Introduction
During natural disasters such as earthquakes, floods, and storms, or man-made disasters such as explosions and fires, rapid and efficient response from relief forces is necessary to save human lives and minimize the damage caused by the disaster. One of the most important limitations crisis managers face is the need to effectively cover relief bases and ensure timely and efficient aid to affected areas. To enhance the efficiency of search and rescue operations and optimize resource management, determining the appropriate location for relief bases and the correct and optimal routing of relief teams is particularly important. The location-routing problem (LRP) during disasters is thus a fundamental and highly significant challenge for relief management in the affected areas. Furthermore, risk analysis and the assessment of potential threats in different areas have been considered as an important part of the decision-making process for locating. Addressing this issue can play a crucial role in improving crisis management and increasing the efficiency of relief systems in emergencies.
To respond to the complexities of the LRP and the possibility of solving it on a large scale, the use of mathematical algorithms is not enough, and there is a need to utilize metaheuristic methods. In this study, an innovative two-echelon model for the LRP is developed, which specifically considers the spatial coverage constraint; i.e. each relief base is allowed to provide services within a designated service radius only. The main goal of this study is to determine the optimal locations for relief bases and simultaneously design suitable routes for relief teams so that it can significantly reduce the total time of relief operations and the related costs. The proposed model also considered the risk analysis and the assessment of potential threats in different areas. In this research, an advanced genetic algorithm (GA) is developed to provide appropriate and efficient solutions for the two-echelon LRP. Inspired by natural evolution and natural selection processes, this algorithm gradually finds optimal or near-optimal solutions by utilizing operations such as selection, crossover, and mutation. 

Methods
We combined the covering tour problem (CTP) with the two-echelon LRP to propose a model named “two-echelon relief covering tour location routing problem” (2E-RCTLRP). To evaluate the accuracy and efficiency of the proposed model and the developed algorithm, five small-scale sample problems were initially designed, and the optimal solutions to them were obtained using GAMS software, version 24.1.3 which has the capability to solve exact optimization problems. The results obtained from the GA were then compared with the exact GAMS results to evaluate the algorithm’s efficiency, accuracy, and convergence rate. The proposed algorithm is designed to allow control over various parameters such as population size, mutation rate, and crossover rate, thereby providing the necessary flexibility to adapt to the different characteristics of the problem. Sensitivity analysis was also performed on various model variables, which were examined in terms of the CTP. The risk analysis was also conducted to identify the locations with higher levels of threat.

Results
The results obtained from the implementation of the developed GA demonstrated the very high efficiency of this method in achieving near-optimal solutions, even similar to the exact solutions obtained from GAMS. These findings indicated the rapid convergence of this algorithm and its ability to manage large problem dimensions. The results of the sensitivity analysis showed that in many cases, the use of two-echelon models led to significantly better results compared to single-level models. In two-echelon models, it is possible to model the decision-making structure more accurately, consider real operational constraints, and better account for spatial and temporal complexities, which is of great importance in emergencies. Furthermore, comparing the solution for the non-synchronization of tours at two levels with the proposed model showed that the proposed approach not only improved the performance but also led to optimal resource management and cost reduction. Based on the risk analysis, the locations with higher threat levels were identified, based on which appropriate planning for resource allocation and base deployment was carried out. Thus, the proposed model comprehensively considers all critical aspects of relief operations in a crisis and provides effective operational solutions.

Conclusion
The proposed two-echelon model and the developed GA can efficiently optimize the locating and routing of relief teams during disasters. The proposed model can provide better decisions for the deployment of relief bases by considering the real constraints of search and rescue operations and incorporating risk and threat analyses, thereby accelerating and improving the relief process. Using metaheuristic methods in large problem dimensions enables the solution of more realistic and operational problems, which can help strengthen crisis management infrastructure in the face of natural disasters. The results of this research can serve as a basis for the development of decision support systems in the field of crisis management. By improving the speed, accuracy, and efficiency of decision-making, the proposed model can significantly contribute to saving lives and property in critical situations. Paying attention to two-level approaches in modeling relief problems, especially in combination with intelligent optimization techniques, is a forward-looking and effective solution for managing crises. The application of these approaches can not only be effective in optimizing relief operations but also improve urban planning, enhance the resilience of areas to crises, and increase the preparedness of relief organizations. Combining metaheuristic algorithms with machine learning methods can enhance the LRP model’s capacity to respond to multidimensional crises and complex scenarios.

Ethical Considerations
Compliance with ethical guidelines
All ethical principles were considered in this study. Since there was no experiment on human or animal samples, the need for an ethical code was waived.

Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Authors' contributions
All authors contributed equally to the conception and design of the study, data collection and analysis, interpretation of the results, and drafting of the manuscript. Each author approved the final version of the manuscript for submission.

Conflicts of interest
The authors declared no conflict of interest.

 

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