The Phi-Four (also embodied as $ \phi^4 $) equation (PFE) is one of the most significant models in nonlinear physics, that emerges in particle physics, condensed matter physics and cosmic theory. In this study, propagating soliton solutions for the PFE were obtained by employing the extended direct algebraic method (EDAM). This transformational method reformulated the model into an assortment of nonlinear algebraic equations using a series-form solution. These equations were then solved with the aid of Maple software, producing a large number of soliton solutions. New families of soliton solutions, including exponential, rational, hyperbolic, and trigonometric functions, are included in these solutions. Using 3D, 2D, and contour graphs, the shape, amplitude, and propagation behaviour of some solitons were visualized which revealed the existence of kink, shock, bright-dark, hump, lump-type, dromion, and periodic solitons in the context of PFE. The study was groundbreaking as it extended the suggested strategy to the PFE that was being aimed at, yielding a significant amount of soliton wave solutions while providing new insights into the behavioral characteristics of soliton. This approach surpassed previous approaches by offering a systematic approach to solving nonlinear problems in analogous challenging situations. Furthermore, the results also showed that the suggested method worked well for building families of propagating soliton solutions for intricate models such as the PFE.

This study presents a new approach to optimize the dynamic evacuation process through a dynamic traffic assignment model formulated using mixed-integer linear programming (MILP). The model approximates the travel time for evacuee groups with a piecewise linear function that accounts for variations in travel time due to load-dependent factors. Significant delays are transferred to subsequent groups to simulate delay propagation. The primary objective is to minimize the network clearance time—the total time required for the last group of evacuees to reach safety from the start of the evacuation. Given the model’s computational intensity, a simplified version is introduced for comparison. Both the original and simplified models are tested on small networks and benchmarked against the Cell Transmission Model, a well-regarded method in dynamic traffic assignment literature. Additional objectives, including average travel time and average evacuation time, are explored. A sensitivity analysis is conducted to assess how varying the number of evacuee groups impacts model outcomes.

The integration of hybrid renewable energy systems (HRES) has introduced both opportunities and challenges in managing multisource power systems such as wind and solar. Accurate forecasting of HRES performance is critical to efficient planning and grid stability. This paper proposes a data efficient hybrid framework that combines Grey Wolf Optimization (GWO) for feature selection with Deep Belief Networks (DBN) for predictive modeling. GWO effectively selects relevant features from high dimensional environmental and system parameters, reducing computational burden and enhancing learning performance. The DBN is then trained on the optimized input set to forecast system performance. Two public datasets capturing wind and solar power production across distinct geographic conditions were used for validation. The proposed model significantly outperforms conventional methods, achieving a mean square error of 0.0207, RMSE of 0.144, and an energy efficiency of 98.32%. These results demonstrate the framework’s potential for deployment in smart grid forecasting environments.

Many industrial companies continue to face global uncertainties in demand and failures in supply. The main purpose of this research is to design and optimize a supply chain network (SCN) that performs under completely uncertain environment. In this paper, three advanced meta-heuristic algorithms based on Broyden-Fletcher-Goldfarb-Shanno (BFGS), POWELL, and Non-dominated Sorting Genetic Algorithm (NSGA-II) are used to solve the optimization problem. A real-life case study for a steel manufacturing integrated supply chain is used to demonstrate the efficiency of the model and the solutions obtained by meta-heuristic algorithms. The objective was to maximize the total profit of supply chain network under disruption conditions. The presented mathematical modeling provides an understandable overview of the system for managers to make appropriate decisions to achieve the maximum profit. Findings revealed that advanced meta-heuristic algorithms were the most efficient technique to solve the proposed model when compared with the traditional method.