Modeling the influence of bacteria concentration on the mechanical properties of self-healing concrete (SHC) for sustainable bio-concrete structures

In this research paper, the intelligent learning abilities of the gray wolf optimization (GWO), multi-verse optimization (MVO), moth fly optimization, particle swarm optimization (PSO), and whale optimization algorithm (WOA) metaheuristic techniques and the response surface methodology (RSM) has been studied in the prediction of the mechanical properties of self-healing concrete. Bio-concrete technology stimulated by the concentration of bacteria has been utilized as a sustainable structural concrete for the future of the built environment. This is due to the recovery tendency of the concrete structures after noticeable structural failures. However, it requires a somewhat expensive exercise and technology to create the medium for the growth of the bacteria needed for this self-healing ability. The method of data gathering, analysis and intelligent prediction has been adopted to propose parametric relationships between the bacteria usage and the concrete performance in terms of strength and durability. This makes is cheaper to design self-healing concrete structures based on the optimized mathematical relationships and models proposed from this exercise. The performance of the models was tested by using the coefficient of determination (R2), root mean squared errors, mean absolute errors, mean squared errors, variance accounted for and the coefficient of error. At the end of the prediction protocol and model performance evaluation, it was found that the classified metaheuristic techniques outclassed the RSM due their ability to mimic human and animal genetics of mutation. Furthermore, it can be finally remarked that the GWO outclassed the other methods in predicting the concrete slump (Sl) with R2 of 0.998 and 0.989 for the train and test, respectively, the PSO outclassed the rest in predicting the flexural strength with R2 of 0.989 and 0.937 for train and test, respectively and the MVO outclassed the others in predicting the compressive strength with R2 of 0.998 and 0.958 for train and test, respectively.

Self-healing concrete is a type of concrete that has the ability to repair cracks and damage on its own 1 , (see Fig. 1).One of the key components of self-healing concrete is the incorporation of bacteria, typically of the species Bacillus or Sporosarcina, along with a calcium-based healing agent, such as calcium lactate or calcium carbonate 2 .The bacteria in self-healing concrete remain dormant until cracks occur in the concrete 3,4 .When water or moisture enters the cracks, it reactivates the bacteria 5 .The bacteria then consume the calcium lactate or calcium carbonate and produce limestone (calcium carbonate) as a byproduct 6 .This limestone fills the cracks, effectively healing the concrete 7 .The concentration of bacteria in self-healing concrete can have an influence on its strength and healing efficiency 8 .Here are some points to consider: Healing capacity: Higher concentrations of bacteria generally result in greater healing capacity 9 .This means that a higher number of bacteria can produce more limestone and effectively fill a larger number of cracks, improving the self-healing capability of the concrete 10 .Crack width: The concentration of bacteria can also impact the ability to heal wider cracks.If the concentration is too low, the bacteria may not be able to produce enough limestone to effectively close wider cracks 11 .In such cases, additional measures may be required to enhance the healing process, such as incorporating fibers or other materials to bridge wider cracks 12 .Strength considerations: While higher concentrations of bacteria can improve the self-healing capacity 13 , they may also have an impact on the strength of the concrete 14 .Excessive bacterial concentrations can potentially interfere with the cement hydration process or affect the overall structural integrity of the concrete 15 .Therefore, it is important to find the right balance between healing efficiency and maintaining adequate concrete strength 16 .Optimization: The optimal concentration of bacteria in self-healing concrete depends on various factors such as the specific bacteria used, the type of healing agent, the crack width expected, and the desired strength requirements 3,17 .Extensive research and testing are necessary to determine the most suitable concentration for a particular application 18 .It's worth noting that the concentration of bacteria is just one factor in the overall design and performance of self-healing concrete 1, 19 .Other factors, such as the selection of healing agents, the mix design, and the curing conditions, also play crucial roles in achieving the desired selfhealing properties while maintaining the required strength and durability of the concrete structure 20 .Overall, the concentration of bacteria in self-healing concrete can influence its healing capacity, ability to close cracks of different widths, and potentially impact its strength 21 .The strength ranges can be either low (below 20 MPa), medium (20-40 MPa), high (40-80 MPa), or ultra-high (80-120 MPa and above) 22 .Careful consideration and optimization of bacterial concentrations are necessary to ensure effective self-healing while maintaining the structural integrity of the concrete 23 .
Figure 1.Structural benefits of bacillus subtilis in concrete.
concrete 40 .This concentration is achieved by adding specific amounts of bacteria-containing solution or powder during the concrete mixing process 7,15 .Activation Reaction: The activation of bacteria in self-healing concrete occurs when water enters the cracks and comes into contact with the dormant bacterial spores 19 .The presence of water triggers the germination of spores and the subsequent growth of bacteria [2][3][4] .Once activated, the bacteria start to consume nutrients present in the concrete, metabolize them, and produce the enzyme urease [6][7][8][9][10][11] .This urease production is an essential part of the self-healing process.The reaction can be summarized as follows: The bacteria use the nutrients, such as urea, as a food source for their growth and metabolism 8,40 .As a byproduct of their metabolism, they release urease enzyme, which initiates the hydrolysis of urea 6,9 .It's worth noting that the specific activation reaction can vary depending on the type of bacteria used and the formulation of the self-healing concrete 10,[20][21][22][23][24] .Different bacterial strains may have different nutrient requirements or metabolic pathways, but the general principle of bacterial growth and urease production remains consistent 7 .Optimizing both the bacteria concentration and the activation reaction is crucial for achieving effective self-healing properties in concrete 8 .Ongoing research aims to refine these parameters and develop standardized guidelines for incorporating bacteria into self-healing concrete [41][42][43] .The use of bacteria in self-healing concrete has the potential to enhance the sustainability of concrete structures in several ways: Extended Lifespan: Self-healing concrete reduces the need for frequent repairs and maintenance, thus extending the lifespan of concrete structures 3,42 .This leads to reduced resource consumption and waste generation associated with the construction and repair processes 44 .Reduced Material Consumption: By autonomously repairing cracks, self-healing concrete minimizes the need for additional materials, such as repair mortars or epoxy resins 45 .This can contribute to resource conservation and reduce the carbon footprint associated with the production and transportation of these materials 3 .Energy Savings: The self-healing process eliminates or reduces the need for manual intervention and repair work, which can be energy-intensive 35,42 .It eliminates the energy required for repair activities, such as drilling, patching, or replacing damaged concrete elements 43 .Improved Durability: Self-healing concrete can enhance the durability and resilience of structures 44 .By sealing cracks promptly, it prevents the ingress of water, chemicals, and other harmful substances that can lead to further deterioration 45 .This reduces the likelihood of structural failures and the need for major repairs or replacements 46 .Reduced Environmental Impact: The self-healing process of bacteria in concrete relies on the use of natural microorganisms 47 .Compared to traditional repair methods that may involve the use of synthetic materials or chemicals, self-healing concrete with bacteria has the potential for a lower environmental impact 48,49 .However, it's important to consider some potential sustainability challenges associated with bacteria use in self-healing concrete: Energy and Resource Requirements: The production and cultivation of bacteria and the incorporation of bacteria into concrete require energy and resources 42,49 .The sustainability benefits of self-healing concrete need to be balanced against the environmental impacts associated with bacterial cultivation and incorporation processes 6,35 .Bacterial Viability: Ensuring the long-term viability and performance of bacteria in concrete structures is a challenge 49 .Factors such as harsh environmental conditions, nutrient availability, and competition with other microorganisms can affect the survival and effectiveness of bacteria over time 48 .Ongoing research is focused on optimizing bacterial strains and formulations to enhance their viability and longevity 49 .Regulatory Considerations: The use of bacteria in construction materials may involve regulatory considerations related to safety, health, and environmental impacts 2,17 .It's important to conduct thorough assessments to ensure that the use of bacteria in self-healing concrete aligns with applicable regulations and standards [43][44][45] .Overall, while bacteria-based self-healing concrete offers potential sustainability benefits, further research and development are needed to optimize its performance, evaluate its life cycle impacts, and address any associated challenges 48 .Hence, this research work is focused on applying the metaheuristic machine learning and the symbolic response surface methodology methods in the prediction of the strengths of the bacterial-based self-healing concrete for use in the design and production of optimized materials-based bio-concrete at optimal rate of bacteria concentration.The research on "Modeling the influence of bacteria concentration on the mechanical properties of self-healing concrete (SHC) for sustainable bio-concrete structures" holds significant importance in several ways: Advancing Sustainable Construction: Self-healing concrete (SHC) offers a promising solution to extend the service life of concrete structures, thereby reducing the need for frequent repairs and replacements.By incorporating bacteria into concrete mixes to facilitate self-healing, the research contributes to the development of sustainable construction practices that minimize resource consumption and environmental impact.Enhancing Structural Integrity: Understanding the influence of bacteria concentration on the mechanical properties of SHC is essential for ensuring the structural integrity and performance of bioconcrete structures.By modeling these relationships, the research can provide valuable insights into optimizing the design and production of SHC to achieve desired mechanical properties and durability.Promoting Innovation in Concrete Technology: The incorporation of bacteria into concrete mixes represents an innovative approach to address common issues such as cracks and deterioration in concrete structures.By studying the effects of bacteria concentration on mechanical properties, the research contributes to the advancement of concrete technology and encourages the adoption of novel materials and methods in construction.Mitigating Maintenance Costs: Self-healing concrete has the potential to significantly reduce maintenance costs associated with concrete structures by autonomously repairing cracks and damage over time.By quantifying the relationship between bacteria concentration and mechanical properties, the research can help optimize SHC formulations to maximize healing efficiency and minimize maintenance requirements, leading to cost savings for infrastructure owners and operators.Improving Long-Term Durability: The durability of concrete structures is crucial for ensuring their long-term performance and resilience against environmental factors such as moisture, chemical exposure, and freeze-thaw cycles.By investigating how bacteria concentration influences mechanical properties, the research contributes to improving the long-term durability of bio-concrete structures, thereby extending their service life (1) Bacteria + Water + Nutrients → Growth + Urease Production and reducing life-cycle costs.Addressing Infrastructure Challenges: Cracking and deterioration are common challenges faced by concrete infrastructure worldwide, leading to safety concerns, service disruptions, and costly repairs.By developing self-healing concrete technologies, the research addresses these challenges proactively, offering a sustainable and cost-effective solution to enhance the resilience and longevity of infrastructure systems.In summary, the research on modeling the influence of bacteria concentration on the mechanical properties of self-healing concrete for sustainable bio-concrete structures has significant implications for advancing sustainable construction practices, improving structural integrity, promoting innovation in concrete technology, mitigating maintenance costs, enhancing long-term durability, and addressing critical infrastructure challenges.More important to consider is the deployment of machine learning to forecast the behavior of the SHC for sustainable design of its properties.

Methodology Data collection and preliminary analysis
The database has been collected from a previous research paper 49 , prepared, shuffled and used in the various models presented in this research paper.Table 1 presents the descriptive statistics of the outputs (compressive strength, slump and flexural strength) and the influential factors.This shows the minimum, maximum, mean, standard deviation, and variance of the collected entries of the SHC.Moreover, for the purpose of visually illustrating the statistical representation of the data, Fig. 2 exhibits a violin plot.This particular graphical depiction comprises both a boxplot and a density plot.The upper and lower boundary lines effectively demonstrate the span between the lower quartile (Q1) and the upper quartile (Q3).Meanwhile, the central line corresponds to the 95% confidence interval.The relationship between the independent variables known as the regressors (C, FA, w/c, B, and CA) and the studied mechanical properties of the bacterial-inspired self-healing concrete is also illustrated through the vibratory nodes of the violin.The width of the violin plots at any given point, which represents the probability density of the data at that value is more robust with FA, w/c, B, and CA showing the contributory strength they possessed in the studied mix.This shows that wider sections indicate higher density, while narrower sections such as in C, indicate lower density.It gives a visual representation of the distribution of the data.Figure 3 shows the correlation matrix between the input and the output parameters.It also presents the internal consistency between the inputs and the outputs.This further shows that it was w/c and B that showed good effects to only CS and SI, with degrees of 0.59 and 0.6, respectively.This further strengthened the need for machine learning predictions to achieve optimized mechanical properties of the bacterial-inspired self-healing concrete, regardless of the lack of internal consistency between the regressors and the targets.

Research plan
Linear multivariate Regression Model (LMR), Response Surface Methodology (RSM), Gray Wolf Optimization (GWO), Multi-Verse Optimization (MVO), Moth-Flame Optimization (MFO), Particle Swarm Optimization (PSO) and Whale Optimization Algorithm (WOA) were deployed to forecast the effect of bacteria concentration on the mechanical properties of medium-strength self-healing concrete (MSSHC) for the production of bio-concrete for the design and construction of green and sustainable structures.GWO, MVO, MFO, PSO, and WOA are advanced metaheuristic techniques in machine learning known for their precision, robustness and speed in model execution.

LMR
Linear multivariate regression (LMR) is a statistical technique used to model the relationship between multiple independent variables and a dependent variable.The flowchart is illustrated in Fig. 4. In LMR, the goal is to find a linear equation that best predicts the value of the dependent variable based on the values of the independent variables.This technique is commonly used in various fields, including economics, finance, social sciences, and more, to analyze and predict the relationship between multiple variables.Linear multivariate regression is an extension of simple linear regression to multiple independent variables.In simple linear regression, we have one dependent variable and one independent variable, whereas in multivariate regression, we have one dependent variable and multiple independent variables.The goal of multivariate linear regression is to estimate the coefficients that best fit the observed data.This is typically done by minimizing the sum of squared differences between the observed and predicted values of the dependent variable.The estimation of coefficients is often done using methods like Ordinary Least Squares (OLS), where the coefficients are chosen to minimize the sum of squared residuals.Multivariate regression analysis allows us to understand the relationship between the dependent variable and multiple independent variables simultaneously.It's widely used in various fields, including economics, social sciences, engineering, and many others, for predictive modeling, hypothesis testing, and understanding the impact of independent variables on the dependent variable.www.nature.com/scientificreports/RSM Response Surface Methodology (RSM) is a collection of statistical and mathematical techniques used to model and analyze the relationship between a set of controlled independent variables and the observed response of a system.The flowchart is illustrated in Fig. 5. RSM is often used in the field of engineering, chemistry, and other physical sciences to optimize processes, improve product quality, and understand the interactions between input variables.One of the key features of RSM is its ability to construct and analyze mathematical models that describe the relationship between the input variables and the system response.These models can help in predicting optimal process conditions and understanding the behavior of complex systems.RSM typically involves conducting a series of experiments to systematically vary the input variables and observe the corresponding changes in the system response.The data gathered from these experiments is then analyzed to develop a predictive model that can be used to optimize the system's performance.Overall, RSM provides a systematic and efficient approach for optimizing processes and understanding the relationships between input variables and system responses.

GWO
Gray Wolf Optimization (GWO) is a nature-inspired optimization algorithm that is based on the social hierarchy and hunting behavior of gray wolves.The flowchart is illustrated in Fig. 6.It is a metaheuristic algorithm used to solve optimization problems and is inspired by the hunting and leadership hierarchy of gray wolf packs.In GWO, the population of candidate solutions is divided into four types of wolves: alpha, beta, delta, and omega.These wolves represent the best solution, the second-best solution, the third-best solution, and the worst solution, respectively.The positions of the wolves are updated iteratively based on the hunting and social behavior of the gray wolves.The algorithm involves simulating the way that a pack of wolves collaborates to hunt and track down prey, with the goal of converging towards an optimal solution.The concept of alpha, beta, delta, and    , the best solution found so far) while maintaining a certain distance to avoid being too close to the light source (to prevent convergence to local optima).Moth Encirclement: Some moths may move too close to the light source.To prevent premature convergence, a fraction of the moths are selected for encirclement, where they are forced to move randomly to explore new regions of the search space.Updating Light Intensity: The intensity of the light source (fitness of the best solution) may decrease over iterations to simulate the diminishing attractiveness of the light source as moths gather around it.Termination Criteria: The algorithm stops when a termination condition is met, such as reaching a maximum number of iterations or finding a satisfactory solution.MFO has been applied to various optimization problems, including both continuous and discrete optimization tasks.Like other metaheuristic algorithms, its performance depends on parameter settings, problem characteristics, and tuning strategies.While it may not guarantee finding the global optimum, MFO often provides competitive results and can be particularly effective for certain types of problems.In PSO, the potential solutions to an optimization problem, called particles, are treated as a swarm.Each particle adjusts its position in the search space according to its own flying experience as well as the flying experiences of other particles in the swarm.The movement of particles is influenced by their own best-known position and the best-known position in the entire swarm.The algorithm iteratively improves the candidate solutions by adjusting the velocity and position of each particle based on its own experience and the experience of its neighbors.As the iterations progress, the particles move through the search space, gradually converging towards the optimal solutions.PSO has been widely applied in fields such as engineering, computer science, finance, and many others, to solve optimization problems, including function optimization, neural network training, and feature selection, among others.Its simplicity, ease of implementation, and effectiveness in finding near-optimal solutions have contributed to its popularity.

WOA
The Whale Optimization Algorithm (WOA) is a nature-inspired optimization algorithm proposed by Seyedali Mirjalili in 2016.The flowchart is illustrated in Fig. 10.It is inspired by the social behavior of humpback whales during hunting.In the Whale Optimization Algorithm, the search process is modeled after the bubble-net feeding behavior of humpback whales.The algorithm mimics the hunting behavior of whales and their communication to encircle and catch prey.WOA operates by iteratively updating the position of a population of candidate solutions,

Performance evaluation
The models are further tested for run efficiency by using selected error metrics as follows; the coefficient of determination (R 2 ), root mean squared errors (RMSE), mean absolute errors (MAE), mean squared errors (MSE), variance accounted for (VAF) and the coefficient of error (CE).The mathematical expressions of the validation indices are presented in Eqs.(2-7). (2)

Response surface methodology analysis for the compressive strength
Factor coding is actual.Sum of squares is Type III-Partial.The Model F-value of 59.90 implies the model is significant.There is only a 0.01% chance that an F-value this large could occur due to noise.P-values less than 0.0500 indicate model terms are significant.In this case A, B, C, D, AB are significant model terms.Values greater than 0.1000 indicate the model terms are not significant.If there are many insignificant model terms (not counting those required to support hierarchy), model reduction may improve your model.The Lack of Fit F-value of 0.25 implies the Lack of Fit is not significant relative to the pure error.There is an 89.82% chance that a Lack of Fit F-value this large could occur due to noise.Non-significant lack of fit is good -we want the model to fit.Adeq Precision measures the signal to noise ratio.A ratio greater than 4 is desirable.Your ratio of 22.320 indicates an adequate signal.This model can be used to navigate the design space.These can be read out from Tables 2 and 3, and Figs.11, 12, 13 and 14.The constraints of the RSM model and the selected solution from the 100 iterations are presented in Tables 4 and 5.The desirability of the optimized compressive strength, color contour configurations and the response surface optimized configuration are presented in Figs. 15, 16, 17, 18 and 19.
(     are scaled to accommodate the units of each factor and the intercept is not at the center of the design space.The above analyses are presented in Tables 6 and 7, Figs. 20, 21, 22 and 23, and Tables 8 and 9, while the desirability of the optimized flexural strength, color contour configurations and the response surface optimized configuration are presented in Figs. 24, 25, 26, 27 and 28.

Response surface methodology analysis for the slump
Factor coding is actual.Sum of squares is Type III-Partial.www.nature.com/scientificreports/intercept in an orthogonal design is the overall average response of all the runs.The coefficients are adjustments around that average based on the factor settings.When the factors are orthogonal the VIFs are 1; VIFs greater than 1 indicate multi-colinearity, the higher the VIF the more severe the correlation of factors.As a rough rule, VIFs less than 10 are tolerable.The equation in terms of actual factors can be used to make predictions about the response for given levels of each factor.Here, the levels should be specified in the original units for each factor.This equation should not be used to determine the relative impact of each factor because the coefficients  are scaled to accommodate the units of each factor and the intercept is not at the center of the design space.
The above analyses are presented in Tables 10 and 11, Figs. 29, 30, 31 and 32, and Tables 12 and 13, while the desirability of the optimized flexural strength, color contour configurations and the response surface optimized configuration are presented in Figs.33, 34, 35 and 36.(9)             intercept in an orthogonal design is the overall average response of all the runs.The coefficients are adjustments around that average based on the factor settings.When the factors are orthogonal the VIFs are 1; VIFs greater than 1 indicate multi-colinearity, the higher the VIF the more severe the correlation of factors.As a rough rule, VIFs less than 10 are tolerable.The equation in terms of actual factors can be used to make predictions about the response for given levels of each factor.Here, the levels should be specified in the original units for each factor.This equation should not be used to determine the relative impact of each factor because the coefficients are scaled to accommodate the units of each factor and the intercept is not at the center of the design space.The above analyses are presented in Tables 14 and 15            www.nature.com/scientificreports/

Metaheuristic models and sensitivity analysis
The GWO, MVO, MFO, PSO, and WOA are the metaheuristic techniques applied in the optimization of the compressive strength, flexural strength, and slump of the SHC and these results are compared with the baseline regression; multilinear regression (MLR).Table 18 shows the detailed prediction performance evaluation of the metaheuristic models alongside the MLR.The performance indices used as shown in Table 18 are the VAF, MSE, RMSE, MAE, CE, and R 2 .In Table 19, the performance indices are ranked in a score analysis with respect to the outputs modeled in this extensive exercise.The score analysis was conducted to identify the most suitable model for each output of the dataset, both during the training and testing phases.To achieve this, a score of "n" was assigned to each model, where "n" represents the total number of proposed models (6 in this case).This score was used to determine the optimal value for each performance indicator.The models were subsequently ranked based on their individual performance indices, which are elaborated in Table 19 and Fig. 48.By aggregating the training and testing scores, an overall score was calculated for each model, providing a comprehensive evaluation of its performance.It can be shown that MVO is ranked 1 st in the prediction of the CS and FS of the SHC, while GWO is ranked 1 st in the prediction of the Sl of the SHC.Comparatively, it can be deduced from previous literature 49 , that the presented metaheuristic techniques present models, which have performed better than the novel ANN used previously.

Sensitivity analysis
The study employed sensitivity analysis to determine the relative influence of each parameter on the output within the model, utilizing the cosine domain method as proposed by Yang and Zang 50 .In order to implement this method, all data pairs were transformed into a shared X-space.To facilitate this technique, it was necessary to construct a data array X by incorporating all available data pairs according to the following procedure 51,52 :   Each of the elements, xi, in the data array X is a vector of lengths of m, that is: The strength of the relation between the dataset, xi and xj, is presented as follows: (12) X = {x 1 1, x 2 2, x 3 3, . . ., x im }  The results as presented in Fig. 49 show that the FA is more sensitive than B, which followed closely to the behavior of the compressive strength (CS) of the SHC with an insignificant difference.For the flexural strength (FS), the cement shows to be more influential and again followed by the bacterial concentration (B).For the  Surface Methodology (RSM).The concrete parameters considered in addition to the bacteria concentration in this model exercise include cement, fine aggregate, coarse aggregate, and water-cement ratio and these were utilized as input variables to predict the outputs; compressive strength, flexural strength and the slump.The performance of the models was also tested by using the coefficient of determination (R 2 ), root mean squared errors (RMSE), mean absolute errors (MAE), mean squared errors (MSE), variance accounted for (VAF) and the coefficient of error (CE).The following can be concluded; • The classified metaheuristic techniques outclassed the RSM due their ability to mimic human and animal genetics of mutation providing highly acceptable values of R 2 and error metrics.• The GWO outclassed the other methods in predicting the concrete slump (Sl) with R 2 of 0.998 and 0.989 for the train and test, respectively.• The PSO outclassed the rest in predicting the flexural strength (FS) with R 2 of 0.989 and 0.937 for train and test, respectively.• The MVO outclassed the others in predicting the compressive strength (CS) with R 2 of 0.998 and 0.958 for train and test, respectively.• The CS, FS, and Sl model sensitivity analysis shows that the bacterial concentration (B) showed to be the second most influential parameter in the production design and behavior of the SHC and should be taken as one of the major decisive constituents needed to produce a reliable healing potential in concrete.• Overall, the GWO, PSO, and MVO having performed within acceptable limits are considered superior to the other models.• The RSM did not generate R 2 value rather the adequate precision in computation has been used to judge its ability to predict the mechanical properties of the bacterial-inspired self-healing concrete, which is considered adequate above 7.0.It further proposed closed-form polynomial relationships between the regressors and the outputs, which can be applied manually in the prediction of the SHC mechanical properties.

Figure 2 .
Figure 2. Violin plot of outputs and effective parameters.

Figure 3 .
Figure 3.The correlation matrix of inputs.
www.nature.com/scientificreports/Responsesurface methodology analysis for the flexural strengthFactor coding is actual.Sum of squares is type III-partial.The model F-value of 7.83 implies the model is significant.There is only a 0.22% chance that an F-value this large could occur due to noise.P-values less than 0.0500 indicate model terms are significant.In this case A, B, C, D, AB, AC are significant model terms.Values greater than 0.1000 indicate the model terms are not significant.If there are many insignificant model terms (not counting those required to support hierarchy), model reduction may improve your model.The lack of fit F-value of 1.43 implies the Lack of Fit is not significant relative to the pure error.There is a 34.68% chance that a Lack of Fit F-value this large could occur due to noise.Non-significant lack of fit is good-we want the model to fit.

Figure 15 .
Figure 15.Desirability level of the selected optimized solution.

Figure 17 .
Figure 17.Compressive strength model interaction between the parameters.

Figure 18 .
Figure 18.Compressive strength model contour for the concrete mixtures.

Figure 19 .
Figure 19.3D configuration of the compressive strength model.

Factor
coding is actual.Sum of squares is Type III-Partial.The Model F-value of 3.40 implies the model is significant.There is only a 3.73% chance that an F-value this large could occur due to noise.P-values less than 0.0500 indicate model terms are significant.In this case B, AB are significant model terms.Values greater than 0.1000 indicate the model terms are not significant.If there are many insignificant model terms (not counting those required to support hierarchy), model reduction may improve your model.The Lack of Fit F-value of 0.96 implies the Lack of Fit is not significant relative to the pure error.There is a 50.20% chance that a Lack of Fit F-value this large could occur due to noise.Non-significant lack of fit is good-we want the model to fit.Adeq Precision measures the signal to noise ratio.A ratio greater than 4 is desirable.Your ratio of 6.406 indicates an adequate signal.This model can be used to navigate the design space.The coefficient estimate represents the expected change in response per unit change in factor value when all remaining factors are held constant.The

Figure 29 .
Figure 29.Slump model predicted versus actual values scatter plot.

Figure 39 .
Figure 39.Splitting tensile strength model residual versus cement optimized plot.

Figure 40 .
Figure 40.Splitting tensile strength model actual versus predicted optimized scatter plot.

Figure 48 .
Figure 48.Comparing the results based on their rank.

Table 1 .
Descriptive statistics of outputs and influential factors.

Table 3 .
Fit Statistics for the compressive strength model.

Table 4 .
Constraints of compressive strength model.

Table 5 .
Selected solution out of the 100 solutions found.

Table 7 .
Fit Statistics for the flexural strength model.

Table 8 .
Constraints for the flexural strength model.

Table 9 .
Selected optimized solution from the 100 solutions found.

Table 11 .
Fit statistics for the concrete slump (Sl).

Table 12 .
Selected optimized slump model from 100 Solutions found.

Table 13 .
Constraints for the slump model.

Table 15 .
Fit statistics for the concrete splitting tensile strength.

Table 16 .
Constraints for the concrete splitting tensile strength model.

Table 17 .
Selected optimized STS solution from 100 solutions found.

Table 18 .
The models' prediction performance evaluation.