Different AI Predictive Models for Pavement Subgrade Stiffness and Resilient Deformation of Geopolymer Cement-Treated Lateritic Soil with Ordinary Cement Addition

Abstract

Laboratory determination of subgrade soil stiffness (T) and modulus of resilient deformation (D) of subgrade lateritic material is a cumbersome exercise that requires sophisticated and expensive triaxial experimental set up. This, in most cases is lacking in most laboratories. The same condition is experienced during the experimentation of suction and capillary rise for the design of hydraulically bound pavement layers; the subgrade or unsaturated soils. To overcome the above costs, time and challenges, intelligent techniques; genetic programming (GP), artificial neural network (ANN) and genetic algorithm (GA) optimized polynomial linear regression (PLR) known as the evolutionary polynomial regression (EPR), have been adopted in this work to propose models for T and D under cemented and uncemented conditions of the lateritic subgrade soil treated with geopolymer cement (G). The parameters considered in this predictive model research work were capillary rise (C), suction (S), unconfined compressive strength (U) and California bearing ratio (B) as well as the varied proportions of geopolymer cement (G). At the end of the model on G-treated uncemented and cemented soil classified as A-2-6 and highly plastic soil, performance indices (SSE and R²) were used to measure the accuracy of the models. For the uncemented case, GP and EPR showed equal accuracy with R² of 99.9% but with SSE of 2.9% and 2.6%, while ANN ended with 99.8% and 3.6% for the stiffness (T) prediction. For the resilient deformation (D) prediction, GP and ANN performed with R² of 99.3% and SSE of 6.9% and 6%, respectively, while EPR has an R² value of 99.2% and SSE of 6.7%. For the cemented case, EPR outclassed GP and ANN with R² of 99.9%; 99% and SSE 2.1%; 4.6% for the stiffness (T^) and deformation (D^) predictions, respectively. Generally, the predictive models showed consistent performance accuracy in predicting uncemented and cemented subgrade soil treated with geopolymer cement.

1 Introduction

Pavement design and construction needs serious research attention due to complexity of the parametric consideration that is involved. Based on its response to load effects, pavement stiffness plays a critical role in reestablishing the stability of pavement systems and this was supported by Tingle and Jersey [1, 2]. Overall, the resilient deformation of geopolymer cement-treated lateritic soils are important property inputs in their mechanistic empirical pavement design for enhanced performance and resistance of the base layers and subgrade against dynamic, impulsive or critical loads [3]. Hence, the load and environmentally associated pavement response and performance are of fundamental concern to transportation geotechnical engineers [4]. Resilient deformation of pavement, its ability to absorb energy when it is deformed elastically, and tendency to release that energy upon unloading is a critical factor when assessing its service life [5]. Stabilized soils account for plastic stress accumulation for durable performance due to their safe resilient modulus [6]. It is also reported that the confining pressure, moisture content and deviator stress levels influence the resilient and permanent deformation of treated soils [7]. For a sustainable pavement structure design and long-term pavement performance maintenance, pavement subgrade strength, a measure of its strength (the stress needed to break or rupture a material) or stiffness (the relationship between stress and strain in the elastic range) or how well a material is able to return to its original shape and size after being stressed should be perfectly analyzed, explained, described and/or predicted using certain influencing parameters such as: suction, capillary rise, California bearing ratio (CBR), stiffness, compressive strength, and modulus of resilient deformation [8,9,10,11].

Owing to the complex nature of the pavement system, pavement material characteristics, and loading conditions, there is induced complexity in analyzing and modelling the mechanism at which geopolymer cement-treated soils response. Hence, accurate prediction of optimal pavement subgrade stiffness and obtainable resilient deformation deserves evolutionary computational techniques where these non-linear interactions are perfectly traced and evaluated. Some researchers have reported the benefits of using many other soil stabilization materials and techniques in modifying pavement subgrade stiffness and hence reduction in its deformation tendencies. For example, Tang et al. [3] reported that the incorporation of geogrids at the interface between a base course layer and subgrade provides separation and minimize the degradation of the base layer caused by the mix of base aggregate and subgrade soil. They also maintained that the geogrids provide lateral restraints and prevent lateral spreading of the base aggregate, as well as increase the elastic modulus of the base layer. This report was because of the increase in the confining stresses of the interlocking and shears interaction between the geogrids and aggregate. Chen et al. [12] confirmed that the use of the geosynthetic materials had showed appreciable benefit to reducing the permanent deformation of subgrade, imposing some effect on the resilient properties of subgrade. Sahaf et al. [13] extensively discussed the benefit of accurate determination of stiffness modulus of pavement subgrade. They described important loading parameters including loading waveform, loading time, and rest time in subgrade layer. They showed that the effect of loading time could be intensified due to the increase in depth and decrease in the quality of the materials. In addition, they argued that owing to the elasto-plastic nature of subgrade material, the rest period should be considered in determination of stiffness modulus.

The use of geopolymer materials for treating soil for optimal pavement subgrade performance has proven an excellent binder that attains high strength by curing at room temperature [14]. Javdanian [15] evaluated for clayey soil specimens stabilized with fly ash and blast furnace slag based geopolymer, the unconfined compressive strength (UCS) of improved fine-grained soils by utilizing a large database of unconfined compressive strength. Many published works have demonstrated that for geopolymer cement-treated subgrade, the super bonding between aluminosilicate and alkali solution, which produce high compressive strength, low shrinkage [16], and resistance toward acid, resistance to fire and enhance the pavement subgrade stiffness and resilient deformation characteristics.

As stochastic algorithms whose search methods model some natural phenomena, evolutionary computational techniques such as genetic programming (GP) sorts a population of computer programs genetically to resolve a problem ([17]). Whereas GP are domain-independent techniques that transmutes an assembly of computer programs into a new scheme by iteration and incorporation of genetic operations, artificial neural networks (ANNs) simulate the way the human brain analyzes and processes information via a collection of connected units or nodes called artificial neuron. An optimization of the ANN and genetic algorithm (GA), a natural selection where the fittest individuals are selected for reproduction to produce offspring or solutions of the next generation referred to as evolutionary polynomial regression (EPR) has also been largely investigated. Many researches in civil engineering have make use of the services of GP, ANN and EPR for making their predictions [14, 18,19,20,21,22].

The pavement subgrade stiffness and resilient deformation can be predicted using the physical properties such as liquid limit, plastic limit, plasticity index, maximum dry density, optimum moisture content, including state variables such as, degree of compaction, and stress variables like confining pressure and deviatoric stress [3]. Zhang et al. [23] proposed an optimized artificial neural network (ANN) approach based on the multi-population genetic algorithm (EPR) to effectively determine the resilient modulus of compacted subgrade soils. Coleri and Ahmet [24] reported the applicability of genetic algorithm and curve shifting methodology to the estimation of the resilient modulus at various stress states for subgrade soils using the results of triaxial resilient modulus tests. Due to the viscoelastic nature of asphalt mixes, the stiffness of these materials depends on temperature, loading time duration, rest period, and loading waveform. The interest in pavement deformations under static and dynamic loads as a measure of pavement performance and the need for a procedure whereby these deflections might be predicted in advance using reliable evolutionary prediction models such as genetic programming, artificial neural networks (ANN) and evolutionary polynomial regression (EPR) techniques fed from suitable experimental investigations of Bui Van et al. [10] has necessitated this study. Despite the numerous investigations on the application of geopolymer materials, attention has not been focused on incorporating ordinary Portland cement with the geopolymer for pavement subgrade stiffness and resilient deformation using multiple sets of soft computing techniques. The present study aims at employing GP, ANN and EPR soft computing techniques for predicting pavement subgrade stiffness and resilient deformation of geopolymer cement-treated lateritic soil with conventional cement addition using the various predictor variables: geopolymer cement proportion, capillary rise, suction, compressive strength, and California bearing ratio to predict stiffness, and modulus of resilient deformation of the geopolymer-cement treated lateritic soil. It is also important to ensure the sustainability of the entire operation; transport geotechnics and soft computing aspects of the work.

Solid waste-based ash and powder materials utilized as geomaterials or as supplementary cementing materials are generated from the combustion and/or pulverization of agro-industrial-wastes obtained from farm, household or industries. Meanwhile, sustainable materials are such that could be used by the present generation to meet their needs without affecting the ability of future generations to meet theirs [25]. Low embodied energy (EE) is used for the calcination by combustion or pulverization of these waste materials such as quarry dust (QD), rice husk (RH), sugarcane bagasse (SB), palm bunch (PB), GGBFS, fly ash (FA), etc. to generate pozzolana-based green construction materials unlike the huge EE expended for the production of conventional materials like OPC [26]. The sustainability of the waste-based construction materials depends on the advantages derived from converting waste to useful supplementary cementing materials (SCM) as well as the use of lower EE for the process. In other words, using activated waste-based ash or powder for the treatment of expansive pavement subgrade in place of the conventional building materials is very sustainable. Conversely, in the sustainability of materials and system, the cost is a vital parameter to consider. Experimental procedures in Civil Engineering could be time, energy and resource consuming [27] and so, there is a need to develop good predictive and intelligent models capable of forecasting the performance of waste-based cementing materials. Some of the predictive models that have been used to design and analyze the performance of construction materials include ANN [21] and Support vector machine [20, 28], genetic programming (GP [21]), etc. The utilization of waste-based materials, which are unconventional construction materials, has the challenge of an inexact understanding of the material properties and behaviors. These inexactitudes are due to the use of conventional materials procedures for the determination of their materials’ properties [28]. Application of different machine learning-based predictive models such as SVM, GP, ANN and EPR techniquesto study the performance of these unconventional materials for building and road subgrade construction is vital to reduce the challenges of conducting repeated experiments to obtain the properties of materials. These developed models are sustainable because they save time, energy and resources that would have been used for recurrent experimental procedures aimed at obtaining the performance of green construction materials. Generally, the present work tries to use intelligent models to predict the behavior of waste-based materials-treated lateritic soil for use for the construction of sustainable pavement subgrades.

2 Materials and Methods

2.1 Materials Preparation

Figure 1 shows the map location at Amaba from where the soil sample was collected, sundried for 5 days, rubber-pestle-mashed to remove lumps, sieved through 6.35 mm aperture sieve and stored in bags for use. The above was done in line with the requirements of BS1377-2 [29]. The quarry dust (QD), fly ash (FA) and ground granulated blast furnace slag (GGBFS) or metallurgical slag (MS) were collected as solid waste materials and prepared for use [10]. The activator needed for the formulation of geopolymer cement (GPC) was generated with NaOH in aqueous solution with mole concentration of 11 M (for eco-friendly handling) and NaSiO₂ mixed in the ratio of 1:1 according to the GPC design requirements of Davidovits [30] and utilized in the work of Bui Van et al. [10]. 5% of the activator solution was mixed with 50% of QD, 25% of FA and 20% of GGBFS by weight to formulate the quarry dust based geopolymer cement (QD-GPC) designated as G in this work for the purpose of clarity. G-treated uncemented soil was achieved by deeply mixing the predetermined proportions of the formulated geopolymer cement with the soil while G-treated cemented soil was achieved by deeply mixing the predetermined proportions of the formulated geopolymer cement with soil and adding 5% of ordinary Portland cement to the mixed G plus soil blend. The X-ray fluorescence test was conducted on the Soil, QD, FA, GGBFS and OPC (Dangote brand of Portland cement) to characterize the materials based on the weight of chemical oxides each contains [10] in accordance with appropriate design standards (ASTM) E1621-13 [31, 32]). Figure 2 shows the schematic path of the methodology of the research work [10].

Fig. 1

Amaba location map of soil sample

Fig. 2

Schematic path of the research methodology [10]

2.2 Experimental Methods and Data Collection and Analysis

2.2.1 Collected Database

Multiple 3 × replicate experiments were conducted and an average of 99 specimens were tested and an average of each set of three (3) was estimated to have 33 outcomes on the following physical and mechanical properties of G-treated lateritic soil [33] (geopolymer cement (G) was added to the soil in the proportions of 0%, 1.5%, 2.5%, 3.5%, …, 40% as presented in the supplementary material):

• Capillary rise at 72 h curing of G-treated soil (C) %,

• Suction at 72 h curing of G-treated soil (S) %,

• Compressive strength at 72 days curing of G-treated soil (U) MPa,

• California bearing ratio at 24 h soaking of G-treated soil (B) %,

• Stiffness at 24 h soaking of G-treated soil (T) GPa, [10],

• Modulus of resilient deformation of G-treated soil (D) kg/m² [10],

• Capillary rise at 72 h curing of G-treated soil modified/cemented with OPC (C^) %,

• Suction at 72 h curing of G-treated soil modified/cemented with OPC (S^) %,

• Compressive strength at 72 days curing of G-treated soil modified/cemented with OPC (U^) MPa,

• California bearing ratio at 24 h soaking of G-treated soil modified/cemented with OPC (B^) %,

• Stiffness at 24 h soaking of G-treated soil modified/cemented with OPC (T^) GPa,

• Modulus of resilient deformation of G-treated soil modified/cemented with OPC (D^) kg/m²

The 33 data items from the multiple experiments, which gave rise to five (5) input parameters and two (2) output parameters were used to in the predictive model exercise in line with the models’ functional equations (Eq. 1 and 2) to predict two sets of treatment conditions (G- treated uncemented lateritic and G- treated cemented lateritic soil).

$$T; \, D \, = \, f \, \left( {G, \, C, \, S, \, U, \, B} \right)$$

(1)

$$T^{ \wedge } ;D^{ \wedge } = f(G,C^{ \wedge } ,S^{ \wedge } ,U^{ \wedge } ,B^{ \wedge } )$$

(2)

2.2.2 Statistical Reliability Analysis

The reliability or the internal consistency of the input parameters with the statistical influence on the entire system for the cases of uncemented and cemented geopolymer cement treated lateritic soil was tested and analyzed with the Cronbach’s alpha reliability technique and the results are presented in Tables 1, 2, 3, 4, 5, 6, 7, 8. In Tables 1, 3, 5, and 7, the Cronbach’s alpha of the consistency of the uncemented input parameters with T and D and those of the cemented input parameters with T^ and D^ were observed to be 0.658, 0.228, 0.581 and 0.219 respectively. According to Pallant [34], the Cronbach’s alpha for any system with items greater than 10 in a scale must be ≥ 0.7 for the internal consistency between the studied parameters to be acceptable. This shows that the consistency of the parameters was not agreeable and Tables 2, 4, 6, and 8 allows an opportunity to improve the Cronbach’s alpha level [34]. In the case of Table 2, the capillary rise (C) item alpha level shows that if C is removed from the entire system, the alpha level will improve from 0.658 to 0.721. However, in the cases of Tables 4, 6 and 8, the item alpha levels to be removed to improve the alpha level of the entire system are all below 0.7 and this means that for the reliability of the parameters in both cemented and uncemented cases to be consistent and influential to the output parameters, the 33 items have to be recoded or reshuffled in accordance with the requirements of Pallant [34]. Finally, the data items were reshuffled and analyzed to achieve maximum consistency as presented in Tables 1 and 2.

Table 1 Cronbach’s alpha reliability statistics of uncemented input parameters (G, C, S, U, and B) with the output, T

Table 2 Item total statistics of uncemented input parameters (G, C, S, U, and B) with the output, T

Table 3 Reliability statistics of uncemented input parameters (G, C, S, U, and B) with the output, D

Table 4 Item total statistics of uncemented input parameters (G, C, S, U, and B) with the output, D

Table 5 Reliability statistics of cemented input parameters (G^, C^, S^, U^, and B^) with the output, T^

Table 6 Item total statistics of cemented input parameters (G^, C^, S^, U^, and B^) with the output, T^

Table 7 Reliability statistics of cemented input parameters (G^, C^, S^, U^, and B^) with the output, D^

Table 8 Item total statistics of cemented input parameters (G^, C^, S^, U^, and B^) with the output, D^

The reshuffled data items recorded in Tables 1 and 2 divided into training set (21 records) and validation set (12 records). Tables 9, 10, 11, 12 summarize their statistical characteristics and the Pearson correlation matrix. Finally, Fig. 3 and 4 show the histograms for both inputs and outputs. The California bearing ratio (B) showed the highest means in both training and validation analysis of the items for the uncemented and cemented conditions while suction showed the least mean as shown in Tables 9 and 10. This shows the high influence of B on the output parameters of the models T and D and T^ and D^. The unconfined compressive strength showed the least variance and standard deviation from the output parameters compared to any other parameter in both the uncemented and cemented conditions. Figure 3 and 4 show good distribution of the items consistent with the output parameters (the targets). In Tables 11 and 12, B and B^ showed the highest correlation with T and D and T^ and D^ respectively for both cement conditions (uncemented and cemented) and this agrees with the mean, standard deviation and variance outcome of the parameters. This correlation is followed by the effect of the geopolymer cement (G) on the targets (T, T^, D, and D^). The third in terms of the influence in correlation is the unconfined compressive strength (U). Capillary rise (C) and suction (S) equally showed good correlation with the targets and this shows that all five (5) input parameters are important and should be given a chance to be eliminated in the full model exercise through step wise approach.

Table 9 Statistical analysis of collected database for uncemented condition

Table 10 Statistical analysis of collected database for cemented condition

Table 11 Pearson correlation matrix for the uncemented condition

Table 12 Pearson correlation matrix for cemented condition

Fig. 3

Distribution histograms for inputs and outputs

Fig. 4

Distribution histograms for inputs and outputs

2.2.3 Predictive Models Research Program

Three different Artificial Intelligent (AI) techniques were used to predict the shear strength parameters of the tested soil samples. These techniques are Genetic programming (GP), Artificial Neural Network (ANN) and Polynomial Linear Regression optimized using Genetic Algorithm which is known as evolutionary polynomial regression (EPR). All the three developed models were used to predict the values of both Stiffness (T), Portland cemented Stiffness (T^), Modulus of resilient deformation (D) and Portland cemented Modulus of resilient deformation (D^) of G-treated soil using the measured geopolymer cement proportion (G), uncemented sets of; Capillary rise (C), Suction (S), Compressive strength (U) and California bearing ratio (B) and cemented sets of; Capillary rise (C^), Suction (S^), Compressive strength (U^) and California bearing ratio (B^). This was executed to study the influence ordinary Portland cement (OPC) addition has on the studied parameters treated with GPC and on the predicted models as a comparative study. Each model on the three developed models was based on different approach (evolutionary approach for GP, mimicking biological neurons for ANN and optimized mathematical regression technique for EPR). However, for all developed models, prediction accuracy was evaluated in terms of Sum of Squared Errors (SSE).The following section discusses the results of each model. The Accuracies of developed models were evaluated by comparing the (SSE) between predicted and calculated shear strength parameters values. The results of all developed models are summarized in Table 4.

3 Results and Discussion

3.1 General Remarks

Table 13 shows the basic properties of the soil used in this research work. The soil was a highly plastic soil though classified as A-2-6 group soil according to AASHTO system of soil classification. The CBR (B) of the soil was observed to be 8%, which falls below the minimum CBR standard for soil to be used as a compacted pavement subgrade material (< 10% according to AASHTO [35]). In Fig. 5, other properties of the soil were presented. Table 14 shows the oxide composition by weight of the experimental materials, which shows that FA, QD. GGBFS and OPC had high pozzolanic ability according to the conditions of ASTM C618 [31]. It is observed in Tables 1 and 2 that the capillary rise (C) and suction (S) reduced with increased geopolymer cement (G) in both the non-cemented and cemented conditions. At 15% by weight addition of the G, the trial test cured at 72 h fell below the minimum allowable critical capillary rise of 25% proposed by Austroads [36]. This is due to increased release of Ca²⁺, Si⁴⁺ and Al³⁺ by the silicate of sodium component of the activator from the FA, GGBFS and QD blend which sped up cation exchange reaction rate [16, 23]. The substantial reduction in suction with increased addition of G was due to the ability of G to react and reduce the porosity of the G-treated uncemented and cemented soil. The UCS (U) also behaved in an improved consistency with increased addition of G due to the increased Ca²⁺ release [10]. This equally increased the intergranular contact between the soil structures and consequently reduced the porosity in the treated soil thereby increasing densification, flocculation and strength gain, which is translated as improved USC (U). Figures 6 and 7 show the effect of the G on stiffness (T) and the modulus of resilient deformation (D) of both uncemented and cemented lateritic soil. It can be observed that the T and D improved substantially with increased G and with 5% addition of cement on the second case. This serves as a good data to predict the behavior of stabilized lateritic subgrade soils for sustainable construction.

Table 13 Basic properties of the lateritic soil

Fig. 5

Grain size distributions of materials [10]

Table 14 Chemical oxides composition of experimental materials [10]

Fig. 6

Subgrade reference soil stiffness of uncemented and cemented lateritic soil treated with geopolymer cement (GPC or GP) [10].*GP is geopolymer

Fig. 7

Modulus of Resilient Deformation (D) of uncemented and cemented lateritic soil treated with geopolymer cement (GPC or GP) [10]. *GP is geopolymer

3.2 Prediction of Output Parameters (T, T^, D, and D^)

3.2.1 Model (1): Using (GP) Technique

The developed GP model has only two levels of complexity. The population size, survivor size and number of generations were 30,000, 10,000 and 50 respectively. Equation (3) and (4) present the output formulas for (T) and (D) respectively, while Figs. 10a and 12a show their fitness respectively. The average error % of total set for (T) and (D) are (2.9%) and (6.9%), respectively, while the (R²) values are (0.999) and (0.993), respectively.

Equation (5), (6) present the output formulas for (T^) and (D^) respectively, while Figs. 11a and 13a show their fitness respectively. The average error % of total set for (T^) and (D^) are (6.9%) and (6.5%) respectively, while the (R²) values are (0.990) and (0.980) respectively, which agrees with El-Bosraty et al. [37] and Ebid [38].

$$T = \frac{U + B}{{10}}$$

(3)

$$D = \frac{1 + U}{{8.25 - 2.2 B}}$$

(4)

$$T^{ \wedge } = 4.5~U^{ \wedge } + 0.16~B^{ \wedge } - 0.27$$

(5)

$$D^{ \wedge } = \frac{{\left( {G/C^{ \wedge } } \right) + 2~e^{{3.66~G^{ \wedge } }} }}{{13}} + 0.055$$

(6)

3.2.2 Model (2): Using (ANN) Technique

A back propagation ANN with one hidden layer and nonlinear activation function (Sigmoid) was used to predict both Stiffness (T) and modulus of resilient deformation (D). The used networks layout and their connation weights are illustrated in Fig. 8. Equations (7) and (9) present the equivalent functions of the developed ANN models. The average error % of total dataset for these equations are (3.6%) and (6.0%) and the (R²) values are (0.998) and (0.993) respectively. The relations between calculated and predicted values are shown in Figs. 10b and 12b.

Fig. 8

Layout for the developed ANN’s and their connection weights

A back propagation ANN with one hidden layer and nonlinear activation function (Sigmoid) was used to predict both stiffness (T^) and modulus of resilient deformation (D^). The used networks layout and their connation weights are illustrated in Fig. 9. Equations (15) and (16) present the equivalent functions of the developed ANN models. The average error % of total dataset for these equations are (3.1%) and (5.2%) and the (R²) values are (0.997) and (0.987) respectively, which agrees with El-Bosraty et al. [37] and Ebid [38]. The relations between calculated and predicted values are shown in Figs. 11b and 13b.

$$T = 0.01 + \frac{0.32}{{1 + e^{ - Y1} }}$$

(7)

$$Y1 = 0.048 + \frac{6.26}{{1 + e^{ - X1} }} - \frac{3.00 }{{1 + e^{ - X2} }}$$

(8)

$${\text{D}} = 0.10 + \frac{0.80}{{1 + {\text{e}}^{{ - {\text{Y}}2}} }}$$

(9)

$${\text{Y}}2 = { } - 1.95 + \frac{8.42}{{1 + {\text{e}}^{{ - {\text{X}}1}} }} - \frac{{0.77{ }}}{{1 + {\text{e}}^{{ - {\text{X}}2}} }}$$

(10)

$$X1 = - 5.31 + 2.48 G^{\prime} - 2.58 C^{\prime} - 2.19 S^{\prime} + 1.67 U^{\prime} + 1.28 B^{\prime}$$

(11)

$$X2 = 0.37 - 1.49~G^{\prime} + 1.02~C^{\prime} + 1.79~S^{\prime} - 1.21~U^{\prime} - 1.97~B^{\prime}$$

(12)

$$G^{\prime} = \frac{G}{0.40}; C^{\prime} = \frac{C - 0.11}{{0.67}}; S^{\prime} = \frac{S - 0.001}{{0.02}}; U^{\prime} = \frac{U - 0.065}{{0.04}}; B^{\prime} = \frac{B - 0.08}{{3.12}}$$

(13)

$$T^{ \wedge } = 0.077 + \frac{{0.645}}{{1 + e^{{ - Y1}} }}$$

(14)

$$Y1 = 7.263 - \frac{4.576}{{1 + e^{ - X1} }} - \frac{6.204 }{{1 + e^{ - X2} }}$$

(15)

$$D^{ \wedge } = 0.190 + \frac{{0.790}}{{1 + e^{{ - Y2}} }}$$

(16)

$$Y2 = ~4.697 - \frac{{3.143}}{{1 + e^{{ - X1}} }} - \frac{{4.604~}}{{1 + e^{{ - X2}} }}$$

(17)

$$X1 = 0.797 - 1.675~G^{\prime} + 0.052~C^{\prime} + 1.323~S^{\prime} - 1.536~U^{\prime} - 2.032~B^{\prime}$$

(18)

$$X2 = 6.363 - 4.551 G^{\prime} + 5.566 C^{\prime} + 4.885 S^{\prime} - 0.630 U^{\prime} - 1.631 B^{\prime}$$

(19)

$$G^{\prime} = ~\frac{G}{{0.40}};~C^{\prime} = ~\frac{{C^{ \wedge } - 0.11}}{{0.29}};~S^{\prime} = ~\frac{{S^{ \wedge } - 0.011}}{{0.026}};~U^{\prime} = ~\frac{{U^{ \wedge } - 0.07}}{{0.045}};~B^{\prime} = ~\frac{{B^{ \wedge } - 0.10}}{{3.21}}$$

(20)

Fig. 9

Layout for the developed ANN’s and their connection weights

Fig. 10

Relation between predicted and calculated (T); a GP, b ANN, and c EPR values using the developed models

Fig. 11

Relation between predicted and calculated (T^); a GP, b ANN, and c EPR values using the developed models

Fig. 12

Relation between predicted and calculated (D); a GP, b ANN, and c EPR values using the developed models

Fig. 13

Relation between predicted and calculated (D^); a GP, b ANN, and c EPR values using the developed models

3.2.3 Model (3): Using (EPR) Technique

Finally, to predict (T) and (D) values, the developed EPR model was limited to quadratic level, for five inputs, there are 21 possible terms $\left( {\mathop \sum \nolimits_{i = 1}^{i = 5} \mathop \sum \nolimits_{j = 1}^{j = 5} X_{i} .X_{j} + \mathop \sum \nolimits_{i = 1}^{i = 5} X_{i} + C} \right)$. GA technique was applied on these 21 terms to select the most effective 3 terms. On the other hand, to predict the values of (T^) and (D^) the developed EPR model was limited to cubic level, for five inputs, there are 56 possible terms $\left( {\mathop \sum \nolimits_{i = 1}^{i = 5} \mathop \sum \nolimits_{j = 1}^{j = 5} \mathop \sum \nolimits_{k = 1}^{k = 5} X_{i} .X_{j} .X_{k} + \mathop \sum \nolimits_{i = 1}^{i = 5} \mathop \sum \nolimits_{j = 1}^{j = 5} X_{i} .X_{j} + \mathop \sum \nolimits_{i = 1}^{i = 5} X_{i} + C} \right)$. GA technique was applied on these 56 terms to select the most effective 4 terms. The outputs are illustrated in Eqs. (21)–(24) and their fitness are shown in Figs. 10c, 11c, 12c, 13c. The average error % (SSE) and (R²) values in the case of uncemented model were improved to (2.6%)–(0.999) and (6.7%)–(0.992) for the total datasets respectively while for the case of the cemented model, the average error% (SSE) and (R²) values were improved to (2.1%)–(0.999) aand (4.6%)–(0.990) for the total datasets respectively, which agrees with El-Bosraty et al. [37] and Ebid [38].

$$T = 0.098 B - 0.591 S + 0.0179$$

(21)

$$D = 6.865 G^{2} - 0.913 C \cdot B + 0.157$$

(22)

$$T^{ \wedge } = 0.068~ + ~3.31~U^{ \wedge } \cdot B^{ \wedge } - 0.055~B^{{ \wedge 2}}$$

(23)

$$D^{ \wedge } = 0.094~ + 23.1~S^{ \wedge } \cdot B^{ \wedge } + \left( {0.249~G - 8.52~S^{ \wedge } } \right)B^{{ \wedge 2}}$$

(24)

The general summary of the performance of the models for cemented and uncemented predicted parameters of the geopolymer cement treated lateritic soil is presented in Tables 15, 16.

Table 15 Performance accuracies of developed models for the uncemented parameters

Table 16 Performance accuracies of developed models for the cemented parameters

4 Conclusions

This research presents three models using three artificial intelligent (AI) techniques (GP, ANN and EPR) to predict values of both stiffness (T) and modulus of resilient deformation (D) of G-treated soil using the measured geopolymer cement proportion (G), and the uncemented parameters; capillary rise (C), Suction (S), Compressive strength (U) and California bearing ratio (B) and the cemented parameters; capillary rise (C^), suction (S^), compressive strength (U^) and California bearing ratio (B^). From the performance accuracy evaluation conducted on the proposed models, the following can be concluded (Tables 17, 18);

• Prediction accuracies of (T) for all models are so close (between 96.4 and 97.4%),which gives an advantage to the simplest model(GP model), while the prediction accuracies of (D) are ranged between 93.1 and 94.0% and the simplest model equation is the (EPR) model. Also, prediction accuracies of (D^) for all models are so close (between 94.5 and 95.5%), which gives an advantage to the simplest model equation (GP model), while the prediction accuracies of (T^) are ranged between 93.0 and 98.0% and the simplest model equation is the (EPR) model.

• (GP) model illustrated that both (T) and (D) values are governed by (B) and (U). On the other hand, the (EPR) model showed that they mainly depend on (B) besides (S) and (G). On the second case (cemented), both (GP) and (EPR) models illustrated that both (T^) values are governed by (B^) and (U^). On the other hand, the (D^) values are mainly dependence on (G) besides (B^), (S^) and (C^).

• Although the ANN model had very simple configurations (one hidden layer with two neurons and sigmoid activation function), but it was still able to capture the relationship between the inputs and the outputs accurately. It almost shared the same accuracy level with (GP) and (EPR) models.

• Since the three models share almost the same level of accuracy, (ANN) models are considered the worst of them due to its complicated equivalent equations.

• GA technique successfully reduced all possible polynomial terms of conventional PLR the only effective terms without significant impact on its accuracy.

• Like any other intelligent regression technique, the generated formulas are valid within the considered range of parameter values, beyond this range; the prediction accuracy should be verified.

• Prediction accuracies of (T) for all models are so close (between 96.4 and 97.4%),which gives an advantage to the simplest model(GP model), while the prediction accuracies of (D) are ranged between 93.1 and 94.0% and the simplest model equation is the (EPR) model. Also, prediction accuracies of (D^) for all models are so close (between 94.5 and 95.5%), which gives an advantage to the simplest model equation (GP model), while the prediction accuracies of (T^) are ranged between 93.0 and 98.0% and the simplest model equation is the (EPR) model.

• (GP) model illustrated that both (T) and (D) values are governed by (B) and (U). On the other hand, the (EPR) model showed that they mainly depend on (B) besides (S) and (G). On the second case (cemented), both (GP) and (EPR) models illustrated that both (T^) values are governed by (B^) and (U^). On the other hand, the (D^) values are mainly dependence on (G) besides (B^), (S^) and (C^).

• Although the ANN model had very simple configurations (one hidden layer with two neurons and sigmoid activation function), but it was still able to capture the relationship between the inputs and the outputs accurately. It almost shared the same accuracy level with (GP) and (EPR) models.

• Since the three models share almost the same level of accuracy, (ANN) models are considered the worst of them due to its complicated equivalent equations.

• GA technique successfully reduced all possible polynomial terms of conventional PLR the only effective terms without significant impact on its accuracy.

• Like any other intelligent regression technique, the generated formulas are valid within the considered range of parameter values, beyond this range; the prediction accuracy should be verified.

Table 17 The used database

Table 18 The used database