# Prediction of Blast Crushing Lumpiness Based on CPO-BP Modeling
## 1 Introduction  
In mining, road construction, water conservancy, hydropower, and tunnel construction, blasting technology plays a vital role as an efficient method of rock breakage and cannot be ignored. In the blasting operation, the rock mass distribution after crushing directly determines the efficiency of the subsequent excavation, loading, and transportation, which is a measure of the core indicators of the effectiveness of blasting. Accurate prediction of the blasting block size not only helps optimize the design of blasting parameters and reduce construction costs but also significantly enhances project safety and economic performance. Since the 1930s, extensive research has been conducted to establish empirical models for predicting blast fragmentation size. Among the earliest, [PERSON] and [PERSON] [1] proposed a semi-empirical formula describing the relationship between particle size and the cumulative percentage of screen residue, providing a foundational approach to estimate rock fragmentation. Subsequently, [PERSON] [2] expanded this model by incorporating explosive characteristics and burden-spacing parameters, although it was later noted that the model may underestimate fragmentation in certain geological settings. Building on these approaches, [PERSON] [3] developed the Kuz-Ram model, which integrates Rosin-Rammler distribution theory with the Kuznetsov formula, and remains widely used for blast design optimization. However, its predictive capability is limited in heterogeneous rock masses where structural features, such as joints and fractures, significantly affect fragmentation outcomes [4]. To address these limitations, more physically based models have been proposed. For instance, the BM-MC model developed by [PERSON] et al. [5] is grounded in stress wave propagation and energy absorption theory, providing a more mechanistic understanding of fragmentation. Similarly, [PERSON] [6] introduced [PERSON]'s energy theory into benchblasting applications, highlighting the energy-fragmentation relationship, though this model does not fully account for discontinuities in rock masses. Later models, such as the crushed zone model (CZM) by [PERSON] et al. [7] and the Kuznetsov Cunningham Ouchterlony (KCO) model by [PERSON] et al. [8], further refined the classification of coarse and fine fragmentation and improved predictive accuracy by integrating updated empirical relationships and field observations. Despite these advancements, traditional models generally struggle to capture the complex, nonlinear interactions between geological conditions and blast design parameters. With the rise of artificial intelligence and machine learning (ML) techniques, researchers have shifted toward data-driven predictive models. Techniques such as multiple regression analysis (MRA) and artificial neural networks (ANN) [9] support vector machines (SVM) and genetic algorithms (GA) and particle swarm optimization (PSO) have been widely applied to blasting data modeling [10; 11]. Notably, [PERSON] et al. [12] utilized MRA to incorporate joint characteristics, explosive energy, and hole geometry into a blast fragmentation model. Similarly, [PERSON] et al. [13] employed a support vector regression (SVR) model to outperform ANN and MRA methods in terms of prediction accuracy and speed, though parameter tuning remained a challenge. In parallel, researchers have explored more complex models incorporating rock structural parameters, Monte Carlo simulations, and heuristic optimization. [PERSON] et al. [14] used fractal theory and random simulations to predict initial block size distributions. [PERSON] et al. [11] combined the firefly algorithm with ANN to forecast fragmentation in iron ore blasting [15], though the model exhibited signs of overfitting due to excessive learning capacity. Recent works by [PERSON] et al. [16] and [PERSON] et al. [17] integrated feature selection (e.g., via random forests) with neural networks (e.g., RBF, GA-BP), yielding higher prediction accuracy than standalone BP networks.