1 Introduction

Concrete plays a very crucial role in the construction industry. But traditional concrete mix designs rely on codal provisions and trial batches in both laboratory and field settings to ensure compliance with specified requirements in terms of strength, workability, and durability. While codal provisions can address the basics of concrete mixture designs, it should be noted that concrete is inherently heterogeneous, thus posing challenges for predicting its diverse properties. Moreover, formulating mixture proportions for optimal performance at a minimal cost remains a complex task. Therefore, it is critically important to predict and optimize concrete mix proportions by leveraging the potential of various machine learning and optimization tools, because such tools can potentially simplify the prediction and optimization of concrete properties by establishing multidimensional nonlinear relationships between raw materials and the desired performance, thus enhancing the efficiency of construction processes.

Several researches have been conducted to predict concrete properties by using various machine learning techniques. For instance, Oh et al. (1999) used ANN to proportion concrete mixes. Rao (2012) successfully used ANN to predict concrete compressive strength at different binder ratios. A new database platform (Compos) was developed by Chen et al. (2015), which implemented stepwise multiple linear regression and BP ANN approaches to identify correlations between concrete properties and the proportion of concrete raw materials. Mustapha and Mohamed (2017) proposed a Weighted SVM Based on High-Performance Concrete Compressive Strength Prediction, finding that the accuracy of the proposed SVM was significantly better than other SVMs for different evaluation measurements. Tafis and Sistonen (2017) explored different machine learning methods for durability and service-life prediction of reinforced concrete structures. Abuodeh et al. (2019) used ANN for interpretation of ultra-high performance concrete’s compressive strength. Marani et al. (2020) also predicted such strength by using tabular generative adversarial networks. Muliauwan (2020) used ANN, SVM and linear regression (LR) to predict concrete properties. In a comparative study on ANN and ANFIS models, Armaghani and Asteris (2021) estimated cement-based mortar materials’ compressive strength. Pandey et al. (2021) used multi-variable linear regression, SVM, Decision Tree Regression and ANN to design concrete mixes, with and without plasticiser of certain desired properties. When using XGBoost, MARS and SVM techniques to determine the shear strength of RC concrete beams, Hayder Riyadh Mohammed et al. (2022) observed that MARS was more powerful than other algorithms. Ly et al. (2021) successfully predicted the strength of rubber concrete with ANN. Shariati et al. (2021) evaluated the shear strength of tilted angular connectors (a kind of applications in steel composites) by using ANN, ANFIS, and ELM. Duan et al. (2021) utilized ICA, XGBoost, SVR, ANN, and ANFIS to predict the recycled aggregate concrete’s compressive strength. Nguyen et al. (2021a) predicted the strength of concrete by using MLP, GBR, XG Boost, and SVM. Ayaz et al. (2021) predicted the compressive strength of geo-polymer concrete combined with fly ash in the methods of ANN and ResNet. Feng De-Cheng et al. (2020) adopted AdaBoost to assess concrete’s strength. Mishra et al. (2020) utilized SVM, ANN, and ANFIS to measure the strength of unreinforced bricks. Similarly, Naser et al. (2019) also used ANN and ANFIS to identify the strength of asphalt rubber. Moayedi et al. (2019) established the friction capacity of driven piles in clay by ANFIS, ANN, SVM and GP, finding that ANFIS can deliver the best result. Naser (2019) predicted the service life of exterior painted surfaces by using ANN. In order to determine the viscosity of asphalt rubber, ANN and ANFIS were implemented by Specht et al. (2014). In addition, Chou et al. (2013) assessed the strength of concrete by utilizing ANN, CHAID, CART, LR, l GENLIN and SVM, concluding that the combination of ANN and SVM can yielded the best result. Several other machine learning applications are also available in the field of geotechnical engineering for evaluating the uniaxial compressive strength (Jahed Armaghani et al., 2018), the bearing capacity of piles (W. Chen et al., 2020), and the concrete-filled steel tube columns (Sarir et al., 2021), among others.

Some researches have also been carried out to optimize the concrete mixture design in different methodologies. For example, Abbasi et al. (1987) designed an experimental method to optimize concrete mixtures for a given workability and compressive strength. Shilstone and James (1990) suggested a quantitative technique for optimizing the proportions of aggregate, while carrying out some adjustments during the experiment. With locally available ingredients, Kasperkiewicz (1994) applied analytical techniques for concrete mix designs, attaining an optimal composition of concrete at the lowest cost. Chang (2001) proposed a densified mixture design algorithm to yield a sort of optimum high-performance concrete, which can deliver high workability and good durability. Soudki et al. (2001) produced statistical analysis results in full factorial experiments, aimed to optimize a concrete mix proportion for hot climate conditions. Nunes et al. (2006) explored self-compacting concrete in a robust mix design methodology. Ahmad (2007) proposed a laboratory trial procedure for an optimum design of concrete mixes at the minimum cost of concrete. In several analytical methods, such as ANN, nonlinear programing, and Genetic Algorithm (GA), Yeh (2007) obtained an optimum mixture of concrete composition for required performance at the lowest cost. By adopting GA, ANN and convex hull as an optimum technique, Lee et al. (2009) determined the minimum cost of concrete under a given strength requirement. Yeh (2009) optimized concrete mixture proportions by using a flattened simplex—Centroid mixture design as well as ANN. Self-consolidated high-strength concrete was optimized by Akalin et al. (2010) using a D-optimal design. By applying the Particle Swarm Optimisation (PSO) for designing high-performance concrete mixtures, Xiaoyong and Wendi (2011) presented an optimum approach to designing concrete mixtures based on experimental and orthogonal methods, while identifying the main factors influencing the compressive strength of concrete in a mix proportion. Jayaram et al. (2010) demonstrated opportunities for elitism-based PSO models to develop high-volume concrete fly compounds and found that when using PSO to design high-performance concrete mixtures, the number of trial compounds, which have desirable structures in the field, can be reduced. Ghiamat et al. (2019) proved that cost optimization with the optimal combination of genetic operators can reduce the construction cost and weight of the bridge superstructure by 13%, mostly thanks to the reduction in the required pre-stressing tendons. Hamed Naseri (2019) reported that PSO is better than GA as a good solution for concrete mix proportions. In an investigation carried out by Sobhani, PSO yielded faster and better results than GA. Feng et al. (2021) used three metaheuristic ANFIS-based algorithms, namely, PSO, ant colony optimization (ACO) and differential evolution optimization (DEO), to predict the super plasticiser demand of self-consolidated concrete mixtures.

Conventional reliability-based design optimisation was carried out in dealing with optimization under uncertainty by Abbasnia et al. (Cheng et al., 2017). Additionally, the robust optimisation has been successfully implemented in stochastic mechanical systems over recent years (Beyer & Sendhoff, 2007; Cheng et al., 2017). Moulick et al. (2019a) presented an efficient robust optimization procedure for concrete mixtures with rice husk ash.

The above literature review identifies a gap in the current research on concrete mix design and property prediction: the absence of comprehensive research to integrate multiple modern machine learning tools, such as SVM, ANN, FIS, ANFIS, and GEP. Although many studies have explored these techniques separately, there is a notable lack of holistic investigations into their combined effectiveness. Furthermore, the literature review highlights a scarcity of research focusing on multi-objective deterministic and robust optimization strategies for striking a balance among compressive strength, workability, and cost minimization in concrete mixtures. Most of the existing studies tend to concentrate on individual aspects, rather than providing a comprehensive approach to concrete-related technologies. Addressing these gaps could significantly enhance the understanding and application of advanced machine learning tools for optimizing concrete properties.

In order to predict different concrete properties, this study would adopt five machine learning techniques—SVM, ANN, Fuzzy FIS, ANFIS and GEP. In addition, two hundred data about concrete mixtures would be considered at various levels of key ingredients—cement, water, fine aggregate, coarse aggregate, and size of coarse aggregate, along with their compressive strength and slump value. 170 of the 200 data would be used for training purpose, while the remaining 30 for testing. Such a division of data is a common practice in the field of machine learning, particularly in supervised learning techniques for accurately training a network (Madhiarasan & Louzazni, 2022). In addition, experiments would be conducted in laboratory, given that 15 data about different mix proportions and their strength and slump values have been obtained. Thus, this approach could definitely boost the reliability and generalizability of results, and this is a feature different from that of other studies. Furthermore, deterministic and robust optimisation would be implemented at an optimum concrete mixture to obtain the maximum compressive strength and slump at the minimum cost.

2 Experiment procedure

2.1 Data collection and experiment program

Concrete mix properties were selected from the available literature for conducting this study (Abdelatif et al., 2018). With a total of 1240 numbers in the data set, data pre-processing is very important for this type of research (Fan et al., 2015; Xiao & Fan, 2014). Therefore, data reduction was carried out, as the row or column value regarding 28 days’ compressive strength and workability discarding the missing values and mix design containing additives. Thus, data reduction was made on the basis of mix design having no additives and no value of 28 days’ compressive strength as 714 and 293, respectively. Afterwards, data scaling was implemented with the normal distribution curve (Gaussian distribution). It shall be noted that as an important concept in statistics, the normal distribution forms the backbone of Machine Learning. It yields a conclusion that approximately 99.7% of data fall within three standard deviations of the mean. The data were transformed from categorical data into numerical data, and they were 200. Finally, the 200 numbers in the mix design dataset were used in this study. The input data include: aggregate size, amount of Ordinary Portland Cement (OPC), water, fine aggregate, and coarse aggregate in kg per cubic meter; and the output is 28 days’ compressive strength (in MPa) and slump (in mm) corresponding to each mix proportion. The boxplots of the mixture content, slump and compressive strength are presented in Fig. 1.

Fig.1
figure 1

Box plot of different concrete ingredients compressive strength and slump

Full size image

For the purpose of validating the present prediction methodology, an extensive experimental program was conducted (Fig. 2), with 15 concrete trial mixes chosen arbitrarily. The OPC cement was used in this experiment. Crushed stone particles obtained from a local hill were used as coarse aggregates, and local river sands were used as fine aggregates. Both aggregates confirm to IS 383–1970. Potable water was used for mixing the constituents of all the tests. However, chemical admixtures were not used in the tests. A total of 15 × 3 = 45 cubes having a dimension of 150 × 150 × 150 mm were casted for obtaining the compressive strength of the concrete. After 24 h, the cubes were removed from the mould, and then cured for 28 days in a curing tank under normal temperatures. Then, the compressive strength of the specimens was measured in a standard compressive testing machine. The average compressive strength of the three cube samples prepared from the same concrete mixture design was marked as the compressive strength of that mixture. Table 1 presents the average test results of 28-day compressive strength for all 15 concrete mixtures, along with each mixture per 1 cubic meter.

Fig. 2
figure 2

Experiment performed in laboratory for validation

Full size image
Table 1 Experimental mixture design data along with measured compressive strength and slump
Full size table

2.2 Different machine learning techniques

In order to predict the concrete mix proportions, SVM, ANN, FIS, ANFIS and GEP were used, which are important predictive tools in civil engineering applications (Chen et al., 2015; Jalal et al., 2021). These methods’ effectiveness in prediction has been proved in many aspects.

2.2.1 Support vector machine (SVM)

SVM, a supervised machine learning model, adopts classification algorithms for two-group classification problems. It was first introduced by Vapnik (Cortes & Vapnik, 1995; Shiuly et al., 2022a, 2022b). After given labelled training data for each category, an SVM model is able to categorize them into new texts. Compared to some newer algorithms, e.g., neural networks, SVM has two main advantages: higher speed and better performance with a limited number of samples (in thousands). This makes SVM very suitable for text classification, where it’s common to have access to a dataset of at most a couple of thousands of tagged samples. SVM has been used in many civil engineering applications; and in recent years, it is often used to predict concrete's compressive strength (Mishra et al., 2020; Mohammed & Ismail, 2022; Muliauwan et al., 2020; Nguyen et al., 2021a, 2021b; Shih et al., 2015; Wang et al., 2014; Yu et al., 2018). The support vector regression, which is a variation of SVM, is also used to build input–output models for concrete. SVM uses an objective function, which drive the function estimation process. When a nonlinear space is reached, a kernel radial-based function (RBF) will be selected as the kernel function for SVM, because it can provide better results than other kernels.

This model underlies the functional relationship between one or more independent variables and a response variable:

$$y(A) = wT \phi (A)+B$$
(1)

where A Є R, y Є R, and ϕ(A): Rn is a process of mapping to a higher dimensional feature space; w indicates the weight; and B is a constant.

2.2.2 Artificial neural network (ann)

Artificial neuron is a computational model inspired by natural neurons (Alam et al., 2020; Desai et al., 2008; Madhiarasan & Louzazni, 2022; Sada & Ikpeseni, 2021; Şahin & Erol, 2017; Shiuly et al., 2020, 2022a, 2022b; Walczak & Cerpa, 2003): Natural neurons receive signals through synapses found in the dendrites membrane of neurons; when signals are found strong enough, neurons would activate and release a signal to the axon; this signal can be sent to another synapse, which may activate other neurons. ANN solves various problems in pattern recognition, predictive performance, memory-related tasks, etc. As it has the ability to automatically learn from a given training pattern, ANN can solve map-related problems by finding the approximate limitations of input data associated with output data. This feature separates it from other conventional specialist systems. An ANN computing system is made up of many synthetic neurons, which play the role of vital units and mimic a parallel process of brain biology for responses. The behaviour of ANN networks is influenced by the communication pattern of neurons, which determine the class of a network as well. As mentioned above, training a network to improve its performance is thought. In precise terms, the structures and weights of network connections would change iteratively, so as to minimize the errors that refer to the entire node of the output layer.

A trained neural network acts as an analytical tool for qualified predictions of results based on any sets of input data which are not involved in the learning process of the network. Its functions are practically simple and easy, nevertheless correct and precise. A neural network is composed of numerous mutually connected neurons grouped in layers. The complicity of a network is determined by the number of layers. Besides the input (first) and the output (last) layers, a network may have one or a few hidden layers. The purpose of the input layer is to accept data from the surroundings. Those data are processed in the hidden layers and then sent into the output layer. The final results from the network are the outputs of the neurons from the last network layer, forming the solution to the analyzed problem.

2.2.3 Fuzzy inference system (FIS)

Fuzzy theories can take intermediate values, such as high, medium, low, very low and similar values. FIS is a set of multi-valued logic, which permits intermediary values to be defined between conventional assessments, e.g., true/false, yes/no, high/low, etc. Ideas like very tall or very fast can be expressed mathematically and processed by computational systems, so as to apply a more human-like thinking procedure in computers’ programming (Berenji, 1992; Shiuly et al., 2022a, 2022b). The term “fuzzy” was first presented by Zadeh (D’Urso & Gil, 2017) in his research paper on fuzzy sets, which introduced a new mathematical discipline, namely fuzzy logic, based on the theory of fuzzy sets. The aim of the logic was to support the consideration and presentation of rough ideas by fuzzy sets. The imprecision is to be understood as grouping certain set members into classes, with the boundaries between them not sharply defined. The theory of fuzzy sets is expected to become a novel methodology suitable enough for helping formulate and solve complex problems in engineering and science, which are often difficult to handle by using "precise" crisp logic, such as binary logic, under which the variables can only be either true or false. The theory of fuzzy sets allows the concept of partial belongingness of an object or a variable in a fuzzy set and, therefore, enables a gradual transition from full membership to a totally non-membership. Thereby, in fuzzy logic, an object or a variable within a domain may partially belong to several fuzzy sets in the same domain simultaneously. Thus, this theory provides a framework for a multi-valued logic, which is essential for capturing the vagueness in a natural linguistic description of any system. Moreover, the underlying fuzzy logic incorporates a variety of rules with certain premises containing fuzzy propositions, which are generally defined in linguistic terms, such as low and high (temperature, pressure, flow, frequency, voltage, etc.), or old, older and very old (person, engine, sensor, measured value, etc.). These related linguistic rules are of IF–THEN art.

Successful prediction by models in various research in the past indicated that fuzzy logic could be a useful modelling tool for engineers and researchers in the area of cement and concrete (Shiuly et al., 2022a, 2022b). Fuzzy logic toolboxes have been used to predict the strength and slump of concrete. The purpose of making this logic is to decision making, understanding, problem solving, planning etc.

2.2.4 Adaptive neuro fuzzy inference system (ANFIS)

Neuro-fuzzy modelling refers to applying various learning techniques developed in the neural network literature to FIS (Jang, 1993; Sada & Ikpeseni, 2021; Shiuly et al., 2022a, 2022b). ANFIS, a hybrid neural network that amalgamates fuzzy logic and neural networks, consists of two main components: the consequent and the antecedent, which work together to formulate some fuzzy rules for shaping a network. Throughout the training process, a hybrid optimization technique is employed to update the parameters of each section. The ANFIS architecture is organized into five layers: The first layer, known as the fuzzification layer, would compute membership degrees for each input and adjust antecedent parameters in a gradient descent algorithm; subsequently, the rule layer, constituting the second layer, would calculate the firing strength of each rule; the third layer, referred to as the normalization layer, would normalize the firing strengths through min–max normalization; moving on to the fourth layer—the defuzzification layer—marks the inception of the consequent part and updates its parameters by using the least square error technique in the forward path; finally, the fifth layer, called the output layer, would aggregate the outputs from the fourth layer. In ANFIS, the antecedent portion spans the initial three layers, while the consequent portion covers the remaining layers. This study employed the concept of adaptive network (which is a generalization of the common back-propagation neural network) to tackle the parameter-identifying problem in FIS. An adaptive network is a multi-layered feed-forward structure, with its overall output behaviour determined by the value of a collection of modifiable parameters. More specifically, the configuration of an adaptive network is composed of a set of nodes connected through directional links, and each node is a processing unit to perform a static node function based on the incoming signals, so as to generate a single node output. Such a node function is a parameterized function, because it has modifiable parameters. It shall be noted that the links in an adaptive network only indicate the flow direction of signals between nodes, with no weights associated with these links. In this study, three ANFIS models were developed to predict the concrete compressive strength, slump, and dry density. In all the three models, water, cement, fine aggregate and coarse aggregate were used as input variables. All the three models were developed by using a grid partition fuzzy interference system, which adopted 3 linear membership functions. It is noteworthy that ANN lacks the ability to make informed decisions, while Fuzzy Logic is criticized for its weakness in learning capabilities.

$${\text{Neural network}}\, + \,{\text{FUZZY}}\, = \,{\text{ANFIS}}$$

As a neural network, it has strong generalization capability and high strength. The only limitation is very high computational cost, due to complex structures; so it is not suitable for a large amount of inputs.

2.2.5 Genetic expression programing (GEP)

In this method, the first human chromosomes are initially randomly generated by software (Bansal et al., 2022; Shiuly, 2018; Shiuly et al., 2020, 2022a, 2022b). In the next step, the chromosomes are transferred and the suitability of the individual population is assessed. Then, with the eligibility considered, individuals are selected randomly, leaving a seed with new traits. In the new generation, the same process is repeated: genetic predisposition, hostility, and reproduction through evolution. For a certain number of generations, or before a specific solution is reached, this process continues (Fig. 3).

Fig. 3
figure 3

Genetec Expression tree working flow chart

Full size image

It is based on the mechanism of natural selection and natural genetics:

  1. a)

    Mainly focus on optimization to get better results.

  2. b)

    Solve complex problems.

  3. c)

    Follow the theory of “Survival of the fittest” (Darwin’s theory).

In the early GA methods, the first human chromosomes are randomly produced by this software; then, chromosomes are distributed, and the suitability of an individual population is assessed; then, when considering suitability, individuals are selected randomly, leaving a seed with new traits. In the new generation one finds the same process—genetic predisposition, hostility, and reproduction through evolution. For a certain number of generations or until a specific solution is reached, this process continues. In order to produce the compressive strength, slump and dry density separately, such as water work, cement, good amount, coarse, small plastic, large plastic; GeneXpro Tools 5.0 (Ferreira, 2001) was used in this study. In all cases, the 30 chromosomes, each containing the three-dimensional genes, were used in calculation.

2.3 Development of the optimisation scheme

In this study, two ANFIS models were developed to predict the concrete compressive strength and slump; and coarse aggregate size, cement, water, fine aggregate, and coarse aggregates were utilised as input variables, while the values of compressive strength and slump were used as output variables, as depicted in Fig. 4(a) and (b), respectively. All the models were developed in a grid partition fuzzy interference system.

Fig. 4
figure 4

ANFIS network for predicting compressive strength and slump

Full size image

In this study, cement, coarse aggregate, fine aggregate, water, and size of coarse aggregate were used as input parameters of the fuzzification layer in ANFIS. Furthermore, the membership degree of all the input parameters are adjusted by using the gradient descent algorithm as cement, water, coarse aggregate, size of coarse aggregate, fine aggregate. The types of all membership functions were triangular. The initialization of the membership function's parameters involves a random assignment, which uses cluster centres derived from the K-means algorithm for input–output training data. Subsequently, these parameters were fine-tuned to ensure compliance with distinguishing ability constraints. In addition, ANFIS' information includes:

Number of nodes: 15; Number of linear parameters: 4; Number of nonlinear parameters: 1; Total number of parameters: 5; Number of training data pairs: 135; Number of checking data pairs: 15; Number of fuzzy rules: 2; Minimal training and testing for both RMSE and R2: indicated in Sect. 3.6. The whole ANFIS model was carried out in MATLAB’s tool box. The model’s accuracy was measure by RMSE and MAE.

2.3.1 The deterministic optimization

The performance of an optimal design depends on design parameters (z) and design variables (x) (Moulick et al., 2019b). The design parameters refer to those that cannot be controlled or are difficult and expensive to control. The design variables refer to specific parameters that need to be optimized, so as to achieve the intended performance. In this study, design variables (x) include: size of coarse aggregate, amount of cement, water, and fine aggregate and coarse aggregate in kg per cubic meter of concrete; the design parameters (z) include the cost of ingredients, compressive strength and slump.

Deterministic optimization was made to obtain the optimal design variables, which will minimize the cost, so as to satisfy the required criteria of 28 days’ compressive strength and slump (workability):

$$\begin{gathered} {\text{minimize}}\quad f({\mathbf{x,z}}):{\text{cost}} \hfill \\ {\text{subjected}}\,{\text{to}}\quad g_{{_{1} }} ({\mathbf{x,z}}): \, \sigma_{c}^{t} { - }\sigma \left( {{\mathbf{x,z}}} \right) \le 0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,g_{{_{2} }} ({\mathbf{x,z}}): \, s_{c}^{t} { - }s\left( {{\mathbf{x,z}}} \right) \le 0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x_{i}^{{\text{L}}} \le x_{i} \le x_{i}^{{\text{U}}} ,\,\,\,\,\,i = 1,2,.....,K. \hfill \\ \end{gathered}$$
(2)

where xiL and xiU indicate the lower and the upper bounds of the ith design variable, respectively; \(\sigma_{c}^{t}\),\(\sigma \left( {{\mathbf{x,z}}} \right)\),\(\, s_{c}^{t}\) and \(s\left( {{\mathbf{x,z}}} \right)\) are the target 28 days’ compressive strength, the obtained 28 days compressive strength, the target slump, and the obtained slump, respectively; \(\sigma \left( {{\mathbf{x,z}}} \right)\) and \(s\left( {{\mathbf{x,z}}} \right)\) are the implicit functions of [x z]. It shall be mentioned that the deterministic optimization problem as presented by Eq. (2) does not reflect the effect of uncertainty in [x z]. However, the objective function and the constraints are the function of [x z]. Thus, the uncertainty in [x z] is expected to spread at the system level, thus inducing the objective function and the constraints of the associated optimization problem.

2.3.2 The robust optimization

2.3.2.1 Robustness of the objective function

The robustness of the objective function is generally measured in terms of the dispersion of the performance function from its mean value (Moulick et al., 2019b). The aim of a model design is to attain the optimal performance as well as low sensitivity of the performance function, with respect to the variation in the design variables and design parameters due to uncertainty. Thus, it is required to optimize the objective function with the dispersion (standard deviation for normal random parameters). Therefore, the robust optimization problem can be modelled as a minimization problem of the mean and standard deviation of the objective function, leading to a criterion of robust design optimisation problem, which can be expressed as:

$${\text{ Find x}},{\text{ to minimize}}\,\,{[}\mu_{f} ,\sigma_{f} ].$$
(3)

where \(\mu_{f}\) and \(\sigma_{f}\) are the mean and the standard deviation of the performance function, respectively. Generally, minimization of the mean and the standard deviation of the performance is required, resulting in a set of Pareto-optimal solution, as presented by Deb et al. (2002). Applying the equation generated by GEP, a multi-objective function was converted to an equivalent single-objective function:

$$\phi {(}{\mathbf{u}}{)} = {(}1 - \alpha {)}{{\mu_{f} } \mathord{\left/ {\vphantom {{\mu_{f} } {\mu_{{_{f} }}^{*} + }}} \right. \kern-0pt} {\mu_{{_{f} }}^{*} + }}\alpha {{\sigma_{f} } \mathord{\left/ {\vphantom {{\sigma_{f} } {\sigma_{f}^{*} }}} \right. \kern-0pt} {\sigma_{f}^{*} }}{;}\quad {0} \le \alpha \le 1$$
(4)

where \(\phi {(}{\mathbf{u}}{)}\) is a new objective function, called the desirability function; parameter \(\alpha\) acts as a weighting factor; \(\mu_{{_{f} }}^{*}\) and \(\sigma_{f}^{*}\) are the optimal values of the mean and the standard deviation for \(\alpha\) and equal to 0.0 and 1.0, respectively. The maximum robustness would be attained when \(\alpha\) becomes 1.0. Suppose u = [x z]. By using the first-order perturbation approach, the mean and the standard deviation of the objective function can be acquired for normal random parameters (Doltsinis et al., 2005), as expressed below:

$$\mu_{{f_{1} }} {(}{\mathbf{u}}{)} \approx f_{1} \left( {{\overline{\mathbf{u}}}} \right),\quad \sigma_{{f_{1} }}^{2} \approx \sum\limits_{i = 1}^{N} {\left( {\left. {\frac{{\partial f_{1} }}{{\partial u_{i} }}} \right|_{{\overline{u}_{i} }} } \right)}^{2} \sigma_{{u_{i} }}^{2} \,$$
(5)

Similarly, in an Uncertain but Bounded system, by using the worst case propagation concept, the nominal value \(\overline{f}\) (i.e., the mean in the normal random case) and the dispersion Δf (i.e., the standard deviation in the normal random case) can be presented as below (Lee & Perk, 2001):

$$\overline{f}_{2} = f\left( {{\overline{\mathbf{u}}}} \right)\quad \Delta f_{2} = \sum\limits_{i = 1}^{N} {\left| {{{\partial f} \mathord{\left/ {\vphantom {{\partial f} {\partial u_{i} }}} \right. \kern-0pt} {\partial u_{i} }}} \right|\Delta u_{i} }$$
(6)

where \({\overline{\mathbf{u}}}\) denotes the nominal value of u, i.e. \({\overline{\mathbf{u}}}\) = (uL + uU)/2. Finally, for a mixed system of Uncertain but Bounded and random parameters, the resulting nominal value and the dispersion of the objective function can be obtained:

$$\mu_{f} = \mu_{{f_{1} }} + \overline{f}_{2} \quad \sigma_{f} = \sigma_{{f_{1} }} + \Delta f_{2}$$
(7)

The formulation above is effective for relatively low levels of uncertainty (up to 25%). However, to deal with non-normal variables, the Monte Carlo Simulation (MCS) approach is proper for estimating the mean and the standard deviation values.

2.3.2.2 Robustness of the constraints

Due to uncertainty in [u], the optimal solution found with deterministic constraint functions may vary. However, the final intended performance obtained by that type of deterministic restraints may become infeasible in the presence of uncertainty in u, as presented by Cheng et al. (2017). To address the viability of constraints under uncertainty, Venanzi et al. (2015) proposed an overall probabilistic possibility formulation for the jth constraint \(g_{j}\):

$$P \, [g_{j} \left( {\mathbf{u}} \right) \le 0] \ge P_{oj} ,\quad j = 1,.........,J$$
(8a)

where Poj is the intended probability for satisfying the jth constraint. In order to decrease the involvement of probabilistic feasibility computation, assuming that gj(u) is normally distributed, the probabilistic feasibility of the restraint can be presented as below (Lee & Park, 2001):

$$\mu_{{g_{j} }} + k_{j} \sigma_{{g_{j} }} \le 0.$$
(8b)

where \(\mu_{{g_{j} }}\) and \(\sigma_{{g_{j} }}\) are the mean and the standard deviation of \(g_{j}\), respectively, computed by the first-order perturbation approach (see Eq. (5)). Further, the designer-specified penalty factor, kj, is used to improve the feasibility of the jth restraint, and it can be found from \({\text{k}}_{j} \, = \,{\Phi }^{{ - 1}} {(}P_{oj} {)}\), where \({\Phi }^{{ - 1}} {(}{\text{.)}}\) is the representation of an inversion of the cumulative density function for the standard normal distribution. In other words, kj signifies the target reliability index. However, for Uncertain but Bounded parameters, the dispersion of constraint can be found in the worst-case uncertainty propagation approach, as represented in Eq. (6). Hence, considering the mixed system of random and unconstrained but bounded parameters, the equivalent mean \(\mu_{{g_{j} }}\) and equivalent standard deviation \(\sigma_{{g_{j} }}\) are measured as:

$$\mu_{g} = g_{{_{j} }} \left( {\mu_{{{\mathbf{x}}_{R} }} ,\mu_{{{\mathbf{z}}_{B} }} ,{\overline{\mathbf{x}}}_{R} ,{\overline{\mathbf{z}}}_{B} } \right){;}$$
(9a)
$$\sigma^{2}_{{g_{j} }} { = }\left\{ {\sum\limits_{i = 1}^{N} {\left( {\left. {\frac{{\partial g_{j} }}{{\partial {\mathbf{x}}_{R} }}} \right|_{{\overline{u}_{i} }} } \right)}^{2} \sigma_{{{\mathbf{x}}_{{R_{i} }} }}^{2} { + }\sum\limits_{i = 1}^{N} {\left( {\left. {\frac{{\partial g_{j} }}{{\partial {\mathbf{z}}_{R} }}} \right|_{{\overline{u}_{i} }} } \right)}^{2} \sigma_{{{\mathbf{z}}_{{R_{i} }} }}^{2} } \right\}{ + }\left\{ {\sum\limits_{i = 1}^{N} {\left| {\frac{{\partial g_{j} }}{{\partial {\mathbf{x}}_{{B_{i} }} }}} \right|_{{\overline{u}_{i} }} \Delta {\mathbf{x}}_{{B_{{_{i} }} }} } { + }\sum\limits_{i = 1}^{N} {\left| {\frac{{\partial g_{j} }}{{\partial {\mathbf{z}}_{{B_{{_{i} }} }} }}} \right|_{{\overline{u}_{i} }} \Delta {\mathbf{z}}_{{B_{{_{i} }} }} } \, } \right\}{,}$$
(9b)

In Eq. (9a, b), xR, xB, zR and zB denote the random design variables, the Uncertain but Bounded design variable, the random design parameter, and the Uncertain but Bounded design parameter, respectively. A close observation of Eq. (9a) can reveal that the equivalent mean of the constraint function is determined by the mean values of probabilistic parameters and the nominal values of Uncertain but Bounded parameters. Likewise, the first part of Eq. (9b) reveals a treatment to probabilistic parameters, and the second part of Eq. (9b) depicts the consideration of Uncertain but Bounded parameters both for the design variables and the design parameters. In the first part, the first-order perturbation approach for probabilistic parameters Eq. (5) was used, while the second part used the worst-case propagation principle for Uncertain but Bounded parameters Eq. (6).

2.3.2.3 The robust optimisation formulation

By combining Eqs. (4) and (8a, b) to meet the requirements of the performance and the constraint feasibility under uncertainty in u, the Robust Optimisation problem can be expressed as:

$$\begin{gathered} {\text{minimize:}}\quad \phi ({\mathbf{u}}) = (1 - \alpha )\frac{{\mu_{f} }}{{\mu_{f}^{*} }} + \alpha \frac{{\sigma_{f} }}{{\sigma_{f}^{*} }}{,}\quad {0} \le \alpha \le 1 \hfill \\ {\text{subjected to:}}\quad \mu_{{g_{j} }} + k_{j} \sigma_{{g_{j} }} \le 0\quad j = 1,2,......,J \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x_{i}^{L} \le x_{i} \le x_{i}^{U} {,}\quad i = 1,2......,K. \hfill \\ \end{gathered}$$
(10)

It shall be mentioned that the individual gradient of the performance function and the restraints need to be evaluated at each updated design point during the optimization process.

2.4 Performance of the machine learning (ML) techniques by statistical analysis

For any predictive relationship, Root Mean Square Error (RMSE) depicts how far the predicted values lie from the actual ones. RMSE can be mathematically computed as follows:

$$\mathrm{RMSE }=\sqrt{\frac{{\sum }_{i=1}^{n}{\left({A}_{i}-{P}_{i}\right)}^{2}}{n}}$$
(11)

where Ai = actual value of ith term, Pi = corresponding predicted value, and n = number of data points.

Mean absolute error (\(MAE\)) can be denoted as:

$${\text{MAE}}=\frac{\sum_{i}^{n}\left|{A}_{i}-{P}_{i}\right|}{n}$$
(12)

Coefficient of determination (\({R}^{2})\) can be presented as:

$${R}^{2}=1-\frac{{\text{RSS}}}{{\text{TSS}}}$$
(13)

where \(RSS\) represents the sum of squared residuals, i.e., \(\sum_{i=1}^{n}{({A}_{i}-{P}_{i})}^{2}\), and \(TSS\) represents the total sum of squares, i.e., \(\sum_{i=1}^{n}{({A}_{i}-\overline{{A }_{i}})}^{2}\).

In addition, Chi-Square (\({\chi }^{2})\) test, a statistical test, can help determine the level of confidence of the proposed relationships. It can be computed as:

$${\chi }^{2}=\sum_{i=1}^{n}\frac{{({A}_{i}-{P}_{i})}^{2}}{{P}_{i}}$$
(14)

where χ2 indicates the Chi-Square value. After computing Chi-Square value, the degree of freedom was determined, which denotes the number of categories reduced by the number of parameters of the fitted distribution. The χ2 of the specific confidence level was compared with the critical value from χ2 distribution of specific degrees of freedom. Finally, the null hypothesis (H0), which signifies that there is no difference between the expected and observed values, can be accepted if χ2 value does not go beyond the critical value of a certain confidence level. However, the null hypothesis was rejected; and an alternative hypothesis (H1), which indicates the difference between the observed and expected values, can be accepted if χ2 value is bigger than the critical value of a certain confidence level.

3 Results and discussion

In this study, two hundred numbers of data about concrete mixtures were explored by changing the levels of key ingredients—size of coarse aggregate, cement, water, fine aggregate, and coarse aggregate, along with their compressive strength and slump value. 170 numbers of data randomly chosen from them were used for training purpose, while the rest 30 for testing. In addition, experiments were conducted to measure 15 numbers of different mix proportions, and their strength and slump values were obtained.

The rationale behind dividing a data set into 85% for training and 15% for testing is a common practice in the field of machine learning, particularly in supervised learning techniques (Madhiarasan & Louzazni, 2022). This division is often known as the “train-test split”, which is mainly aimed to strike a balance between having enough data to train a model efficiently and retaining adequate data to determine the model's performance accurately. Specifically, the rationale behind this proposal is:

Sufficient Training Data: If a model is trained more, it can learn better patterns. Thus, by assigning 85% of a data set for training, the model would have admittance to a sufficient amount of information to learn from, so it can perform better.

Generalization: The primary motive of machine learning techniques is to create a model that can be generalized well to new, unseen data. If a large amount of data are used for testing (e.g., 50% or more), the model may not have enough data to learn from during training, which will result in poor generalization.

Adequate Testing: Assigning 15% of a data set for testing permits for a substantial amount of data to determine a model's performance accurately. With too few test data (e.g., 5% or less), the evaluation result might not well illustrate a model's ability to perform on new data.

However, regarding the specific splitting percentages of 85% for training and 15% for testing, there is in fact no strict rule to dictate the exact percentages. The splitting may vary, depending on such factors as dataset's size, question's complexity, and the like. In practice, the division is often in the range of 70%–90% for training and 10%–30% for testing. A common practise is the 80–20 split; however, 85–15 is also feasible. It’s important to note that in addition to the train-test splitting, there is a third type of data in division, known as the validation set, which is used during the model training process to fine-tune hyper parameters and restrict the occurrence of over-fitting. It shall be mentioned that the fifteen sets of validation data is generated in laboratory separately in the laboratory. Ultimately, the choice of data splitting should be made carefully on the basis of the nature of a certain data set, the amount of data and the objectives of machine learning techniques. Joseph and Roshan (2022) have also opined the same logic for splitting data sets. In addition, Nguyen et al. (2021b) has demonstrated that data slitting would not affect the prediction results too much.

3.1 Support vector machine (SVM)

In this study, the coarse Gaussian Kernel function was used for SVM. Prediction Speed was 5200 obs/s and Training time was 1.1814s. The whole process was carried out in Matlab Toolbox.

3.2 Artificial neural network (ANN)

In this study, two ANN networks were used for predicting the compressive strength and slump. Each network consisted of five input parameters, namely coarse aggregate size, cement, sand, coarse aggregates, and water. The output parameters for the three networks are 28 days’ compressive strength and slump. The MATLAB ANN toolbox (Demuth & Beale, 2002) was used to perform experiments. To search for the most appropriate network for a given set of training data, trial methods and errors were used. In this study, the most efficient network was selected with a correlation coefficient (R), whose value varies from − 1 to + 1. R close to 1 or − 1 means a strong positive or negative relationship, respectively; while R close to 0 means no relationship. In this study, Mean Square Error (MSE) was used as the performance function. A trial and error procedure was adopted to select the best two networks. Tables 2 and 3 show the trial and error procedure adopted to select the finest network for estimating the compressing strength and slump, respectively. The trial and error procedure was conducted manually. The finest network for predicting the compressive strength and slump are presented in Fig. 5(a) and (b), respectively, in the pictorial form. It can be obviously noted that the ANN models can simultaneously predict the strength and slump, thus eliminating the need to develop separate models. However, it yielded a poor R value, which is not desirable.

Table 2 Trial used to find finest network in ANN for predicting compressive strength
Full size table
Table 3 Trial used to find finest network in ANN for predicting slump
Full size table
Fig. 5
figure 5

a Network used in ANN for predicting compressive strength. b Network used in ANN for predicting slump

Full size image

3.3 Fuzzy inference system (FIS)

Successful predictions by FIS models in the existing research indicate that fuzzy logic could be a useful modelling tool for engineers and researchers in the area of cement and concrete. This study also tried to predict the strength and slump of concrete by using FIS toolboxes. Specifically, coarse aggregate size, amount of cement, water, fine aggregate, and coarse aggregate were used as the input variables, while the strength and slump were used as the output variables in the two networks of FIS.

15 Membership functions for input and output parameters were used for fuzzy modelling, and the type of membership functions was chosen as "trimf" (triangular membership function) (Fig. 6).

Fig. 6
figure 6

Steps of FIS network used for predicting compressive strength and slump

Full size image

3.4 Genetic expression programing (GEP)

To assess whether the compressive strength, slump and dry density were a function of coarse aggregate size, cement, sand, and coarse aggregates, GeneXpro Tools 4.0 (www.genexprotools.soft112.com/) were adopted in this study. In all cases, the 30 number chromosomes, each containing the three-dimensional genes of the head size (i.e., eight), were used in the calculation. “Addition” was used as a link function to predict the compressive strength and slump. For the maximum correlation coefficient obtained to measure the strength, the slump was 96,565 and 122,349 iterations, respectively. In both cases, the number of genes was 3. The trees of the genetic algorithm producing a compressive strength and slump are presented in Fig. 7a and b. The compressive strength and slump equations are presented in Eqs. 11 and 12, respectively, where d(1), d(2), d(3), d(4) and d(5) indicate the size of coarse aggregate, amount of cement, water, fine aggregate and coarse aggregate, respectively. These two equations are very useful in predicting the above-mentioned features of concrete. In addition, they can be used as a basis for the optimisation process.

$$\begin{aligned} y\left( 1 \right) = & \left[ {\left\{ {\left( {d\left( 2 \right) - d\left( 4 \right)} \right) + d\left( 2 \right)} \right\}*\left\{ { \left( { - 46.3750570834068} \right) - d\left( 1 \right)} \right\}*\left\{ {\frac{3.87863402806583}{{(\left( {d\left( 2 \right)* 10.8301795938042} \right)}}} \right\}} \right] \\ & + \left[ {\frac{{\left\{ {\left( {d\left( 2 \right) - d\left( 1 \right)} \right) - d\left( 3 \right)} \right\}}}{{\left\{ {\frac{d\left( 4 \right)}{{d\left( 3 \right)}}} \right\}}}} \right] - \left[ {\frac{{\left( {d\left( 1 \right) - \left( { - 16.1558986905887} \right)} \right)}}{{\left\{ {\frac{d\left( 5 \right)}{{d\left( 3 \right)}}} \right\}}}} \right] \\ & + \left[ {\left\{ {\frac{d\left( 2 \right)}{{\left( {\left( {d\left( 3 \right) - d\left( 2 \right)} \right)*\left( { - 7.89030981708951} \right)} \right) - \left( {d\left( 2 \right) - \left( { - 31.3645369854215} \right)} \right)}}} \right\}*d\left( 1 \right)} \right] \\ \end{aligned}$$
(15)
$$\begin{aligned} y{ }\left( 2 \right) = & { }\left[ {\left\{ {\frac{{\left( {\left( {0.450301792461842{*}d\left( 5 \right)} \right){* }7.89544358653523} \right)}}{7.89544358653523}} \right\}{ *}\left\{ {\frac{{\left( {\frac{{\left( { - 11.0172670250786} \right)}}{2.18198406609944}{ }} \right)}}{d\left( 2 \right)}} \right\}} \right] \\ + & { }\left[ {\frac{{\left\{ {\left( {\left( {d\left( 4 \right) + d\left( 4 \right)} \right) + d\left( 5 \right)} \right) - \frac{d\left( 2 \right)}{{\left( { - 2.63698393180232} \right)}}} \right\}}}{{\left\{ {\left( {{ }\left( { - 21.2197122639063} \right) - d\left( 2 \right)} \right) - \left( {d\left( 3 \right){* }\left( { - 2.63698393180232} \right)} \right))} \right\}}}} \right] \\ + & { }\left[ {\left\{ {\frac{{\left( {d\left( 5 \right) + d\left( 5 \right)} \right)}}{d\left( 1 \right)}} \right\} + { }\left( { - 10.3960394115875} \right) + \left\{ {\frac{{\left( { - 10.3960394115875} \right)}}{d\left( 1 \right)}{ } + { }\left( { - 10.3960394115875} \right)} \right\}} \right] \\ + & { }\left[ {\frac{{\left[ {\left\{ {\frac{d\left( 1 \right)}{{d\left( 5 \right)}}{*}d\left( 4 \right)} \right\} + \left\{ {\left( { - 6.20188909573656} \right) + d\left( 1 \right)} \right\}} \right]}}{{\left\{ {\frac{d\left( 5 \right)}{{\left( {d\left( 4 \right) + d\left( 4 \right)} \right)}}} \right\}}}} \right] \\ \end{aligned}$$
(16)
Fig. 7
figure 7

a Generated GEP tree for prediction of compressive strength. b Generated GEP tree for prediction of slump

Full size image

The partial dependence plot is presented in Fig. 8. It can be observed that with the increase of cement, the compressive strength and slump would increase rapidly. However, with increased water, the compressive strength would decrease and the workability would increase. Furthermore, with increased sand, both the strength and workability would decrease; with increased coarse aggregate, the strength and workability would increase (Figs. 9, 10).

Fig. 8
figure 8

The partial dependence plot of strength and slump from GEP

Full size image
Fig. 9
figure 9

Actual verses predicted compressive strength curve for testing

Full size image
Fig. 10
figure 10

Actual verses predicted slump curve for testing

Full size image

3.5 Comparative analysis of the machine learning techniques used

The actual versus predicted curves for training and validation are presented in Figs. 11 and 12, respectively. Figure 13a depicts the RMSE values of training for the compressive strength and slump. Figure 13b depicts the RMSE for testing the compressive strength and slump. Figure 13c illustrates the MAE for the compressive strength and slump in the case of training data set. Figure 13d shows the MAE for the same validation dataset. Figure 13e and f demonstrate the R2 value obtained for training and testing, respectively. Furthermore, Chi square value are plotted in Fig. 13g and h for testing and validation, respectively.

Fig.11
figure 11

Actual verses predicted compressive strength curve for validation

Full size image
Fig. 12
figure 12

Actual verses predicted slump curve for validation

Full size image
Fig. 13
figure 13

The statistical analysis result for testing and validation of strength and slump

Full size image

It shall be noticed that the degrees of freedom for the 30 numbers of the training data set and the 15 numbers of the validation data set are 29 and 14, respectively. Thus, at the 95% level of confidence, the Chi square values are 42.557 and 23.685 for testing and validation, respectively. Figure 13(g) demonstrates that for the testing value, the compressive strength obtained by all five methods reached the 95% confidence level. However, the slump value fails to achieve the same level in all other five methods. Nevertheless, Fig. 13(h) corroborates that both strength and slump values successfully achieved the 95% confidence level for the validation data set.

In summary, all the results above clearly signify that ANFIS yielded the best results among all the five methods. Obviously, ANFIS is a type of adaptive network, which includes both ANN and FIS. As a supervised learning algorithm, ANN uses a historical dataset for forecasting the future values. On the other hand, in FIS, the control signal is produced from firing the rule base. Thus, ANFIS can be used most successfully in predicting the concrete compressive strength and slump with the available data. However, GEP generates equations that are very much helpful for others to predict concrete's properties; and the generated equations can be used as a basis equation for the optimisation process.

3.6 Results after optimisation

In order to obtain an objective function in this present process for performing optimisation, which estimates compressive strength as a function of aggregate size, cement, water, fine aggregate and coarse aggregate, generated GEP Eqs. 11, 12 has been used which is described in previous section. Another objective function, which predicts the cost of mixture, was constructed based on the cost of materials taken from the US market, as seen in Eq. 13. The cost of cement, sand coarse aggregate and water per kg are 0.124, 0.006, 0.0075 and 0.000013 US dollars, respectively.

$$y(3) = 0.124*d(2)+0.006*d(3)+0.0075*d(4)+0.000013*d(5)$$
(17)

In order to maximize the strength of concrete and slump and to minimize the cost [i.e., to minimize (1/strengtℎ), (1/slump) and (cost)], Eqs. 1, 2, 3 were minimized in this study for the given constraints.

Available range constraints:

  1. 1)

    \({CA}_{max}\) < Size of CA < \({CA}_{min}\)

  2. 2)

    Cemax < Ce < Cemin

  3. 3)

    FAmax < FA < FAmin

  4. 4)

    CAmax < CA < CAmin

  5. 5)

    Wamax < Wa < Wamin

These values of the lower limit [20 350 130 505 1015] and upper limit [40 410 195 865 1365] were taken based on the minimum and maximum values of the size of CA,weight of cement, water, fine aggregate and coarse aggregate used.

  1. 2)

    Ratio constraint

It has been found that, generally, the water/cement ratio shall be 0.35 at the minimum (El-Gazery & Ali, 2019; Emadi & Modarres, 2022; Kang & Yan, 2011; Wong et al., 2020). Below this ratio, hydration of cement would not take place, and honey comb structure would be created. Therefore, the following constraint was introduced: \(Wa/Ce>0.37\).

  1. 3)

    Absolute weight constraint

All the weights were converted with respect to the total volume of mixtures as 1 m3. The whole program was carried out in MATLAB 2014. The optimisation result is presented in Table 4, which is very helpful for obtaining the maximum strength and slump at the minimum cost.

Table 4 Value obtained by multi objective optimisation maximization of strength, slump and minimization cost
Full size table

In the case of Robust Optimisation, the above mentioned methods were followed. The design parameter, basis function, upper bound, and lower bound were used as deterministic optimisation. Furthermore, uncertainty of obtaining the strength and slump with respect to aggregate size, cement, sand, and coarse aggregate were also considered (Moulick et al., 2019a). The mixture design was evaluated at the maximum strength and maximum slump with the lowest cost, as presented in Table 5. This methodology is very much conducive to obtaining the optimum mix proportion for finding the maximum strength and workability at the lowest cost, while considering all the uncertainties. Moreover, experiments were conducted with the same mix proportion considered. However, due to the lack of time, 7 days’ compressive strength was achieved as 29.5 MPa, which should be multiplied by 1.5 to obtain 28 days’ strength—the result is 44.25 MPa (Table 5).

Table 5 Obtained Robust Optimisation result
Full size table

4 Conclusion

The construction industry relies heavily on concrete; thus, it's crucial to precisely forecast and optimize the strength and workability of concrete mixtures, while diminishing the cost. In this study, concrete's compressive strength and workability in terms of slump were predicted by adopting support vector machine (SVM), artificial neural network (ANN), fuzzy inference system (FIS), adaptive fuzzy inference system (ANFIS) and genetic expression programing (GEP), while considering two hundred pieces of data about concrete's mix designs, covering coarse aggregate size, cement, water, fine aggregate and coarse aggregate. Among these data, 85% were used for training purpose, and 15% for testing purpose in all the five machine learning methods. In addition, in order to validate the five methodologies, some experimental investigations were conducted on fifteen concrete mixes. This approach increases the reliability and generalizability of results. The R and MSE values were computed in all the five methods both for training and validation. The results clearly signify that ANFIS can yield the best results among all the five methods in terms of both training and validation. Thus, it can be used successfully for predicting concrete's compressive strength and slump based on the available data. However, GEP generated the equations very much helpful for others in predicting concrete's strength and slump. It is evitable that, one of the main obstacles in the construction industry is that concrete is very much sensitive to even a small variation in the constitutive materials' mix proportions and other external factors. Therefore, robust optimisation was carried out by considering the uncertainty of strength, slump and cost, while using the basis equations obtained from GEP to determine the optimum proportions of concrete mixtures, so as to get the maximum concrete strength and workability at the lowest cost. This study contributes much to the concrete industry, on-site engineers, etc. in predicting the strength and workability for particular coarse aggregate size, cement, water, fine aggregate, and coarse aggregate. It can also help them obtain the maximum strength and workability at the minimum cost. However, certain practical factors, such as the availability and cost of materials, or specific requirements of individual projects, may limit the applicability of the optimized mix designs.

It is demonstrated in this study that the deterministic optimization of concrete mixes' design using Genetic Algorithm (GA) can be realized with some basic functions generated by Genetic Expression Programing (GEP), so as to obtain the optimum concrete mix proportion and determine the maximum compressive strength and slump at the lowest cost. This study also revealed that robust optimization of concrete design mix by considering all the uncertainty using Genetic Algorithm (GA) depending on the basis functions generated by Genetic Expression Programing (GEP), can be used to predict the maximum compressive strength and slump at the lowest cost with the least uncertainty.

This study was carried out by using only 200 data sets, which may not be sufficient to capture the full range of variability in designing concrete mixes, particularly given the number of variables that were examined. Furthermore, concrete mixes' proportion types and the quantity of admixtures were not used, and the gradation of coarse and fine aggregate was not considered. Besides, fibres and fibre reinforcement polymers in concrete were not used in this study (Sarir et al., 2016). Moreover, each machine learning method has its limitation, which can impact the performance of this study’s model. Despite some deficiencies, this study is useful for predicting and optimising desired properties of concrete.