for the optimum solutions. No. 10 UHPC, which has a compressive strength of 121 MPa, a carbon emission of 453 kg, a slump flow of 657 mm, and a cost of 1001.18 Yuan/m³, is the best choice out of these with the highest TOPSIS score. It is suggested as the finest choice that is available. Six groups of optimal concrete mix proportions show a significant degree of cost and carbon emission similarity. The strengths of these groups range from 120 to 121 MPa, slump flows from 608 to 657 mm, expenses of around 1001 Yuan, and carbon emissions of approximately 453 kg. The optimal solution obtained by the AGE-MOEA method has similarity with the solution of NSGA-III, thus verifying the correctness of the NSGA-III model. On the other hand, comparing the AGE-MOEA results to the NSGA-III results reveals an excessive amount of concentration. Different costs and carbon emissions of UHPC may be selected according to the requirements of the specific project. The uniform distribution of the Pareto solution is not achieved by the AGE-MOEA method. This can be the result of limitations in the gathered data set. Therefore, in order to increase the multi-objective optimization models’ resilience, future research should concentrate on gathering larger data sets.

6.3 Experimental verification

Eleven sets of proportions for the concrete mix, derived from two multi-objective optimization methods, are tested in this experiment. Tab.19 contains a list of the 28 d compressive strength and slump flow test results. The ML model’s prediction accuracy is high because the difference between the tested and predicted values is only 10%. The outcomes demonstrate that the two chosen ML models have the ability to forecast compressive strength and slump flow with accuracy. The prediction model’s accuracy will increase as more pertinent information about the proportion of concrete mix is gathered.

7 Conclusions

This research develops a novel multi-objective optimization and UHPC performance prediction methodology that is both environmentally friendly and economically viable. To develop UHPC that satisfies the particular needs of various projects, this framework combines ML performance prediction models, multi-objective optimization methods, and multi-criteria decision-making methodologies. Through the quantitative analysis, the life-cycle cost and carbon emissions of concrete are quantified and evaluated in the study. For performance prediction, a variety of ML models are combined and tuned. Next, the mix ratio of UHPC is intelligently optimized using NSGA-III and AGE-MOEA to optimize its compressive strength, flowability, cost, and CO₂ emissions. The following conclusions have been drawn from the study.

1) The raw material manufacturing, transportation, mixing, and scrap recycling stages of UHPC are all quantitatively assessed following the life cycle evaluation. After calculating the life cycle cost and carbon emissions, an objective function for further optimization is established.

2) The RFE approach is utilized to screen and analyze the gathered database in order to exclude the less significant characteristic factors. The mechanical characteristics and fluidity of UHPC are fitted using six ML methods. The ML prediction model’s generalization and prediction accuracy are greatly enhanced by a range of hyperparameter tuning techniques.

3) The compressive strength of UHPC may be more accurately predicted by using XGBR with the Bayesian optimization approach. When it comes to slump flow prediction, RFR has a superior prediction impact. It therefore possesses good generalization and prediction capabilities.

4) Multi-objective optimization of the mix ratio is conducted in order to account for compressive strength, slump flow, life-cycle carbon emissions, and life-cycle cost. With a variety of UHPC mixtures to choose from and a series of optimum solutions generated by the NSGA-III algorithm and the AGE-MOEA technique, decision-makers may utilize the TOPSIS approach to determine the best UHPC in accordance with project requirements.

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