Optimization of Fly Ash—Slag One-Part Geopolymers with Improved Properties

One-part geopolymer concrete/mortar is a pre-mixed material made from industrial by-products and solid alkaline activators that only requires the addition of water for activation. Apart from being environmentally friendly, it also reduces complexity and improves consistency in the mixing process, leading to more efficient production and consistent material properties. However, developing one-part geopolymer concrete with desirable compressive strength is challenging because of the complexity of the chemical reaction involved, the variability of the raw materials used, and the need for precise control of curing conditions. Therefore, 80 different one-part geopolymer mixtures were compiled from the open literature in this study, and the effects of the constituent materials, the dosage of alkaline activators, curing condition, and water/binder ratio on the 28-day compressive strength of one-part geopolymer paste were examined in detail. An ANN model with the Levenberg–Marquardt algorithm was developed to estimate one-part geopolymer’s compressive strength and its sensitivity to binder constituents and alkaline dosage. The ANN model’s weights and biases were also used to develop a CPLEX-based optimization method for achieving maximum compressive strength. The results confirm that the compressive strength of one-part geopolymer pastes increased by increasing the Na2O content of the alkaline source and the slag dosage; however, increasing the Na2O content in alkaline sources beyond 6% by fly ash weight led to decreasing the compressive strength; therefore, the optimum alkaline activator dosage by weight of fly ash was to be 12% (i.e., 6% Na2O). The proposed ANN model developed in this study can aid in the production and performance tuning of sustainable one-part geopolymer concrete and mortar for broader full-scale applications.


Introduction
Concrete is the world's second most consumed commodity after water and the most used construction material globally. This results in a colossal environmental footprint with considerable carbon emissions and depletion of natural resources. Around 8% of all CO 2 emissions worldwide are related to concrete, and most of those emissions come from the manufacture of cement [1]. According to some estimates, 4.2 billion tons of cement are produced annually worldwide, causing about 4 billion tons of CO 2 emission into the atmosphere [2]. The manufacture of one ton of ordinary Portland cement (OPC) emits about 0.8 to 1 ton of CO 2 . Thus, there is growing pressure on the concrete industry to develop different binders to reduce the need for OPC. To produce environmentally friendly concrete, it is necessary to develop viable alternatives to OPC that emit little or no CO 2 [3,4].
One of the potential options to lessen the environmental impact of OPC binders is the development of low-carbon binders [5,6]. Aluminosilicate materials react with water slowly, but when exposed to hydrolysis and condensation reactions in an alkaline solution, maintenance applications. Using the proposed informational model, users would have an insight into the influential parameters on 28-day compressive strength, and also, with the aid of the CPLEX-based optimization method, they can optimize the input parameters to achieve the maximum compressive strength. The objective is to enhance the prediction accuracy of the user-friendly one-part geopolymer, which would make it more convenient for application in diverse building projects, particularly in harsher settings like coastal areas.

Literature Review
The conventional method of using an alkaline activator solution in geopolymer production involves the use of highly caustic sodium-or potassium-based hydroxide, silicates, carbonates, or their combinations. This makes it dangerous to handle, store, and transport, requiring additional safety precautions that can slow down production and increase costs. To overcome these challenges, researchers have explored the use of solid activators to produce a user-friendly one-part geopolymer that only requires the addition of water. Some of the notable materials used as alkali sources include sodium hydroxide combined with various silica sources such as fly ash, rice husk ash, micro silica, calcium hydroxide, different grades of sodium metasilicate, and red mud. In recent years, many studies have focused on investigating the mechanical properties and durability of one-part geopolymer mortars and concrete.
In their study, Askarian and colleagues [8] created one-part hybrid concrete mixes using a combination of ordinary Portland cement (OPC) and geopolymers. They added solid potassium carbonate, which made up 7.5% of the total geopolymeric raw materials, as the primary activator. The researchers blended fly ash and ground granulated blastfurnace slag with the geopolymeric raw materials in various proportions and found that the addition of OPC decreased workability and setting time. However, it notably enhanced early age and ultimate compressive strength due to the rapid reaction of OPC with alkali activators.
Muthukrishnan et al. [14] conducted a study on the rheochemical approach to analyze the early strength development resulting from alkali reactions and formulate a suitable 3D printable one-part geopolymer concrete. The researchers evaluated the impact of different design parameters, such as activator content, thixotropic additive (Magnesium Alumino Silicate-MAS), and retarder (sucrose) dosage, on the rheological properties of the concrete. The findings indicated that the one-part geopolymer formulation exhibited improved printing characteristics when the binder contained 0.75 wt% MAS, 10 wt% activator, and 1.5 wt% sucrose.
Muhammad Riaz Ahmad et al. developed a new type of energy-efficient and sustainable concrete based on industrial waste materials and vegetal aggregate for hygrothermal and low load-bearing applications. They conclude that the vegetal concrete mixtures containing red mud exhibited higher capillary and water absorption as compared to other mixtures. Moreover, all concrete mixtures were classified as good to excellent moisture buffer materials.
Dongthe et al. [15] studied the solid activator, the synthetic sodium metasilicate pentahydrate against water, and a hybrid sodium silicate and sodium hydroxide activator solution to develop a high-strength one-part geopolymer mortar. They conclude that the solid activator using sodium silicate pentahydrate outperformed the often-used liquid activator in terms of the compressive strength of the mortar. Nevertheless, the compressive strength decreased, and efflorescence increased significantly once the metasilicate content exceeded Na 2 O% = 6%.
Wangthe et al. [16] investigated the early-age properties of one-part fly ash/ground granulated blast-furnace slag (FA/GGBS) geopolymer through the utilization of hybrid activators, such as anhydrous sodium metasilicate (Na 2 SiO 3 ), sodium carbonate (Na 2 CO 3 ), and sodium aluminate (NaAlO 2 ). They indicated that Na 2 SiO 3 -activated one-part geopolymer released high reaction heat and achieved a faster setting. Such shortcomings could be improved by partially replacing Na 2 SiO 3 with Na 2 CO 3 in a solid form. Besides, incorporat-ing slight NaAlO 2 decreased the self-flow of geopolymer paste, whereas the slump-flow properties remained unchanged.
The previous research mainly acknowledged that to meet the dual requirements of convenience and eco-friendliness in practical engineering, the synthesis of one-part fly FA/GGBS geopolymer binders' selection and content of different hybrid combinations of solid activators, including anhydrous Na 2 SiO 3 , Na 2 CO 3 , and NaAlO 2 , etc. is essential. Flowability, setting time, strength, and effect of one-part geopolymer paste molar ratios at different activator dosages were examined to determine the fundamental properties of one-part geopolymer paste cured in the ambient.

Database Description
The most comprehensive 28-day compressive strength data for various one-part geopolymer pastes was collected from accessible, pertinent data in the open literature [17][18][19][20][21][22][23][24][25][26][27][28]. The database comprises 80 FA/GGBS binder-based one-part geopolymers made with an anhydrous sodium silicate alkaline source, see Table 1. The source publications investigated the effects of different parameters on the compressive strength of one-part geopolymers, including the percentage of fly ash and GGBS and the Na 2 O dosage of the alkaline source. Although the variation of water/binder ratio and curing temperature was not considerable, their effects on compressive strength were examined. The range of input/output parameters of the studied dataset is shown in Table 2. To activate the binder components in fly ash/GGBS-based geopolymers, granular sodium metasilicate (Na 2 SiO 3 ) anhydrous (50% Na 2 O, 46% SiO 2 , and 4% H 2 O) is often employed as a solid activator. Since, unlike two-part geopolymers, there is no need to create a NaOH solution before mixing, using granular alkaline activators in one-part geopolymer systems is more straightforward and faster than using the commonly used and caustic alkaline solutions. The granular sodium metasilicate anhydrous was used at 3 to 25% by weight of the binding materials in the studied database.
In the prepared database, pure GGBS was used without further treatment as the main resource of calcium materials in geopolymer production. Low calcium fly ash was also used as another source of precursor materials. Fly ash and GGBS are used in geopolymer concrete because they can react with an alkaline activator to form a binder that can replace Portland cement. Geopolymer concrete made with these materials has several advantages over traditional concrete, including higher strength, lower permeability, and better resistance to chemical attack [29]. GGBS also contains silicates and alumina, which are necessary for the formation of geopolymer binders, and its high specific gravity can increase the density of the concrete and make it more resistant to erosion. Additionally, the use of fly ash and GGBS in geopolymer concrete is a sustainable and environmentally friendly approach to construction that reduces waste and produces a high-quality, durable material. The chemical components of the fly ash and GGBS (generally available in the market) were analyzed by studied references using X-Ray fluorescence (XRF), see Table 3. Table 3. Physical and chemical features of binder materials reported by [20].

Mixing Procedure and Test Methods
Preparing one-part geopolymer paste followed the ASTM C305-14 [30] recommended procedures. Table 1 lists the 80 various geopolymer pastes prepared with fly ash and GGBS precursor materials and various Na 2 O content of alkaline sources. Based on these experimental tests, the granular sodium metasilicate anhydrous and binding materials were mixed for about two minutes using a mechanical mixer. After adding potable tap water to the dry mixture, mixing resumed for an additional three minutes to achieve homogeneity and consistency. Figure 1 shows the block diagram for one-part geopolymer paste production. The studied literature also acknowledged that the compressive strength tests were performed following the guidelines of the ASTM C109-109M [31] standards. these experimental tests, the granular sodium metasilicate anhydrous and binding rials were mixed for about two minutes using a mechanical mixer. After adding p tap water to the dry mixture, mixing resumed for an additional three minutes to ac homogeneity and consistency. Figure 1 shows the block diagram for one-part geopo paste production. The studied literature also acknowledged that the compressive str tests were performed following the guidelines of the ASTM C109-109M [31] standa

Compressive Strength Results and Discussion
The 28-day compressive strength results of all studied mixture designs are pres in Table 1. The effects of the Na2O, fly ash, and GGBS contents on the compressive str of one-part geopolymer paste are depicted in Figure 2. The results indicate that incre the GGBS and Ca2O contents increased the compressive strength of the one-part geo mer paste. Lower Ca2O content led to insufficient alkali and negatively impacted th tem's geopolymerization process. The result indicates that raising the granular al activator content by weight of the fly ash beyond 12% slightly decreased the compr strength and workability of the one-part geopolymer cement pastes [9]. The opt

Compressive Strength Results and Discussion
The 28-day compressive strength results of all studied mixture designs are presented in Table 1. The effects of the Na 2 O, fly ash, and GGBS contents on the compressive strength of one-part geopolymer paste are depicted in Figure 2. The results indicate that increasing the GGBS and Ca 2 O contents increased the compressive strength of the one-part geopolymer paste. Lower Ca 2 O content led to insufficient alkali and negatively impacted the system's geopolymerization process. The result indicates that raising the granular alkaline activator content by weight of the fly ash beyond 12% slightly decreased the compressive strength and workability of the one-part geopolymer cement pastes [9]. The optimum alkaline activator dosage by weight of fly ash was found to be 12% (i.e., 6% Na 2 O) to attain the highest compressive strength at 28 days of curing.
However, the compressive strength of one-part geopolymer paste was negatively affected by the fly ash content. Prior studies have established that geopolymer concrete produced with class C fly ash, which has a high calcium concentration, exhibits higher compressive strength at ambient temperatures than geopolymer concrete made with class F fly ash [32]. Because geopolymer concrete produced with class F fly ash cannot achieve structural integrity at room temperature, it is normally heat cured to 60 • C. At high temperatures, geopolymer concrete produced with class F fly ash outperforms geopolymer concrete manufactured with class C fly ash in terms of mechanical strength. In addition to the calcium content, the particle size distribution, specific surface area, and chemical composition of fly ash can also affect the performance of geopolymer concrete. For instance, fly ash with a higher specific surface area can result in higher compressive strength due to the increased reactivity and better distribution of the geopolymer precursors. Moreover, the chemical composition of fly ash can vary depending on the source and type of coal used, which can influence the geopolymerization reaction and the resulting properties of the geopolymer concrete. Therefore, careful selection and characterization of fly ash is critical to ensure the desired performance and consistency of geopolymer concrete.
The low correlation coefficients in Figure 2 acknowledge that the investigated variables interact with each other in a complex way and therefore is difficult to capture this relationship with a simple regression equation. In such cases, alternative statistical methods may be more appropriate for modeling the relationship between the investigated variables and 28-day compressive strength. For example, non-linear regression models or machine learning algorithms, such as decision trees or neural networks, could potentially capture the non-linear relationship and underlining mechanism between these variables.

Artificial Neural Networks
Researchers are increasingly using artificial intelligence approaches such as genetic algorithms, adaptive regressions, fuzzy logic, and artificial neural networks (ANNs) instead of traditional or classical methods such as linear regression, time-series analysis, etc. [33][34][35]. While these classical methods have been widely used and are still valuable in certain contexts, artificial neural networks (ANNs) have shown to be more powerful in modeling complex non-linear relationships and are, therefore, increasingly being used in many fields, including image recognition, natural language processing, and financial modeling. ANNs are a type of machine learning method that draws inspiration from the structure and operation of the human brain. ANNs are composed of interconnected nodes or neurons that carry out information processing in parallel. ANNs are typically used when the relationship between the input and output is complex or when using another available approach requires considerable investment in time and money. In feed-forward networks, one of the most often used types of ANNs, neurons are arranged in layers containing an input layer, one or more hidden layers, and an output layer. Using network weights and biases, neurons in the hidden layer are linked to those in the preceding and following layers. Moreover, to reduce prediction errors, ANNs should be trained using an optimization approach in which the training function would modify the network weights matrix for each epoch.
The backpropagation (BP) learning algorithm has been successfully utilized with various numerical optimization approaches to accelerate network convergence. Combining the Levenberg-Marquardt, a common non-linear least squares optimization algorithm, into the BP algorithm was proven to be highly effective. The Levenberg-Marquardt has higher convergence, generalization, and precision than other algorithms, and fewer iterations (epochs) are needed to attain lower error rates [34]. Despite being a quick and effective optimization technique, the Levenberg-Marquardt approach has the limitation that it could be trapped in a local minimum [36].

Artificial Neural Networks
Researchers are increasingly using artificial intelligence approaches such as genetic algorithms, adaptive regressions, fuzzy logic, and artificial neural networks (ANNs) instead of traditional or classical methods such as linear regression, time-series analysis, etc. [33][34][35]. While these classical methods have been widely used and are still valuable in certain contexts, artificial neural networks (ANNs) have shown to be more powerful in modeling complex non-linear relationships and are, therefore, increasingly being used in many fields, including image recognition, natural language processing, and financial modeling. ANNs are a type of machine learning method that draws inspiration from the structure and operation of the human brain. ANNs are composed of interconnected nodes or neurons that carry out information processing in parallel. ANNs are typically used when the relationship between the input and output is complex or when using another available approach requires considerable investment in time and money. In feed-forward networks, one of the most often used types of ANNs, neurons are arranged in layers containing an input layer, one or more hidden layers, and an output layer. Using network weights and biases, neurons in the hidden layer are linked to those in the preceding and following layers. Moreover, to reduce prediction errors, ANNs should be trained using an optimization approach in which the training function would modify the network weights matrix for each epoch.
The backpropagation (BP) learning algorithm has been successfully utilized with various numerical optimization approaches to accelerate network convergence. Combining the Levenberg-Marquardt, a common non-linear least squares optimization algorithm, into the BP algorithm was proven to be highly effective. The Levenberg-Marquardt has higher convergence, generalization, and precision than other algorithms, and fewer iterations (epochs) are needed to attain lower error rates [34]. Despite being a quick and effective optimization technique, the Levenberg-Marquardt approach has the limitation that it could be trapped in a local minimum [36].

Generation of Training Model and Statistical Metrics
Correlation or dependency denotes any mathematical relationship, regardless of causation, between two random variables. Correlation measures the strength of the linear association between two variables. A correlation matrix is a tabular representation of the correlation coefficients between the input variables, displaying the relationship between each pair of variables in a table cell. A correlation matrix is useful for summarizing input data for further analysis. Figure 3 illustrates the correlation matrix of the input/output parameters utilized in the present study. Equation (1) was used to normalize each parameter in the range of 1 to −1 while considering each input parameter's domain and preventing any divergence in the results. X n is the normalized value of the parameter, where X max is its maximum value, and X min is its minimum value. X is the variable's original (nontransformed) value. Table 2 provides the maximum and minimum values for each input parameter. The results of the correlation matrix acknowledge that GGBS and the Na 2 O contents had significant effects on the 28-d compressive strength.
into the BP algorithm was proven to be highly effective. The Levenberg-Marquardt has higher convergence, generalization, and precision than other algorithms, and fewer iterations (epochs) are needed to attain lower error rates [34]. Despite being a quick and effective optimization technique, the Levenberg-Marquardt approach has the limitation that it could be trapped in a local minimum [36].

Generation of Training Model and Statistical Metrics
Correlation or dependency denotes any mathematical relationship, regardless of causation, between two random variables. Correlation measures the strength of the linear association between two variables. A correlation matrix is a tabular representation of the correlation coefficients between the input variables, displaying the relationship between each pair of variables in a table cell. A correlation matrix is useful for summarizing input data for further analysis. Figure 3 illustrates the correlation matrix of the input/output parameters utilized in the present study. Equation (1) was used to normalize each parameter in the range of 1 to −1 while considering each input parameter's domain and preventing any divergence in the results. Xn is the normalized value of the parameter, where Xmax is its maximum value, and Xmin is its minimum value. X is the variable's original (nontransformed) value. Table 2 provides the maximum and minimum values for each input parameter. The results of the correlation matrix acknowledge that GGBS and the Na2O contents had significant effects on the 28-d compressive strength.  In all the networks created in this study, the hyperbolic tangent function and the Levenberg-Marquardt training algorithm were employed. Following the Kolmogorov technique [37], if the system has a wide enough range of neurons, an ANN using the BP algorithm with one or two hidden layers can easily calculate any continuous function to any level of accuracy [38]. In this regard, the ith neuron in the network offers a total that collects the bias (b i ) as well as its weighted input (w ij ) to develop (n i ) as network input which is provided in Equation (2). In this equation, b i is the ith neuron bias; p j is the input vector; w ij denotes the strength of the connection from the jth input to the ith neuron.
In order to predict the 28-day compressive strength of one-part geopolymer pastes, an artificial neural network (ANN) was developed using the MATLAB ANN toolbox. The network consisted of five neurons in the input layer, one neuron in the output layer representing compressive strength, and two hidden layers containing various numbers of neurons. The feed-forward neural network was trained using the trainlm function to create an effective network. The transfer functions used were "tansig" or "hyperbolic tangent sigmoid" for the hidden layers and "purelin" or "Linear" for the output layer.
The optimal number of neurons for the hidden layers was determined through a trialand-error process by running the MATLAB code iteratively to obtain the best-performing network. The MATLAB code included a nested loop in determining the best number of neurons for both hidden layers. Initially, a range of 4 to 15 neurons for each loop was selected based on the preliminary code running. The input layer of the network consisted of five parameters, and each neuron in the hidden layers received a distribution of these input parameters multiplied by different weights. The initial magnitude of the bias and weights was presumed for the first iteration. The bias magnitude was related to the hidden layer neurons used as inputs to the specific neurons, and the outputs from each neuron were transferred through an activation function. The output layer neuron then received the magnitude multiplied by specific weights, and errors were determined by comparing the actual values with the model outputs. The bias and weights were updated based on the given learning algorithm for the next iteration, and the process continued until the model converged to the desired degree [39].
In this research, the data were divided randomly into three sets: training (70%), testing (15%), and validation (15%). This specific partition was chosen after testing several other options to ensure the network could be trained effectively without overfitting. During the training process, the aim is to minimize the error function by finding an optimal set of weights and biases that produces outputs similar to the desired values.
After numerous iterations, the best-performing ANN model was identified as the one using the trainlm function in MATLAB with 5 and 7 neurons in the hidden layers. The performance of the developed models was evaluated using statistical indices, including RMSE, MSE, MAE, and correlation coefficient R, with the results shown in Figure 4. From the figure, it is evident that the network with 5 and 7 neurons in the hidden layers yielded the best RMSE values, which were 0.049.
After numerous iterations, the best-performing ANN model was identified as using the trainlm function in MATLAB with 5 and 7 neurons in the hidden lay performance of the developed models was evaluated using statistical indices, in RMSE, MSE, MAE, and correlation coefficient R, with the results shown in Figure  the figure, it is evident that the network with 5 and 7 neurons in the hidden layers the best RMSE values, which were 0.049. After developing the ANN model, each input variable was assigned a specifi cient, which was obtained using separate codes from the MATLAB script. Table 4 d the bias and weights of the optimal model, which was trained using the trainlm alg In an ANN trained with this algorithm, the bias and weights of the optimal mo essential parameters that influence the network's performance. The trainlm algori type of backpropagation algorithm that is commonly used for supervised learnin such as classification and regression. During training, the algorithm adjusts the and biases of the network to minimize the difference between the predicted outp the actual outputs for a given set of input data. The optimal values of the weig biases are those that result in the lowest error or loss on the training data. Once work is trained, the bias and weights of the optimal model are used to make pre on new, unseen data. The bias represents the threshold for activation of the neuron network, and the weights determine the strength of the connections between the n The proposed topology of the feed-forward network with two hidden layers put variables (neurons), and one output parameter is depicted in Figure 5. The (I sents the input vector (from I1 until In), and (O) represents the output vector. T represent the connections. After developing the ANN model, each input variable was assigned a specific coefficient, which was obtained using separate codes from the MATLAB script. Table 4 displays the bias and weights of the optimal model, which was trained using the trainlm algorithm. In an ANN trained with this algorithm, the bias and weights of the optimal model are essential parameters that influence the network's performance. The trainlm algorithm is a type of backpropagation algorithm that is commonly used for supervised learning tasks such as classification and regression. During training, the algorithm adjusts the weights and biases of the network to minimize the difference between the predicted outputs and the actual outputs for a given set of input data. The optimal values of the weights and biases are those that result in the lowest error or loss on the training data. Once the network is trained, the bias and weights of the optimal model are used to make predictions on new, unseen data. The bias represents the threshold for activation of the neurons in the network, and the weights determine the strength of the connections between the neurons. The proposed topology of the feed-forward network with two hidden layers, five input variables (neurons), and one output parameter is depicted in Figure 5. The (I) represents the input vector (from I1 until In), and (O) represents the output vector. The lines represent the connections. Overall, using multiple key phases, as shown in Figure 6, the ANN was used to forecast the 28-day compressive strength of developed geopolymer mixes. In the beginning, the data were split into two groups with a 7:3 ratio, with 70% of the data used to create the training dataset and the remaining 30% used to create the testing & validation dataset. Second, the ANN model was constructed using a training dataset (two hidden layers). Third, in order to validate and assess the effectiveness of the suggested ANN model, the projected outcomes were contrasted with the experimental data using several metrics, including mean absolute error (MAE), root mean square error (RMSE), and coefficient of determination (R 2 ). Overall, using multiple key phases, as shown in Figure 6, the ANN was used to forecast the 28-day compressive strength of developed geopolymer mixes. In the beginning, the data were split into two groups with a 7:3 ratio, with 70% of the data used to create the training dataset and the remaining 30% used to create the testing & validation dataset. Second, the ANN model was constructed using a training dataset (two hidden layers). Third, in order to validate and assess the effectiveness of the suggested ANN model, the projected outcomes were contrasted with the experimental data using several metrics, including mean absolute error (MAE), root mean square error (RMSE), and coefficient of determination (R 2 ).

Multiple Linear Regression Model (MLR)
Determining the connection between two or more variables is a common tas gineering. Regression analysis is one of the effective statistical techniques that has ently piqued scientists' interest in this field. Regression modeling is generally tho as the process of fitting models to data. Linear predictor functions are generated in regression model to determine the relationship between the input/output param is important to note that several input variables are often used in regression anal plications, creating "multiple linear regression" or MLR functions. In this instanc analyzes the observed data and fits it into a linear equation to determine the cor between two or more input variables. Multiple linear regression involves summ data and investigating the relationship between variables [40,41]. The general form multiple regression models is given in Equation 7 below, where Y is a dependent v β0 is a constant, and βj is a regression coefficient (i = 1, 2,…, n).

= + ∑
Simple regression analyses use only one independent variable (Xj), while m regression analyses use two or more variables [42]. It is worth noting that if a da lies on the fitted line entirely, the vertical deviation would be equal to zero. This multiple linear regression model demonstrates the correlation between the one-p polymer paste characteristics and the experimentally measured 28-day comp

Multiple Linear Regression Model (MLR)
Determining the connection between two or more variables is a common task in engineering. Regression analysis is one of the effective statistical techniques that has consistently piqued scientists' interest in this field. Regression modeling is generally thought of as the process of fitting models to data. Linear predictor functions are generated in a linear regression model to determine the relationship between the input/output parameters. It is important to note that several input variables are often used in regression analysis applications, creating "multiple linear regression" or MLR functions. In this instance, MLR analyzes the observed data and fits it into a linear equation to determine the correlation between two or more input variables. Multiple linear regression involves summarizing data and investigating the relationship between variables [40,41]. The general formula for multiple regression models is given in Equation (7) below, where Y is a dependent variable, β 0 is a constant, and β j is a regression coefficient (i = 1, 2, . . . , n).
Simple regression analyses use only one independent variable (X j ), while multiple regression analyses use two or more variables [42]. It is worth noting that if a data point lies on the fitted line entirely, the vertical deviation would be equal to zero. This study's multiple linear regression model demonstrates the correlation between the one-part geopolymer paste characteristics and the experimentally measured 28-day compressive strength.

Multiple Linear Regression Model (MLR)
The following Equation (8)  Concerning the above equation, the value of R 2 was calculated as 0.61. This value demonstrates that MLR could not provide a sufficiently accurate approximation of the compressive strength of the studied geopolymer paste.
Stepwise regression is a statistical technique that is commonly used to identify the most important variables in a regression model. In stepwise regression, variables are added or removed from the model one at a time, based on their statistical significance, as measured by the p-value, until a final model that includes only the most important variables is reached. In stepwise regression analysis, variables are added or removed based on their p-values, with the threshold typically set at a significance level of 0.05. Variables with p-values less than 0.05 are considered significant and are added to the model, while variables with p-values greater than 0.05 are considered not significant and are removed from the model. This process is repeated until no more variables can be added or removed, resulting in a final model with a set of significant predictor variables.
In order to validate or reject the MLR analysis, a stepwise regression was performed in the study. After conducting the stepwise analysis and removing the variables fly ash, water/binder ratio, and curing temperature step-by-step, as they had high p-values, the following equation was obtained: 28-day Compressive Strength = (GGBS × 20.62) + (Na 2 O content × 37.99) − 15.96 (9) In this equation, the p-value is less than 0.05, indicating that the coefficients are satisfactory. However, the value of R-square and the RMSE was calculated at 0.563 and 16.8, respectively, indicating that the MLR analysis method unable to capture the underlying mechanism of the data. Figure 7 also shows the histogram of residuals of the stepwise regression performed in this study, demonstrating the frequency distribution of the differences between the predicted values and the actual values of the dependent variable. Residuals are the differences between the observed values of the dependent variable and the predicted values based on the regression equation. The histogram of residuals is used to evaluate the assumption of the normality of residuals. The horizontal axis of the histogram of residuals represents the values of the residuals, which are the differences between the predicted values and the actual values of the dependent variable. These residuals are typically standardized, with a mean of 0 and a standard deviation of 1, making it easier to compare residuals across different models or datasets. The vertical axis of the histogram represents the frequency of the residuals at each value. This frequency represents the number of data points that have a residual in the corresponding range. The height of each bar in the histogram indicates the number of data points with a residual value in that particular range. Figure 7 is unsymmetric and not bell-shaped, indicating that the residuals are not normally distributed. The pattern of residuals is also irregular that confirm the model may not be appropriate for the data or that there may be some violation of assumptions. To further assess the performance of the two models, the authors also conducted statistical analysis and presented the results in Table 5. The statistical analysis include various performance measures, such as mean absolute error, root mean square error, an coefficient of determination. These measures determined that the ANN model exhibite excellent agreement with the actual experimental data, while the MLR model showe comparatively lower performance. The experimental data (represented by the discontinuous blue line) and the predicte values (represented by the continuous red line) from the training and testing data of th ANN algorithms are shown in Figure 9. The predicted compressive strength of the AN model matched well with the target values, which is supported by both the training pa (70% of data) and the validation/testing part (30% of data) for the ANN algorithms. To further assess the performance of the two models, the authors also conducted a statistical analysis and presented the results in Table 5. The statistical analysis included various performance measures, such as mean absolute error, root mean square error, and coefficient of determination. These measures determined that the ANN model exhibited excellent agreement with the actual experimental data, while the MLR model showed comparatively lower performance. The experimental data (represented by the discontinuous blue line) and the predicted values (represented by the continuous red line) from the training and testing data of the ANN algorithms are shown in Figure 9. The predicted compressive strength of the ANN model matched well with the target values, which is supported by both the training part (70% of data) and the validation/testing part (30% of data) for the ANN algorithms.
The following equation developed by the ANN model (extracted from MATLAB) can be employed to estimate the compressive strength of one-part geopolymer pastes.
Compressive strength = 1.16 * (fly ash) + 1.98 * (GGBS) + 0.81 * (Na 2 O content) − 0.25 * (w/b ratio) − 0.08 * (curing temperature) − 1.13 (10) The experimental data (represented by the discontinuous blue line) and the predicted values (represented by the continuous red line) from the training and testing data of the ANN algorithms are shown in Figure 9. The predicted compressive strength of the ANN model matched well with the target values, which is supported by both the training part (70% of data) and the validation/testing part (30% of data) for the ANN algorithms. The following equation developed by the ANN model (extracted from MATLAB) can be employed to estimate the compressive strength of one-part geopolymer pastes.

Multi-Objective Optimization Using CPLEX Tool
The development of an optimal predictive model is accomplished by: (i) establishing a model for predictions of the compressive strength of one-part geopolymer pastes using multiple linear regression (MLR) and artificial neural network (ANN), (ii) formulating the compressive strength using the best MLR's and ANN's models, and (iii) optimization of the formulated model to identify an optimal (maximum) value for the compressive strength.
Due to advancements in computer technology, the development of efficient algorithms and their application, as well as mathematical progress, many effective solutions have been found for Mixed Integer Programming (MIP) problems. IBM ILOG CPLEX can be used to address mathematical programming problems that require some or all variables to have integer values. These problems are known as MIPs, as the objective function and constraints may involve continuous and discrete variables, such as integers. The CPLEX Optimizer tool can generally solve linear optimization problems (LP), problems with a quadratic objective (QP), and problems with quadratic constraints (QCP). Mixed

Multi-Objective Optimization Using CPLEX Tool
The development of an optimal predictive model is accomplished by: (i) establishing a model for predictions of the compressive strength of one-part geopolymer pastes using multiple linear regression (MLR) and artificial neural network (ANN), (ii) formulating the compressive strength using the best MLR's and ANN's models, and (iii) optimization of the formulated model to identify an optimal (maximum) value for the compressive strength.
Due to advancements in computer technology, the development of efficient algorithms and their application, as well as mathematical progress, many effective solutions have been found for Mixed Integer Programming (MIP) problems. IBM ILOG CPLEX can be used to address mathematical programming problems that require some or all variables to have integer values. These problems are known as MIPs, as the objective function and constraints may involve continuous and discrete variables, such as integers. The CPLEX Optimizer tool can generally solve linear optimization problems (LP), problems with a quadratic objective (QP), and problems with quadratic constraints (QCP). Mixed Integer Linear Programs (MILPs) refer to MIPs with linear objectives, while Mixed Integer Quadratic Programs (MIQPs) refer to MIPs with quadratic objective terms.
CPLEX provides a range of optimizers for solving linear programming problems, which can be accessed through its Con-cert Technology, Callable Library, or Interactive Optimizer. The LP problems are expressed in a certain format [43]: . . a m1 X 1 + a m2 X 2 + . . . + a mn X n ∼ b m With these bounds where the relation~may be greater than or equal to, less than or equal to, or simply equal to, the upper bounds u i , and lower bounds l i may be positive infinity, negative infinity, or any real number. When a linear optimization problem is stated in that conventional form, its coefficients and values are customarily referred to by these terms: Objective function: c 1 , . . . , c n , coefficients constraint coefficients: a 11 , . . . , a mn , right-hand side: b 1 , . . . , b m , upper bounds: u 1 , . . . , u n , lower bounds: l 1 , . . . , l n , Variables or unknowns: x 1 , . . . , x n .
The variables of the objective function in the simplest linear optimization problem are mathematically continuous, meaning that there are no gaps between actual values. CPLEX implements optimizers based on simplex algorithms (both primal and dual simplex), primal-dual logarithmic barrier algorithms, and a sifting technique to resolve such linear programming issues. Computations for an experimental study of the mathematical model proposed in this study were carried out on a personal computer with AMD Ryzen 7 2700X Eight-Core Processor 3.70 GHz having a Windows 10 operating system with 16 GB RAM. The Multi-Objective MIP was conducted on CPLEX 12.9, and the optimal (maximum) value of compressive strength for the geopolymer mixture design was obtained. The decision variables are considered integer values. The objective function and related constraints are as follows: Decision variables Compressive strength : max ∑ n i=1 C i X i = (1.16 * fly ash) + (1.98 * GGBS)+ (0.81 * Na 2 O) + (−0.25 * WB) + (−0.08 * curing temperature) − 1.13 (12) Constraints 15 ≤ fly ash ≤ 60, Fly ash + GGBS = 100, Fly ash ≤ GGBS, The objective of Equation (12) is to maximize the compressive strength of the one-part geopolymer paste mixture. The constraint set by Equation (13) assigns fly ash from 15 to 60%, and the constraint specified by Equation (14) sets GGBS between 20 and 70%. Meanwhile, the constraint set by Equation (15) is defined simply because the cumulative values of fly ash and GGBS must be equivalent to 100%. For this optimization, the water/binder ratio value was selected between 0.30 (minimum) and 0.40 (maximum), as shown by Equation (16). The minimum values of Na 2 O and curing temperature were set as 1.5% and 20 • C, while the maximum values were selected as 12.50% and 60 • C, respectively, as shown by Equations (18) and (19). Solving the mathematical model using CPLEX concerning the set constraints provides the optimal mixture having maximum compressive strength. Concerning all parameters, objective functions, and constraints, the CPLEX calculates the maximum compressive strength value as 72.35 MPa, where the optimal values of each input parameter are shown in Table 6.

Sensitivity Analysis
The previous section established the optimal values of FA, GGBS, Na 2 O content, w/b ratio, and curing temperature for the ANN model to attain the highest compressive strength of the one-part geopolymer paste. This section focuses on performing a sensitivity analysis to assess the influence of the input parameters on the 28-day compressive strength. Through this analysis, the extent to which the output of the model can be affected by changes in the input parameters is determined [44]. In general, there are two main categories of sensitivity analysis: local sensitivity analysis (LSA) and global sensitivity analysis (GSA).
Local sensitivity analysis (LSA) is a method used to evaluate the sensitivity of a model's output to small perturbations in the input parameters around a specific point. It helps to identify which input parameters have the most significant impact on the model's output at a specific point in the input space. On the other hand, global sensitivity analysis (GSA) examines the sensitivity of a model's output to variations in input parameters across the entire parameter space. It is used to identify which input parameters are the most influential over a wide range of inputs and determine how these parameters affect the model's output. Both LSA and GSA are important tools for understanding the behavior of a model and can be used to inform model calibration, parameter estimation, and uncertainty quantification. Local sensitivity analysis (LSA) is based on the first-order derivative of the model output with respect to the input parameters. Global sensitivity analysis (GSA) is based on the variance decomposition of the model output with respect to the input parameters.
In general, local sensitivity analysis (LSA) is concerned with analyzing the effects of individual input variables on the overall system performance, while global sensitivity analysis (GSA) examines the impact of input variables across their entire range of values and assesses the uncertainty in system performance due to interactions between variables or when variables are varied independently. Due to the nonlinear nature of the current study, GSA was deemed more appropriate for evaluating the influence of input variables on overall system performance [45].
The sensitivity analysis is shown in Figure 10. The results indicate that the Na 2 O and GGBS contents significantly influenced the 28-day compressive strength. For instance, decreasing the GGBS by 50% led to a sharp decrease in compressive strength, estimated at around 31 MPa, while a significant increase in compressive strength of 59.5 MPa was estimated for a 50% increase in GGBS. The Na 2 O content also played an important role in compressive strength; the larger the Na 2 O content, the higher the compressive strength. On the other hand, a lower water/binder ratio led to higher compressive strength, as expected.

Concluding Remarks
In this research, by compiling 80 different mixtures of one-part geopolymer pastes, the effects of constituent materials, Na2O content of the alkaline sources, curing condition, and water/binder ratio on the 28-day compressive strength were examined in detail. ANN model using the Levenberg-Marquardt algorithm was also developed to estimate the compressive strength and its sensitivity to the input variables. Using the weights and biases of the ANN model, a CPLEX-based optimization methodology was developed to optimize the binder constituent and alkaline sources to achieve the highest compressive strength. Based on this work, the following observations and conclusions can be drawn:

•
The results confirm that there is a direct relationship between the GGBS and Ca2O content and the 28-day compressive strength. The sensitivity analysis confirmed that a 50% decrease in GGBS content leads to an estimated compressive strength of around 31 MPa, while a 50% increase leads to an estimated compressive strength of 59.5 MPa. The Na2O content also strongly influences compressive strength, with higher Na2O content resulting in higher strength. Lower water-to-binder ratios are also associated with higher compressive strength, consistent with expectations.

•
The R 2 value of the MLR model is 0.61, where the coefficients of the variables in the equation show their relative contribution to the compressive strength, with Na2O content having the highest coefficient. The negative coefficients for water/binder ratio indicate that increasing this variable decreases the compressive strength. However, the model is not accurate enough to provide a precise estimation of the compressive strength of the paste. -50% -30% -10% 0 10% 30% 50% Figure 10. Sensitivity analysis of variables for compressive strength of one-part geopolymer paste.

Concluding Remarks
In this research, by compiling 80 different mixtures of one-part geopolymer pastes, the effects of constituent materials, Na 2 O content of the alkaline sources, curing condition, and water/binder ratio on the 28-day compressive strength were examined in detail. ANN model using the Levenberg-Marquardt algorithm was also developed to estimate the compressive strength and its sensitivity to the input variables. Using the weights and biases of the ANN model, a CPLEX-based optimization methodology was developed to optimize the binder constituent and alkaline sources to achieve the highest compressive strength. Based on this work, the following observations and conclusions can be drawn:

•
The results confirm that there is a direct relationship between the GGBS and Ca 2 O content and the 28-day compressive strength. The sensitivity analysis confirmed that a 50% decrease in GGBS content leads to an estimated compressive strength of around 31 MPa, while a 50% increase leads to an estimated compressive strength of 59.5 MPa. The Na 2 O content also strongly influences compressive strength, with higher Na 2 O content resulting in higher strength. Lower water-to-binder ratios are also associated with higher compressive strength, consistent with expectations.

•
The R 2 value of the MLR model is 0.61, where the coefficients of the variables in the equation show their relative contribution to the compressive strength, with Na 2 O content having the highest coefficient. The negative coefficients for water/binder ratio indicate that increasing this variable decreases the compressive strength. However, the model is not accurate enough to provide a precise estimation of the compressive strength of the paste.

•
The proposed ANN model can adequately estimate the compressive strength of fly ash slag-based one-part geopolymer paste (with R 2 = 0.94 and RMSE = 0.07). Such an assessment of the relative significance of various features and their impact on com-pressive strength can assist material scientists/designers in making reliable decisions about the source materials to use for achieving the required strength performance. • The mathematical model was solved using CPLEX with set constraints to determine the optimal mixture for maximum compressive strength. The resulting optimal input parameters to achieve the maximum compressive strength value of 72.35 MPa were calculated fly ash at 12.30%, GGBS at 21.6%, Na2O content at 12.50%, water/binder ratio at 0.3, and curing temperature at 20 • C.

•
The developed CPLEX mathematical model can be employed as a reliable tool for preparing geopolymer past with optimal mixture design and compressive strength. This technique can lead to an efficient input variable selection and a reduction in training time without compromising model accuracy.

•
The shortcomings of this research include a limited sample size, a narrow range of input parameters, and reliance on the specific type of binders (fly ash and GGBS). To address these issues, future research could involve a larger and more diverse sample size, a wider range of input parameters, and the use of various types of binders and alkaline sources. Additionally, incorporating more advanced machine learning techniques, such as deep learning, could enhance the accuracy of the predictive model and its optimization.