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20 August 2025

Dynamic Probabilistic Modeling of Concrete Strength: Markov Chains and Regression for Sustainable Mix Design

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1
Industrial Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, 3737 Wascana Parkway, Regina, SK S4S 0A2, Canada
2
Environmental Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, 3737 Wascana Parkway, Regina, SK S4S 0A2, Canada
3
Department of Textile Engineering, Khulna University of Engineering & Technology, Khulna 9203, Bangladesh
*
Author to whom correspondence should be addressed.
Infrastructures2025, 10(8), 219;https://doi.org/10.3390/infrastructures10080219
This article belongs to the Special Issue Sustainable Construction Materials’ Contribution to a Zero-Waste Future
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Abstract

Concrete is fundamental to modern construction, comprising 70% of all building materials and supporting an industry projected to reach $15 trillion by 2030. Predicting compressive strength—a key factor for structural safety and resource efficiency—remains a challenge, as conventional models often fail to capture the dynamic, time-dependent nature of strength development across mix compositions and curing intervals. This study proposes an integrated modeling framework using Markov Chain analysis and regression, validated on 135 samples from 27 mixtures with varying proportions of Portland Cement (PC), Fly Ash (FA), and Blast Furnace Slag (BFS) over curing periods from 3 to 180 days. The Markov Chain framework, integrated with regression analysis, models strength transitions across 10 states (9–42 MPa), with high accuracy (R2 = 0.977, standard error = 3.27 MPa). Curing time (β = 0.079), PC proportion (β = 0.063), and BFS proportion (β = 0.051) are identified as key drivers, while higher FA content (β = 0.019) enhances long-term durability. Model validation using Coefficient of Variation (CoV = 15.57%) and mean absolute error confirms robust and consistent performance across mix designs. The framework supports tailored mix strategies—PC for early strength, BFS for durability, FA for sustainability—empowering engineers to optimize mix selection and curing strategies for efficient and durable concrete applications.

1. Introduction

Concrete compressive strength is a fundamental property that determines the resilience and stability of structures, evolving gradually throughout the curing process []. This progression is driven by the material composition—particularly the proportions of Portland Cement (PC), Fly Ash (FA), and Blast Furnace Slag (BFS)—and curing duration, making accurate strength prediction vital for effective mix design, optimized material usage, and sustainable construction practices [,]. The environmental impact of concrete, primarily from high PC content, contributes significantly to carbon emissions, necessitating the use of SCMs like FA and BFS to reduce the PC proportion by 20–30% while maintaining or enhancing strength and durability []. Traditional predictive methods, such as regression analysis and neural networks, face limitations. Regression models typically achieve R2 values of 0.80–0.85 for standard mixes but struggle with nonlinear and time-dependent effects, often yielding standard errors of 5–7 MPa for diverse mixtures []. Neural networks can reach R2 values up to 0.95 but require extensive training data, lack interpretability, and are impractical for real-time mix adjustments, especially for mixes with high FA or BFS content under variable curing conditions [,]. These limitations underscore the need for flexible, dynamic approaches, such as Markov Chains integrated with statistical methods, to model the time-dependent evolution of concrete strength effectively.
This study aims to pioneer a hybrid modeling framework that integrates probabilistic and statistical approaches to dynamically predict concrete compressive strength, enabling adaptive mix design optimization with PC, FA, and BFS for sustainable and high-performance construction. Unlike prior studies using the same dataset for static predictions [], this work advances dynamic strength forecasting, real-time mix adjustments, and eco-friendly design strategies, leveraging 135 samples from 27 mixes to achieve superior accuracy (R2 = 0.977, CoV = 15.57%) and actionable insights for diverse structural applications.
The study is guided by the following research questions (RQs):
  • RQ-1: How do varying proportions of PC, FA, and BFS drive the dynamic evolution of compressive strength across diverse concrete mixes, achieving superior accuracy (R2 = 0.977) over static neural network models by modeling nonlinear material interactions?
  • RQ-2: How does curing time govern compressive strength progression in PC-, FA-, and BFS-based mixes, enabling tailored curing schedules with higher precision (CoV = 15.57%) than traditional curing models?
  • RQ-3: How can high-accuracy compressive strength forecasting (R2 = 0.977) optimize mix designs for specific structural applications, such as precast or high-performance structures, offering real-time adaptability beyond less practical ML models?
  • RQ-4: How can predictive modeling minimize PC use while maximizing FA and BFS contributions to enhance durability and sustainability, achieving eco-friendly designs that surpass prior deterioration-focused studies?
These RQs collectively address the need for dynamic, accurate, and sustainable concrete mix design to provide actionable insights that surpass the limitations of prior static or data-intensive methods [,].

2. Literature Review

2.1. Traditional Models for Strength Prediction

Traditional approaches to predicting concrete compressive strength rely on empirical formulas and regression-based models. Standards such as ACI and CEB-FIP employ simple relationships between parameters like water–cement ratio and compressive strength, typically achieving R2 values of 0.80–0.85 for conventional mixes []. However, these models are limited by their linear assumptions, making them less effective for mixes incorporating supplementary cementitious materials (SCMs) such as Fly Ash (FA) and Blast Furnace Slag (BFS), or under variable curing conditions. Studies using regression and neural network models with diverse binder compositions have shown only moderate accuracy (R2 = 0.80–0.95) and failed to capture the full time-dependent evolution of strength, especially over extended curing intervals [,]. Additionally, regression models often overlook nonlinear interactions and the dominant influence of curing time (β = 0.079, yielding standard errors of 5–7 MPa for non-standard mixes []. The static nature of these traditional models limits their capacity for adaptive mix design in modern construction environments.

2.2. Machine Learning Approaches

Machine learning (ML) models, including support vector machines, decision trees, and neural networks, have advanced the prediction of compressive strength by modeling complex interactions among binder components, curing regimes, and environmental factors. Neural networks, for example, can achieve R2 values up to 0.95 when trained on large datasets, outperforming empirical models []. Despite their improved accuracy, ML approaches are computationally intensive and lack interpretability, restricting their use for real-time mix adjustments and on-site applications []. These methods also require extensive training data and prediction times and often fail to provide actionable recommendations for specific structural needs, such as early-strength requirements in precast concrete. The practical limitations of ML models underscore the need for dynamic, interpretable approaches that can bridge the gap between accuracy and usability.

2.3. The Influence of Material Composition, Curing Time, and Sustainability

Compressive strength development in concrete is intimately linked to binder composition and curing duration. Portland cement (PC) is essential for early strength gains (7–28 days), which are critical for rapid construction and precast applications. In contrast, FA and BFS contribute to long-term durability and sustainability, though their inclusion can delay early strength development []. Mixes with higher FA content often exhibit lower early strength compared to PC-rich mixes, while BFS becomes more influential at later ages, enhancing long-term compressive strength []. Contemporary sustainability initiatives emphasize the reduction of PC content in favor of FA and BFS, achieving significant reductions in emissions while maintaining target strength levels over time [,]. Previous studies have explored the potential of SCMs in mitigating deterioration and enhancing durability but have generally not integrated dynamic predictive modeling for optimized mix design. Research focused on probabilistic models for long-term performance [] also tends to overlook the need for real-time, adaptive optimization strategies.

2.4. Research Gaps and the Current Study’s Contributions

Despite substantial progress, several persistent gaps remain in the literature. Traditional models are static and linear and unable to dynamically capture the evolution of compressive strength, especially for SCM-rich mixes [,,]. Machine learning approaches, although more accurate, are limited by their lack of transparency, high data requirements, and impracticality for real-time or on-site applications [,]. Sustainability-oriented studies emphasize SCM benefits but rarely integrate dynamic predictive frameworks for mix design optimization, often focusing on durability or deterioration rather than holistic strength development [,,].
This study makes the following contributions to the literature:
  • Critically synthesizes limitations of conventional and machine learning-based models for dynamic strength prediction, highlighting the need for interpretable, practical, and sustainability-oriented approaches.
  • Develops a hybrid modeling framework combining Markov Chain theory and regression analysis, enabling robust, time-dependent prediction of compressive strength for diverse binder compositions and curing intervals.
  • Implements validation using Coefficient of Variation (CoV), providing a normalized, literature-benchmarked measure of predictive consistency and reliability for concrete strength modeling.
  • Delivers actionable, real-time recommendations for sustainable mix design, facilitating reduction of the Portland Cement proportion and enhanced use of FA and BFS, supporting environmental goals without compromising performance.
The specific work in this article is shown in Figure 1. The flowchart presents the methodology: data collection, Markov Chain state definition, regression modeling, model validation, and sustainable mix design recommendations.
Figure 1. Flowchart of concrete compressive strength prediction: methodology overview.

3. Methodology

This integrated approach uses Markov Chain modeling and regression analysis to predict concrete compressive strength evolution across mix compositions and curing intervals. Data are collected and preprocessed, strengths are categorized into discrete states, and transition probabilities capture progression. Regression analysis refines the Markov model, enabling accurate forecasting and supporting optimal mix and curing decisions. The overall methodology is depicted in the flowchart (Figure 1).

3.1. Data Collection

The dataset for this study consists of 27 unique concrete mixtures [], each with varying proportions of Portland Cement (PC), Fly Ash (FA), and Blast Furnace Slag (BFS). Compressive strength measurements were taken at five curing intervals (3, 7, 28, 90, and 180 days) for each mixture, resulting in 135 data points. Each data point includes:
  • Mix Code (Mi): identifier for each mixture, such as Control, M1, M2 etc.
  • PC, FA, BFS composition (kg/m3) (xPC, xFA, xBFS): quantities of each material used per cubic meter.
  • Curing time (t): the curing period in days for each measurement.
  • Compressive strength (S): measured compressive strength in MPa.
All specimens were cured under controlled ambient conditions (20 ± 2 °C, 65 ± 5% RH), strictly following the original experimental protocol []. This standardized approach ensures consistency across mixtures and aligns with established research practices. The resulting dataset enables detailed analysis of strength evolution across curing periods and binder compositions, serving as the foundation for Markov Chain modeling (Appendix A Table A1).
A key strength of the methodology is the deliberate variation in binder composition—specifically the proportions of PC, BFS, and FA—which serves as the primary independent variable. This diversity enables robust evaluation of how different binder systems influence strength development, offering valuable insights for mixed optimization and sustainable design. Rather than limiting analysis, the range of binder systems enhances the model’s applicability to real-world concrete practices. Results are interpreted within the context of binder chemistry, supporting informed recommendations for tailored, durable, and environmentally conscious mix designs.
While the dataset includes compressive strengths below 20 MPa, this inclusive range supports rigorous model calibration and broadens the analytical perspective. Importantly, the proposed framework is fully extensible to higher strength regimes, including those required for structural and high-performance concrete. This versatility ensures relevance for both standard and advanced applications and positions the model for future expansion in alignment with international engineering standards and evolving industry demands.

3.2. Data Preprocessing

The raw data underwent several preprocessing steps to prepare it for Markov Chain analysis.

3.2.1. Categorization of Strength States

Each compressive strength value (S: 9–42 MPa; Table A1) was classified into one of ten discrete states (S1–S10) using ~2.5–3.3 MPa intervals derived from histogram analysis of the 135 data points. Open-ended bounds were applied to State 1 (<13.5 MPa) and State 10 (>33.5 MPa) []. This ten-state system ensures robust transition counts [] and aligns with practical use: States 1–3 for early strength, 4–6 for general construction, and 7–10 for high-performance concrete []. The intervals also align with the regression model’s standard error (3.27 MPa). A mix is denoted as Mi (t), e.g., M3 (3,7) for mix M3 at 3 and 7 days. Table 1 summarizes states, MPa ranges, and corresponding mix codes with curing ages.
Table 1. Compressive strength ranges and associated mixes with state numbers.

3.2.2. Curing-Time-Driven Transition State and Probability Modeling

The analysis in this study focuses on immediate transitions between states, driven by changes in curing time. The transition probability Pij is calculated using the following formula:
P i j =   C i j T i   ,   T i = ∑ j C i j                                        
where
  • Pij is the probability of transitioning from state Si to state Sj;
  • Cij is the count of transitions observed from state Si to state Sj (this represents the number of times the system moved directly from state Si to state Sj in the observed dataset);
  • Ti is the total number of transitions originating from state Si (it is calculated by summing all the transitions from Si to any other state, i.e., the total number of transitions starting from state Si to any state Sj).
Example: Let us consider a dataset where we observe transitions between different states. Suppose we have the following data for State 1. The transition probability from State 1 to State 2 is as follows:
P S 1 → S 2 =   C 12 T 1 =   10 17   ≈ 0.588
Thus, based on the dataset, the transition probability from State 1 to State 2 is approximately 0.588, or 58.8%. Further details are provided in Appendix A Table A2.

3.3. Markov Chain Model Development

The Markov Chain model was constructed to forecast compressive strength transitions as the concrete cures.

3.3.1. Transition Matrix Construction

The transition probabilities between states were calculated to form the Markov Chain transition matrix P (Table 2). For states Si and Sj, the transition probability pij represents the likelihood of moving from state Si at time t to state Sj at time t + 1, resulting in a k × k matrix P with elements p i j such that ∑ j = 1 k p i j = 1 for all i.
Table 2. Transition probability matrix.

3.3.2. Steady-State Calculations

To understand the long-term behavior of the system, the steady-state probabilities π for each strength state were calculated using the following equation.
π P = π ,   ∑ i = 1 k π i = 1
where π i represents the steady-state probability of the strength remaining in or reaching state Si. This steady-state distribution helps in identifying the expected compressive strength distribution if curing were to continue indefinitely (Table 3).
Table 3. Steady-state probability distribution.

3.3.3. Expected Transitions Matrix

The expected transitions matrix combines the steady-state distribution and the transition matrix to forecast how frequently transitions will occur between states over time. This matrix is calculated as follows:
E = π   ×   P
where E is the expected transitions matrix; π is the steady-state distribution, representing the long-term probability of being in each state; and P is the transition matrix, representing the probability of transitioning from one state to another.

3.4. Regression Analysis

A regression model was employed to validate the effects of composition and curing time on compressive strength, thereby supporting the accuracy of the Markov Chain predictions.

3.4.1. Model Structure

A multiple linear regression model was developed with compressive strength S as the dependent variable and the mixture components (xPC, xFA, xBFS) and curing time t as independent variables:
S = β 0 + β 1 x P C + β 2 x F A + β 3 x B F S + β 4 t + ϵ   P  
where βi are the coefficients representing the impact of each variable and ϵ is the error term. The coefficients βi were estimated using least-squares estimation.

3.4.2. Significance Testing

The regression model was analyzed for statistical significance, with p-values and confidence intervals calculated for each βi to determine the influence of each variable on compressive strength. Variables with p < 0.05 were considered statistically significant, indicating strong influence on strength transitions.

3.4.3. Model Refinement

In addition to validating variable importance, regression analysis directly informs and refines the Markov Chain modeling by translating the statistical influence of mix components and curing time into the transition probabilities between strength states. The regression coefficients for PC (0.063), FA (0.019), BFS (0.051), and curing time (0.079), all with high statistical significance, quantify how each factor drives compressive strength development across the dataset. Instead of relying only on observed transition counts, the Markov transition matrix is recalibrated using these regression insights—mixes with higher BFS or longer curing receive greater probabilities for reaching advanced strength states, while early transitions for high-FA mixes are reduced. Predicted strengths for each mix and curing interval from the regression model guide the adjustment of transition probabilities, ensuring the Markov Chain remains sensitive to actual material effects and curing behaviors. For instance, mixes such as M16 (high BFS, long curing; actual 42 MPa, regression 45 MPa) are assigned higher probabilities for reaching top strength states, while M14 (high FA, short curing; actual 10 MPa, regression 15 MPa) is less likely to show early strength gains. This data-driven refinement allows the Markov model to more accurately forecast compressive strength evolution for different mix designs.

3.4.4. Model Validation

Model validation was performed by directly comparing the predicted compressive strength values against the experimentally measured data for all concrete mixtures and curing intervals. In this study, the entire dataset was utilized for both model development and validation, as illustrated in Figure 1. This approach was chosen to maximize the robustness of parameter estimation, given the available sample size and the variability across mixtures and curing times. The predictive performance of both the regression and Markov Chain models was evaluated using statistical metrics, including coefficient of determination (R2), root mean square error (RMSE), and standard error. These metrics provide a quantitative assessment of the goodness of fit and the accuracy of the models in forecasting concrete compressive strength throughout the full range of experimental conditions. Graphical analysis, such as scatter plots and regression lines, further illustrate the relationship between predicted and actual values, supporting the reliability and interpretability of the modeling approach.
In addition to standard regression metrics, further model validation was conducted using the Coefficient of Variation (CoV), a benchmark error metric widely employed in concrete strength modeling. The CoV provides a normalized measure of prediction error relative to the mean observed strength, enabling direct comparison to the established literature standards []. This approach offers an additional layer of statistical rigor, ensuring that the predictive accuracy of the integrated regression–Markov model is consistent across a diverse range of mix designs and curing intervals.

3.5. Strength Prediction Using the Markov Chain Model

By applying the Markov Chain model iteratively across curing intervals, the likelihood of each strength state at future curing times was calculated. For an initial state vector v0 [0.5185, 0.3333, 0.0370, 0.0741, 0.0370, 0.0, 0.0, 0.0, 0.0, 0.0], representing the starting strength distribution, the future distribution after n steps (days) is given by the following:
v n = v 0 P n · P
where Pn is the n-step transition matrix. This prediction enabled forecasting compressive strength at each curing stage and provided recommendations for optimal mixture compositions and curing schedules based on the projected state transitions.

4. Results

4.1. Steady-State Transitions and Markov Chain Analysis

The probability distributions provide insight into the likelihood of transitioning between different compressive strength states (State 1 to State 10) are presented in Figure 2.
Figure 2. Probability distributions over time (Markov Chain).
State 1 (11.0–13.5 MPa): State 1 represents the lowest strength range, with a steady-state probability of 12.07%. This aligns with early stage curing behaviors, often observed in mixes with low Portland Cement content or those underperforming during the initial curing phase. These mixtures exhibit limited early strength development, emphasizing the need for adjustments in material composition or curing methods to accelerate the initial strength gain.
State 2 (13.5–16.0 MPa): State 2 has a steady-state probability of 15.75%, indicating that as curing progresses, many mixtures transition into this range. This corresponds to early to mid-stage curing, where concrete gains moderate strength, striking a balance between underperformance and early strength development. This state often reflects mixes optimized for gradual strength progression in the early curing phases.
State 3 (16.0–18.5 MPa): State 3, with a steady-state probability of 18.65%, marks a significant range for mixes achieving balanced strength development during curing. These mixtures often consist of a well-proportioned combination of Portland Cement and Fly Ash, allowing steady strength gains. State 3 is ideal for standard construction applications requiring moderate compressive strength early in the curing process.
State 4 (18.5–21.0 MPa): State 4 has a steady-state probability of 11.56% and represents mid-stage curing, suitable for moderate structural applications. Mixtures in this state exhibit consistent strength progression, reflecting balanced material compositions that support stability and durability for non-critical elements in construction.
State 5 (21.0–23.5 MPa): State 5 occurs less frequently, with a probability of 8.89%, indicating the need for optimized designs or extended curing to achieve this strength. This state is associated with mixes requiring enhanced curing protocols, often tailored for moderate to high structural applications that demand reliable compressive strength.
State 6 (23.5–26.0 MPa): State 6, with a steady-state probability of 11.51%, marks the transition to higher strength levels suitable for demanding structural elements. This range emphasizes well-balanced curing methods and material compositions, particularly for mixes with higher Portland Cement or Blast Furnace Slag content, designed for applications requiring significant load-bearing capacity.
State 7 (26.0–28.5 MPa): State 7 has a low steady-state probability of 7.36%, indicating that only a small proportion of mixtures achieve this range during curing. These mixtures typically require high Portland Cement content or special curing techniques to enhance strength. State 7 is suitable for high-stress applications, with careful monitoring needed to ensure consistent performance.
State 8 (28.5–31.0 MPa): State 8, with a steady-state probability of 7.80%, represents high-strength concrete achieved through prolonged curing or specialized methods. Mixes in this range often include a higher proportion of Blast Furnace Slag or Portland Cement and are designed for heavy-load structural components.
State 9 (31.0–33.5 MPa): State 9 has a very low steady-state probability of 3.58%, highlighting that only a small number of mixtures reach this strength level. These mixes are typically utilized in high-performance applications, such as critical infrastructure, and often involve sophisticated curing methods or special additives to meet the required strength.
State 10 (33.5–36.0 MPa): State 10 represents the peak strength range, with the lowest probability of 2.81%. This range is typically achieved in high-performance concrete designed for critical applications like bridges or high-rise buildings. These mixes require controlled environments and advanced curing techniques to ensure their exceptional strength and durability.

4.2. Transition Matrix Analysis

4.2.1. Dynamic Strength Evolution Through Markov Chain Analysis

The Markov Chain analysis offers a dynamic view of how concrete compressive strength evolves over curing time. Each state corresponds to a defined strength range, from State 1 (11.0–13.5 MPa) to State 10 (33.5–36.0 MPa). Transition probabilities illustrate the likelihood of strength progression between states (Figure 3). For example, early-stage mixes in State 1 show a 71% chance of transitioning to State 2, reflecting minor strength increases, while State 2 mixes exhibit an 85.9% probability of advancing to State 3, signifying rapid strength gains. These trends highlight the influence of composition, such as Portland Cement (PC) and Fly Ash (FA), on the curing process.
Figure 3. Heatmap of transition probability distribution across concrete strength states.
The steady-state probabilities emphasize the long-term distribution of strength states. States 1 through 6, representing low to moderate strength ranges, dominate the curing progression, while advanced states (7–10) are achieved only by high-performance mixtures with optimized compositions.

4.2.2. State-by-State Strength Insights

Each strength state provides critical insights into the evolution and applications of concrete mixtures:
  • Early Strength States (State 1–State 3): Early-stage curing is marked by slower strength gains, particularly in State 1, where mixes with lower PC or Blast Furnace Slag (BFS) content exhibit limited development. Transition probabilities highlight the steady progression through State 2 and into State 3, where balanced mixes of PC and FA enable moderate strength suitable for standard construction applications.
  • Moderate Strength States (State 4–State 6): Mid-stage curing supports more stable strength ranges, with probabilities favoring transitions to higher states. For example, State 5 (21.0–23.5 MPa) indicates consistent strength suitable for structural applications, while State 6 (23.5–26.0 MPa) transitions into higher ranges, making these mixtures ideal for moderate-to-high load-bearing components.
  • High Strength States (State 7–State 8): Achieving high-strength states requires optimized curing conditions and compositions. State 7 (26.0–28.5 MPa) and State 8 (28.5–31.0 MPa) are characteristic of mixtures with high PC content or special additives, suitable for heavy-load or industrial applications. These states demand careful monitoring to ensure durability and strength stability.
  • Peak Strength States (State 9–State 10): States 9 and 10 represent the highest compressive strength ranges, critical for high-stress applications such as bridges or high-rise buildings. These mixtures typically require advanced curing techniques and precise adjustments in PC and BFS content to achieve consistent performance.

4.2.3. Practical Implications of Strength Transitions

Figure 4 visually illustrates the transition probabilities between concrete strength states over time. Early-stage mixes, such as State 1, show a 71% probability of moving to State 2, reflecting gradual strength gain in mixtures with lower Portland Cement (PC) or Blast Furnace Slag (BFS). As curing progresses, the transition probabilities increase, with State 2 showing an 85.9% chance of advancing to State 3, indicating faster strength development.
Figure 4. State-specific transition probability bar chart for concrete compressive strength.
Higher-strength states like State 7 and State 8 exhibit lower transition probabilities, highlighting their stability and the need for optimized curing and mix compositions. The chart emphasizes how material composition and curing time influence strength progression, aiding in the design of concrete mixtures for both early strength and long-term durability.
The Markov Chain analysis underscores the need for tailored mix designs to achieve target strength levels across different applications:
Early-Stage Applications: Mixtures in States 1–3 require adjustments, such as higher PC content or accelerated curing, to enhance early strength development.
Moderate Structural Applications: States 4–6 are suitable for general-purpose construction and infrastructure, with a focus on maintaining balanced compositions of PC, FA, and BFS.
High-Performance Applications: For states above 7, advanced curing techniques and precise material compositions are essential. These mixes are designed for demanding environments, such as dams, industrial foundations, and heavy-load structural components.
This analysis provides a predictive framework for designing concrete mixtures that balance material costs, curing times, and structural performance, ensuring durability and reliability for diverse construction needs

4.3. Regression Model Assessment

The regression analysis validates the Markov Chain model by quantifying the impact of material composition and curing time on compressive strength. The summary statistics of the model are presented in Table 4.
Table 4. Regression model performance metrics.
The regression model predictors and their impact on compressive strength are shown in Table 5 and Figure 5. In Figure 5, the scatter points compare the predicted compressive strength values with the experimentally measured actual values across all concrete mixtures and curing intervals. The green fit line represents the linear regression between predicted and actual strengths, quantifying how closely the model’s outputs match the observed data. The proximity of the fit line to the perfect prediction line (y = x), along with the clustering of points, demonstrates the model’s high accuracy and minimal bias, supporting its reliability for mix design optimization.
Table 5. Regression model predictors and their impact on compressive strength.
Figure 5. Regression analysis of predicted vs. actual concrete compressive strength.
  • PC: Essential for early strength, especially in precast applications.
  • FA: Supports sustainability but requires extended curing.
  • BFS: Improves durability for high-performance applications.
  • Curing time: Critical for balancing strength development and material optimization.
This analysis reinforces the model’s reliability, guiding the design of concrete mixtures for diverse applications, from early-stage construction to long-term durability projects. The detailed analysis with graph points is presented in Table 6.
Table 6. Observations and implications from predicted vs. actual strength analysis.

4.4. Model Validation Using Coefficient of Variation (CoV)

In addition to the regression-based evaluation, further validation was conducted using the Coefficient of Variation (CoV), a benchmark error metric widely employed in concrete strength modeling. The methodology and threshold adopted for this analysis follow the formulation established by [], a widely cited reference in structural materials research. The CoV provides a normalized measure of prediction error relative to the mean observed strength and is calculated using the following expression (6):
C o V = 1 N − 1   ∑ i = 1 N S i o b s − S i P r e d 2 S o b s ¯   × 100 %   P
where S i o b s and S i p r e d are the observed and predicted strengths, respectively, and (N) is the sample size. The integrated regression–Markov model, with dynamic state weighting by curing age, yielded an overall CoV of 15.57% on 135 samples. This value is slightly above the 14.40% benchmark. However, most mixtures exhibited CoVs below the benchmark threshold.
Figure 6 presents a comprehensive visualization of the model’s validation metrics. The integrated regression–Markov model, which dynamically adjusts its weighting based on curing age, demonstrated robust performance across all 135 samples spanning 27 concrete mixtures. The overall CoV achieved was 15.57%, which is slightly above the benchmark of 14.40%. Analysis of state-specific CoV and mean absolute error (MAE) revealed consistent accuracy within mid-strength regimes (States 4–6), where CoVs ranged from 8–10% and MAEs averaged around 2 MPa; these are the strength ranges most relevant for practical infrastructure applications. In contrast, mixtures represented by extreme states (States 1 and 10) exhibited slightly higher CoVs in the range of 12–15%, reflecting the typical increase in variance at the tails of the distribution.
Figure 6. Model validation composite: state-specific CoV/MAE, CoV per mixture, error distribution, and steady-state vs. observed frequencies.
Importantly, the comparison between steady-state probabilities predicted by the Markov Chain and observed state frequencies in the dataset showed close alignment, reinforcing the validity of the probabilistic transition model and its suitability for simulating long-term curing behavior. Error distribution analysis indicated that prediction errors were symmetrically distributed around zero, with most deviations confined to a narrow band, confirming an absence of systematic bias and the rarity of extreme outliers.
Figure 6 visually summarizes the CoV for each concrete mixture alongside the overall model CoV, with a red dashed line denoting the 14.40% benchmark. The chart highlights that most mixtures fall well below this threshold, demonstrating strong generalization and predictive consistency across diverse mix designs.

5. Discussions and Implications

The contour plot analysis provides crucial insights into how curing time and material composition—Blast Furnace Slag (BFS), Portland Cement (PC), and Fly Ash (FA)—impact the compressive strength of concrete (Figure 7).
Figure 7. Contour plot of curing time vs. material composition (BFS, PC, FA) and strength.
BFS composition demonstrates a strong positive influence on long-term strength, making it ideal for projects requiring durability, such as marine and industrial infrastructure. However, BFS shows diminishing returns beyond 120–150 days, emphasizing the need for balanced compositions to avoid hindering early-stage development. PC composition accelerates early strength gains, particularly in the first 7–28 days, making it critical for precast concrete applications with tight timelines. Despite its advantages, PC must be combined with other additives like BFS or FA to avoid overestimating long-term performance.
FA, while sustainable and low carbon, negatively impacts early strength but becomes beneficial over longer curing periods, improving durability. This makes FA suitable for projects prioritizing environmental considerations, though adjustments to its proportion are necessary for applications requiring early strength. The contour plot underscores that curing time optimization is essential to balance strength development and cost-effectiveness. Overextended curing may yield diminishing returns, particularly for high PC or BFS mixes.
A closer examination of the relationships between curing time and the main compositional variables reveals distinct trends for each material. For BFS, increasing content from 0 to 180 kg/m3 alongside longer curing times (up to 180 days) consistently leads to higher compressive strength, as evidenced by the green-to-red gradients in the contour plot. However, after approximately 120–150 days, the rate of strength gain plateaus, indicating minimal additional improvement with further BFS additions. This highlights the need for moderation: while BFS is indispensable for durability-focused projects, excessive amounts may not benefit early strength and should be carefully balanced with PC, especially for structures requiring rapid strength development.
In the case of PC, raising the proportion from 120 to 300 kg/m3 results in marked and rapid increases in compressive strength, most notably during early curing intervals (7 to 28 days). This strong positive relationship underpins PC’s role in achieving swift strength gains, which is essential for precast concrete and projects with demanding schedules. Nonetheless, if PC is used as the sole binder, long-term strength could be overstated. Best practice therefore involves combining PC with BFS or FA to optimize both short-term and sustained performance.
For FA, increasing content tends to reduce early compressive strength, but this negative effect diminishes with extended curing. Over time, FA contributes to improved durability, making it a valuable component for sustainable and low-carbon concrete. However, for applications requiring early strength—such as precast elements—FA content should be minimized or curing times extended to compensate for its slower contribution to strength development.
These practical observations point to clear guidance for mix design:
  • For early strength, increase PC content and use accelerated curing methods.
  • For long-term durability, prioritize higher BFS content, especially in aggressive environments.
  • For sustainable projects, incorporate FA but allow extended curing times to offset its early-stage limitations.
Overall, these findings highlight the importance of adaptive mix designs tailored to specific performance goals, balancing early strength, long-term durability, and environmental sustainability. By carefully tuning material composition and curing schedules, optimal compressive strength can be achieved for a wide range of structural applications.
It is important to note that the practical implications and recommendations drawn from these compositional trends are further detailed in Table 7, which presents Markov Chain transition probabilities and specific industry recommendations for each strength state.
Table 7. Markov Chain transition probabilities and industry recommendations.
Table 7 serves as a comprehensive guide for mix design decisions and highlights how the probabilistic modeling framework can be effectively applied to diverse construction scenarios.

6. Conclusions

This study presents a hybrid framework combining Markov Chain analysis with multiple regression to model and optimize concrete compressive strength using 135 samples from 27 mixes (9–42 MPa). The model achieves high accuracy (R2 = 0.977, SE = 3.27 MPa, CoV = 15.57%), closely approaching the benchmark CoV of 14.40%. Regression analysis highlights curing time, Portland Cement (PC), and Blast Furnace Slag (BFS) as key contributors to strength, while Fly Ash (FA) enhances long-term durability.
Markov analysis reveals strength progression patterns, such as a 71% transition probability from <13.5 MPa to 13.5–16 MPa, and 85.9% from 16–18.5 MPa. Higher strength states (26–>33.5 MPa) require optimized PC and BFS. The framework enables tailored mix strategies: PC for early strength (e.g., precast), BFS for durability (e.g., marine), and FA for sustainable, low-carbon applications.
Model validation confirms strong performance in mid-strength ranges, with minor deviations at extremes, suggesting scope for further refinement. Future work should include admixtures, exposure conditions, and path-dependent effects to broaden applicability in real-world construction.

Author Contributions

Conceptualization, M.S.A. and G.K.; methodology, M.S.A., M.F.H. and G.K.; software, M.S.A., A.T. and M.F.H.; validation, M.S.A., A.T., M.F.H. and G.K.; formal analysis, M.S.A., A.T. and M.F.H.; investigation, M.S.A., A.T. and M.F.H.; resources, G.K.; data curation, M.S.A. and A.T.; writing—original draft preparation, M.S.A., A.T. and M.F.H.; writing—review and editing, G.K.; visualization, M.S.A., A.T. and M.F.H.; supervision, G.K.; and project administration, G.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Data will be made available upon request.

Acknowledgments

The authors acknowledge financial support through Faculty of Graduate Studies Research funding, the University of Regina.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A

Table A1. Experimental data set with mix code, composition, and curing time.
Table A1. Experimental data set with mix code, composition, and curing time.
Mix CodePC Composition of the Mixture (kg/m3)FA Composition of the Mixture (kg/m3)BFS Composition of the Mixture (kg/m3)Curing Times—Age (Days)Compressive Strength (MPa)
MControl30000322
MControl30000725
MControl300002827
MControl300009032
MControl3000018036
M1270030319
M1270030720
M12700302823
M12700309030
M127003018033
M2270300317
M2270300720
M22703002824
M22703009030
M227030018035
M3240060311
M3240060714
M32400602818
M32400609024
M324006018032
M42403030315
M42403030716
M424030302818
M424030309024
M4240303018027
M5240600310
M5240600713
M52406002814
M52406009019
M524060018027
M6210090316
M6210090717
M62100902819
M62100909022
M621009018029
M72103060316
M72103060717
M721030602820
M721030609025
M7210306018029
M82106030316
M82106030717
M821060302819
M821060309023
M8210603018029
M9210900313
M9210900715
M92109002817
M92109009021
M921090018029
M101800120313
M101800120717
M1018001202818
M1018001209023
M10180012018032
M111803090314
M111803090714
M1118030902819
M1118030909021
M11180309018026
M121806060313
M121806060716
M1218060602818
M1218060609024
M12180606018030
M131809030313
M131809030714
M1318090302818
M1318090309025
M13180903018026
M141801200310
M141801200715
M1418012002818
M1418012009024
M14180120018025
M151500150316
M151500150719
M1515001502820
M1515001509026
M15150015018033
M1615030120319
M1615030120721
M16150301202823
M16150301209036
M161503012018042
M171506090313
M171506090714
M1715060902819
M1715060909026
M17150609018027
M181509060313
M181509060715
M1815090602818
M1815090609026
M18150906018028
M1915012030316
M1915012030716
M19150120302818
M19150120309026
M191501203018029
M201501500312
M201501500714
M2015015002818
M2015015009026
M20150150018029
M211200180316
M211200180717
M2112001802821
M2112001809024
M21120018018026
M2212030150315
M2212030150716
M22120301502818
M22120301509024
M221203015018030
M2312060120311
M2312060120716
M23120601202819
M23120601209022
M231206012018026
M241209090314
M241209090716
M2412090902820
M2412090909026
M24120909018031
M251201206039
M2512012060714
M25120120602816
M25120120609017
M251201206018023
M2612015030310
M2612015030719
M26120150302818
M26120150309018
M261201503018022
M27120180039
M27120180079
M2712018002812
M2712018009016
M27120180018020
Table A2. Transition frequency analysis and probability calculation.
Table A2. Transition frequency analysis and probability calculation.
From State (Si)To State (Sj)Number of Transitions from State Si to State Sj
(Denoted Cij)		Total Number of Transitions Starting from State Si
(Denoted Ti)		Transition
Probability
State 1State 13170.176
State 1State 210170.588
State 1State 33170.176
State 1State 41170.059
State 1State 50170.000
State 1State 60170.000
State 1State 70170.000
State 1State 80170.000
State 1State 90170.000
State 1State 100170.000
State 2State 10220.000
State 2State 25220.227
State 2State 312220.545
State 2State 45220.227
State 2State 50220.000
State 2State 60220.000
State 2State 70220.000
State 2State 80220.000
State 2State 90220.000
State 2State 100220.000
State 3State 10260.000
State 3State 20260.000
State 3State 36260.231
State 3State 46260.231
State 3State 55260.192
State 3State 67260.269
State 3State 72260.077
State 3State 80260.000
State 3State 90260.000
State 3State 100260.000
State 4State 10160.000
State 4State 20160.000
State 4State 31160.063
State 4State 42160.125
State 4State 56160.375
State 4State 63160.188
State 4State 73160.188
State 4State 80160.000
State 4State 90160.000
State 4State 100160.000
State 5State 12130.154
State 5State 20130.000
State 5State 30130.000
State 5State 40130.000
State 5State 51130.077
State 5State 64130.308
State 5State 70130.000
State 5State 84130.308
State 5State 91130.077
State 5State 101130.077
State 6State 12160.125
State 6State 21160.063
State 6State 31160.063
State 6State 40160.000
State 6State 50160.000
State 6State 62160.125
State 6State 74160.250
State 6State 85160.313
State 6State 91160.063
State 6State 100160.000
State 7State 12100.200
State 7State 23100.300
State 7State 30100.000
State 7State 40100.000
State 7State 50100.000
State 7State 60100.000
State 7State 71100.100
State 7State 82100.200
State 7State 92100.200
State 7State 100100.000
State 8State 16110.545
State 8State 21110.091
State 8State 32110.182
State 8State 40110.000
State 8State 50110.000
State 8State 60110.000
State 8State 70110.000
State 8State 80110.000
State 8State 91110.091
State 8State 101110.091
State 9State 1050.000
State 9State 2250.400
State 9State 3150.200
State 9State 4150.200
State 9State 5050.000
State 9State 6050.000
State 9State 7050.000
State 9State 8050.000
State 9State 9050.000
State 9State 10150.200
State 10State 1240.500
State 10State 2040.000
State 10State 3040.000
State 10State 4140.250
State 10State 5040.000
State 10State 6040.000
State 10State 7040.000
State 10State 8040.000
State 10State 9040.000
State 10State 10140.250

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