Quick-Response Model for Pre- and Post-Disaster Evacuation and Aid Distribution: The Case of the Tula River Flood Event

Abstract

Background: In the context of humanitarian logistics, efficiently evacuating people from disaster-stricken areas is a complex challenge. This study focuses on the Tula River region in Hidalgo, Mexico, exploring the evacuation and support of individuals in temporary shelters. Despite the fact that the topic has been addressed in the literature, it is necessary to have quick response methods that can be used by decision-makers to adapt and utilize existing spaces as temporary shelters, in addition to knowing how to evacuate people. Methods: Addressing this void, a methodology to minimize evacuation and aid distribution costs is introduced. Leveraging existing algorithms, particularly Integer Linear Programming, the model determines shelter activation and utilizes the Vehicle Routing Problem to assess aid delivery strategies. Results: The research identifies optimal evacuation routes from 13 affected areas to 34 shelters and analyzes aid distribution costs under various demand scenarios: original, increased, and decreased by 10%, based on the number of transport units allocated and Google Maps distances. It also evaluates the costs associated with humanitarian aid distribution under varying collection strategies, involving state and municipal governments. Conclusion: This approach provides a decision-making foundation and can be adapted for similar analyses in other communities during extreme events.

1. Introduction

1.1. Problem Statement

Currently, hazard events pose a significant threat to humanity. According to the United Nations, 90% of extreme events are linked to climate change, indicating a potential increase in their frequency year after year [1]. These events, stemming from destructive origins, result in substantial economic impacts and human losses. In 2021, 401 disasters occurred worldwide, with the Pacific region bearing the brunt of these incidents [2]. Various types of hazard events exist, with earthquakes, droughts, storms, floods, volcanic activity, extreme temperatures, landslides, and forest fires among the most common. Among these, storms and floods were the most frequent, accounting for 223 out of the 401 disasters in 2021 [3]. As shown in Figure 1, floods ranked as the most common hazard events in Latin America and the Caribbean between 2000 and 2019 [4].

Due to the increased frequency of hazard events, the humanitarian supply chain (HSC) has gained significant attention in social, academic, and governmental circles due to its relevance in managing HSC operations, logistics coordination, traditional logistics, performance metrics, and modeling [5]. The primary aim of the HSC is to study various problems, such as facility location, aid distribution, inventory management, and mass evacuation of people, with the overarching goal of safeguarding affected individuals and providing essential resources for their survival.

Facility location pertains to the positioning of warehouses and distribution centers for emergency resources [6,7]. Distribution studies delineate the formation of the distribution network, the choice of transportation modes, and the required capacity to assist people [8,9]. Inventory-related issues revolve around determining resource levels, availability, cost minimization, and demand forecasting [10]. Additionally, addressing the mass evacuation of people encompasses shelter location, evacuation point concentration, transport selection, traffic management, and the time required for evacuation [11,12].

Despite the growing interest in the HSC, the pre-disaster phase has received relatively less attention. However, this phase is of great significance in operations and disaster management, particularly concerning the mass evacuation of people [6,13]. In the context of post-disaster decision-making, numerous proposals exist in the literature for optimizing the distribution of humanitarian aid using various modeling approaches. Nevertheless, they are often designed for diverse scenarios with numerous facilities, leading to complex decision-making processes and unresolved issues. Consequently, future research should explore the characterization of victims’ needs and the systematic examination of multilevel service networks [12].

Effective evacuation planning in disaster scenarios must account for both the pre-disaster and post-disaster phases, ensuring the efficient utilization of resources, including public transportation, and addressing the needs of individuals with mobility difficulties and disabilities [14,15,16,17,18]. Notably, various HSC initiatives have been undertaken in Mexico across different domains, such as food banks, vaccine distribution, and food and donation collection, using diverse solution approaches. This study focuses on a case study methodology for evacuating people in a flood scenario. It employs a mathematical optimization model utilizing integer linear programming (ILP). Additionally, this study utilizes the vehicle routing problem (VRP) to assess various strategies for delivering humanitarian aid.

In the State of Hidalgo, Mexico, the Tula River poses a continuous flood threat to the municipalities that it traverses. Although annual flood events are not uncommon in this region, one of the most severe floods in recent memory occurred on 6 September 2021, which was classified as a natural disaster. Unfortunately, this disaster claimed the lives of 15 people and left over 31,000 homes severely affected [19,20]. According to government authorities, the disaster was precipitated by heavy rains in Mexico City and the State of Mexico, leading to the saturation of the drainage system. As a result, the National Water Commission activated the Joint Operation Protocol for Deep Drainage of the Valley of Mexico, causing the Tula River to swell, overflow its banks, and significantly increase its flow. This overflow affected several municipalities, including Tula de Allende, Ixmiquilpan, Tlaxcoapan, Tezontepec de Aldama, Chilcuautla, Tasquillo, Tlahuelilpan, Tepeji del Río, and Mixquiahuala [21].

The drainage system of the metropolitan area in the Valley of Mexico is interconnected through a network of tunnels and canals, facilitating the discharge of water into the Mezquital Valley via the Tula River. However, this setup has adverse consequences for managing urban stormwater. The problem extends beyond just engineering considerations and encompasses a broader perspective that involves public policy [22]. Effectively managing the flood risk stemming from the drainage system of the metropolitan area of the Valley of Mexico requires a comprehensive approach. This approach should encompass geographical and interdisciplinary dimensions, including the construction of defense structures and diversion gates, river realignment, and dredging. Moreover, it should integrate additional measures aimed at reducing the severity of the damage, such as improved damage control and mitigation efforts. Public awareness and preparedness are crucial aspects, ensuring that the population is well-informed about flood risks and is aware of evacuation and safety zones to protect human lives [23].

1.2. Problem Description

This paper presents a comprehensive strategy to evacuate residents from the municipalities impacted by the Tula River prior to a disaster, and to offer humanitarian assistance in shelters after the disaster. This approach aims to minimize costs and collaboratively plan the essential resources, including determining the number and placement of shelters based on calculations of the affected population and establishing routes for delivering aid to these shelters. Importantly, this strategy is grounded in real-world data, incorporating distances and costs, which can serve as valuable decision-making tools for both state and federal governments.

As will be observed in Section 2, there are few works that provide real-world case studies combining both pre- and post-disaster stages. In fact, only Mollah et al. [24] work presents such a case, but their solution method is complex. Although the solution methods utilized in this work have been used before, the motivation here is to explore a straightforward approach that allows for the planning of which shelters to open, how to evacuate the most vulnerable individuals, and how to distribute humanitarian aid. Following a review of the state of the art, the most significant contributions can be summarized as follows:

• A simple decision-making model is proposed.
• Real demographic and social data are employed.
• Consideration is given to the most vulnerable individuals, specifically those over 60 years of age and people with physical disabilities.
• Both the pre- and post-disaster stages are taken into account.

The methodology involved data collection, building the evacuation model, solving the model, and generating options for aid distribution; the research methodology is detailed in Section 3. Section 4 describes the results and discussion, while Section 5 presents the conclusions. In the next section, the literature review is presented.

2. Literature Review

In recent years, there has been a substantial surge in publications pertaining to humanitarian supply chains (HSCs). These publications have addressed various key aspects, including facility location, distribution models, inventory management, and mass evacuation procedures. Furthermore, different mathematical models have been put forth for the evacuation of people, employing a range of algorithmic solutions tailored to the unique characteristics of the specific problem at hand. Within the context of HSCs, there exist conditions of uncertainty, particularly in relation to the demand for assistance. Addressing such uncertainty is crucial when devising evacuation models.

In the field of humanitarian logistics, various models have been developed to address diverse objectives. These objectives include the allocation of ambulances under uncertain conditions, the minimization of vehicle numbers, travel cost reduction, limiting maximum latency, optimizing shelter locations, and efficient distribution of aid. Additionally, challenges related to facility location, distribution, inventory management, and mass evacuation have been tackled using multi-objective optimization and multi-period distribution planning, often employing metaheuristic techniques like the Non-Dominated Sorting Genetic Algorithm III, simulated annealing (SA), and variable neighborhood search (VNS) due to the inherent complexity [24,25,26,27,28,29].

The humanitarian supply chain (HSC) is characterized by four distinct phases: prevention, preparedness, response, and reconstruction and recovery. Decisions made in the pre-disaster phase significantly influence those in the post-disaster phase. The location of distribution centers, the availability of inventory, and the positioning of shelters all play pivotal roles in the distribution of relief supplies and the evacuation of people. Concerning both the pre-disaster and post-disaster phases, problems related to stock prepositioning, facility placement, evacuation planning, and hospital and distribution center locations have been addressed. These complex challenges have been approached using various metaheuristics, including classical approaches (CAs), pattern search algorithms (PSAs), genetic algorithms (GAs), and Non-Dominated Sorting Genetic Algorithm III [15,28,29,30].

One of the most critical issues within the HSC domain is the efficient mass evacuation of people for providing timely care to victims. The literature offers proposals to minimize the costs of rescue operations while giving priority to injured individuals. These proposals also cover scheduling and assigning rescue teams, taking into account factors like fatigue and deprivation time. Metaheuristic methods, such as genetic algorithms (GAs) and particle swarm optimization (PSO), have been employed to solve these models [16,17,18].

Among the recent contributions, Shehadeh and Tucker [31] focused on determining the location of humanitarian aid warehouses in both the pre- and post-disaster phases, considering uncertainty, and positioning inventory. Seraji et al. [32] optimized the placement of emergency shelters and coordinated the movement of aid vehicles using two-stage multi-objective metaheuristics.

Table 1. Models proposed for the attention of natural disasters.

Author(s)	Problem	Phase (Pre- or Post- Disaster)	Solution Method (Approach)	Case Study
Babaei and Shahanaghi [25]	Multilevel location allocation and emergency routing problem in uncertain conditions	Post-disaster	Monte Carlo simulation and simulated annealing	No
Molina [26]	Multi-objective capacitated vehicle routing problem to deal with the lack of available vehicles and the need for quick evacuation	Post-disaster	MultiStart algorithm with intelligent neighborhood selection (NSGA-II)	No
Molla et al. [24]	Relief distribution during flooding and the evacuation of people	Pre- and post-disaster	Mixed-integer programming and genetic algorithms	Yes
Jha et al. [27]	Humanitarian relief chain that includes a relief goods supply chain and an evacuation chain in case of natural disaster	Pre- and post-disaster	Mixed-integer programming and NSGA-II	No
Doodman et al. [28]	Relief item prepositioning and multi-period distribution planning	Post-disaster	Multi-objective two-stage stochastic programming and TH method	No
Seraji et al. [32]	Location of shelters in the first phase and demands after accommodation in the second phase	Pre- and post-disaster	Two-stage stochastic programming	No
Zhu et al. [16]	Emergency relief routing optimization considering equity and priority issues	Post-disaster	Metaheuristic based on ant colony optimization	Yes
Mansoori et al. [18]	Minimize (a) the total number of people not transferred to hospitals and homeless not evacuated, and (b) total relief commodities	Post-disaster	Multi-objective mathematical model	Yes
Nayeri et al. [17]	Allocate and schedule rescue teams considering deprivation costs and times	Pre-disaster	Multi-objective mixed-integer programming using the LP method and a metaheuristic algorithm	No
Shehade and Tucker [31]	Determine where to open warehouses and how much inventory to preposition in each	Pre- and post-disaster	Two-stage stochastic programming and distributionally robust optimization (DRO) models	No
Madani et al. [33]	Multi-echelon and multi-objective relief network, location of hospitals, local warehouse hybrid centers, and evacuation of people	Pre- and post-disaster	Hybrid non-dominated sorting genetic algorithm and variable neighborhood search	No
Agarwal et al. [30]	Logistics stock prepositioning and evacuation planning	Pre- and post-disaster	Mathematical model for decision making with pattern simulated annealing and genetic algorithms	Yes
Seraji et al. [32]	Optimize the location of emergency shelters and coordinate the movement of relief vehicles considering distributive injustice and dissatisfaction	Pre- and post-disaster	Two stage multi-objective metaheuristics: multi-objective vibration damping and NSGA II	No

provides a summary of the articles found in the literature that explore approaches in both pre- and post-disaster scenarios, along with the solution methods, some of which have been applied in real-world cases.

Other works involving individuals with disabilities include the one by Yazdani and Haghani [34], where they presented a case study in Australia for evacuating the elderly using optimization algorithms combined with behavioral training. The same authors also introduced an emergency plan for relocating vulnerable individuals to shelters in response to heatwaves, presenting a method that they tested in a set of randomly generated locations [35].

While more methods with broader scope exist, involving larger networks and traffic modeling, and considering human behavior [36], in the case of the Tula River, the network only encompasses 50 possible shelters, 9 assembly points, and 2 distribution centers for aid, also without different route options. Thus, integer programming models are sufficient for making quick decisions. The concept of nearest and safest allocation, as outlined by Southworth [37] and Barrett et al. [38], is employed, and this planning is conducted in advance of flooding.

3. Research Methodology

The methodology to address the problem proposed in this work is composed of four stages: (1) data collection, (2) evacuation model, (3) model solution, and (4) distribution of humanitarian aid (Figure 2).

Step (1)—Data Collection: The initial phase involves complete data collection to establish a solid foundation of understanding of the problem for further analysis. The data collected cover the calculation of the population that requires evacuation, the identification of possible shelters, the location of distribution centers for humanitarian aid, and the calculation of distances between affected areas and shelters.

Step (2)—Evacuation Model: This step involves the formulation of the optimization model using integer linear programming. It begins with the calculation of transportation costs for the evaluation of people and the distribution of aid, as well as the calculation of the budget for the qualification of the shelters. These costs are the basis for the optimization of the model; subsequently, the variables involved in the mathematical model are established, and the objective function and the set of restrictions are formulated.

Step (3)—Model Solution: The third phase includes the search for tools to provide a solution to the proposed model and establish the procedure to perform the sensitivity analysis.

Step (4)—Distribution of Humanitarian Aid: This phase considers the use of the VRP technique to determine the collection routes for the distribution of humanitarian aid from the distribution centers to the shelters. In addition, the routes for the evacuation of people are determined and the time required for said evacuation is calculated.

3.1. Data Collection

In the initial phase, the municipalities situated closest to the Tula River were identified. Subsequently, the 13 municipalities that were deemed to be the most severely affected were chosen. The municipalities considered in the State of Hidalgo were Alfajayucan, Atitalaquia, Atotonilco de Tula, Ixmiquilpan, Mixquiahuala, Tasquillo, Tepeji del Río, Tezontepec de Aldama, Tlahuelilpan, Tlaxcoapan, Tula de Allende and Zimapán, and for the State of Mexico we considered the municipality Apaxco.

The method for estimating the number of people requiring evacuation relied on data sourced from the 2020 Population and Housing Census, which was conducted by the National Institute of Statistics and Geography (INEGI 2021) [39]. Notably, this analysis considered two specific groups: individuals aged 60 years or older, and those with physical disabilities, as these populations are the most vulnerable during a disaster. The calculation of the number of people to be evacuated for each municipality involved summing the percentages of people aged over 60 and people with physical disabilities, and then multiplying the total by the municipality’s overall population. Additional details can be found in

Table 2. Calculation of the evacuee population.

Municipality	Total Population	Pensioners or Retired People (%)	People with Disabilities (%)	Quantity to Evacuate
Alfajayucan	19,162	1.13	6.4	1475
Apaxco	39,898	6.1	4.9	3509
Atitalaquia	31,525	10	4.6	4603
Atotonilco de Tula	62,470	5.9	4.6	5559
Ixmiquilpan	98,654	5.1	5.6	10,556
Mixquiahuala	47,222	4.7	5.6	4864
Tasquillo	17,441	4	6.9	1901
Tepeji del Río	90,546	7.7	4.6	11,137
Tezontepec	55,134	2.1	5	3915
Tlahuelilpan	19,067	3	5.4	1602
Tlaxcoapan	28,626	3.4	3.4	1947
Tula	115,107	10.3	5.2	17,842
Zimapán	39,927	2.4	5.1	2995

.

Regarding the selection of temporary shelters, public spaces like auditoriums, schools, and churches were considered, with a preference for locations situated at a significant distance from the Tula River. Initially, 50 such sites were identified. However, after a thorough analysis of infrastructure and distances, the list was refined to 37. These selected facilities were subsequently integrated into the mathematical optimization model. For a comprehensive list of these shelters and their respective capacities, please refer to

Table A1. Selection of properties for the location of shelters. Source: own work.

Municipality	Property Name	Occupant Capacity
Alfajayucan	Bachillerato Alfajayucan TES	2000
Apaxco	Escuela Primaria Lázaro Cárdenas del Rio	2000
	Escuela Angel Gabriel	2000
Atitalaquia	Escuela Primaria Urbana General Ma. Anaya	2000
	Escuela Primaria Amado Nervo	2000
Atotonilco de Tula	Escuela Primaria General Primero de Mayo	2000
	Escuela Secundaria General Felipe Ángeles	2000
	Escuela Secundaria Técnica No. 58	2000
	Colegio de Bachillerato del Estado de Hidalgo	2000
Ixmiquilpan	Escuela Primaria Miguel Hidalgo Loma Hidalgo México	2000
	Escuela Primaria General Ignacio Zaragoza	2000
	Escuela Primaria Francisco I. Madero	2000
	Escuela Secundaria General Libertadores de America	2000
	Escuela Primaria “Teniente Juan de la Barrera”	2000
Mixquiahuala	Escuela Primaria Urbana General Ma. Anaya	2000
	Escuela Primaria Amado Nervo	2000
Tasquillo	Escuela Primaria Alvaro Obregón	2000
Tepeji del Río	Escuela Primaria Maria Angeles	2000
	Escuela Melchor Ocampo (Morelos)	2000
	Escuela Melchor Ocampo (Tlaxinacalpan)	2000
	Escuela Primaria Lázaro Cárdenas	2000
	Colegio Simón Bolívar	2000
	Iglesia Cristiana Maranatha	2000
Tezontepec	Escuela Secundaria General Niños Heroes	2000
Tlahuelilpan	Escuela Primaria Urbana General Ford 72	2000
Tlaxcoapan	Escuela Primaria Pedro Ma. Anaya	2000
Tula	Unidad Deportiva de Tula	4500
	Auditorio Municipal de Tula	4000
	Auditorio Infonavit San Marcos	4000
	Escuela Primaria Francisco Sarabia	2000
	Capilla Parroquia Sagrado Corazón de Jesús	2000
	Catedral de Tula (San José)	2000
	Escuela Primaria Benito Juarez	2000
	Escuela Primaria Miguel Hidalgo y Costilla	2000
Zimapán	Escuela Primaria Club de leones	2000
	Escuela Secundaria General Joséfa Bustamante	2000

in Appendix A.

To determine the locations of distribution centers, two distinct strategies were evaluated: The first strategy involved utilizing municipal presidencies as collection points, from which aid would be dispatched to various shelters. This approach was made feasible by strategically stockpiling a certain amount of food during the disaster preparedness phase. In the second strategy, the team considered the Government Palace, the National System for the Integral Development of the Family (DIF), and the Tula Presidency as distribution centers. This approach assumes that there exists a disaster relief fund that could be employed to directly dispatch aid kits to the shelters without necessitating intermediary distribution centers. The shelters, affected areas, and distribution centers are shown on the map in Figure 3.

To calculate the distances between the affected areas and the temporary shelters, we relied on real-time data obtained through Google Maps coordinates. This methodology was also applied to determine the distances between the distribution centers and the shelters. More detailed information is accessible through the provided repository [40].

3.2. Evacuation Model

In the second phase, the transportation costs were calculated. These costs were derived from selecting a particular type of transportation (NISSAN/URVAN/2015) with a capacity of 15 passengers. To compute the cost per trip, the fuel cost was used, which depended on the distance traveled, and a fixed cost was added for renting the vehicle. The fuel consumption per kilometer was determined based on data provided by the Ministry of Environment and Human Resources, while the fixed cost was obtained through price consultations with transportation providers [41].

The budget calculation for equipping a shelter involved selecting a closed facility and determining the necessary items for its preparation. The essential requirements included portable toilets, sinks, a semi-industrial kitchen, aluminum cookware, basic utensils, plastic sinks, mattresses, bedspreads, blankets, waste containers, tents, tables, and chairs. The calculation was made to accommodate approximately 150 to 200 people, resulting in a budget of MXN 159,147. Detailed information can be found in

Table A2. Items required to outfit a shelter.

Item	Unit Price	Amount	Total
Portable toilets	MXN 2500.00	2	MXN 5000.00
Portable sink	MXN 800.00	3	MXN 2400.00
Semi-industrial kitchen	MXN 1600.00	1	MXN 1600.00
Aluminum saucepan	MXN 1200.00	3	MXN 3600.00
Aluminum pot	MXN 900.00	2	MXN 1800.00
Basic utensils	MXN 500.00	1	MXN 500.00
Plastic sink	MXN 1200.00	3	MXN 3600.00
Hostel mattresses	MXN 1500.00	60	MXN 90,000.00
Bedspreads and blankets	MXN 150.00	60	MXN 9000.00
Trash bins	MXN 1400.00	5	MXN 7000.00
Installation of tents	MXN 6249.00	3	MXN 18,747.00
Tables	MXN 1300.00	3	MXN 3900
Total			MXN 159,147.00

in Appendix A.

The formulation of the optimization model employed integer linear programming (ILP) for evacuating people. In this model, the decision variables included the assignment of individuals to be evacuated from each affected area to each temporary shelter and the quantity of aid to be dispatched from each distribution center to each temporary shelter. The primary objective was to minimize transportation costs.

Table 3. Notation used in the linear programming model for the evacuation problem.

Sets:
$\mathfrak{i}ϵ \mathfrak{l}$	Set of municipalities affected
$j ϵ J$	Set of temporary shelter
$k ϵ K$	Set of distribution centers
$i,j ϵ Z$	Set of origins $i$ and destinations $j$
$k,j \in V$	Set of origins $k$ and destinations $j$
Parameters:
$C_{ij}$	Cost of transportation from the affected municipality $i$ to shelter $j$
$C_{kj}$	Transportation cost from distribution center $k$ to shelter $j$
$f_j$	Fixed cost of opening shelter $j$
$d_{ij}$	Distance from origin $i$ to destination $j$
$d_{kj}$	Distance from origin $k$ to destination $j$
${\mathrm{p}}_{\mathrm{i}}$	Population to evacuate
$q$	Fixed cost of renting a van to transport people
$A_j$	Shelter capacity
$B_k$	Distribution center capacity
$ca$	Capacity of the van to transport people
r	Percentage of vans considered for rental
$pr$	Budget available for disaster care
Variables:
$x_{ij}$	Number of people evacuated from the affected municipality $i$ to shelter $j$
$y_j$	Binary variable indicating whether the camp is open or not

provides the notation used in the ILP model proposed for the evacuation problem.

The model is as follows:

$$\begin{gathered} \mathrm{m}\mathrm{i}\mathrm{n}=\sum _{kj\in V}{C}_{kj}{x}_{kj}+\sum _{j\in J}{f}_j{y}_i+\left(\sum _{i\in I}\frac{p_i}{ca}\ast r\right)\ast q \\ subject to \end{gathered}$$

(1)

$$\sum _{i\in I}\sum _{j\in J}{x}_{ij}\le A_i$$

(2)

$$\sum _{i\in I}\sum _{j\in J}{x}_{ij}=p_i$$

(3)

$$\sum _{k\in K}\sum _{j\in J}{x}_{kj}=\sum _{i\in I}\sum _{j\in J}{x}_{ij}$$

(4)

$$\sum _{k\in K}\sum _{j\in J}{x}_{kj}\le B_{k }$$

(5)

$$\sum _{ij\in Z}{c}_{ij}{x}_{ij}+\sum _{kj\in V}{C}_{kj}{x}_{kj}+\sum _{j\in J}{f}_j{y}_i+\left(\sum _{i\in I}\frac{p_i}{ca}\ast r\right)\ast q\le pr$$

(6)

$$y_i=\left\{1 if the shelter opens, 0 otherwise\right\}$$

(7)

Equation (1) represents the objective function that minimizes the evacuation and the humanitarian aid distribution costs, considering the distances from the affected municipalities to the shelters and the distances from the distribution centers to the shelters. Additionally, the rental cost of the transport units is considered for evacuating people to the shelters. Equation (2) ensures that the number of people evacuated from municipality $i$ is less than the capacity of shelter $j$. Equation (3) establishes that the total number of people who need to be evacuated must be sent to a shelter. Equation (4) ensures that the number of humanitarian aid kits that must be sent from distribution center $k$ to shelter $j$ is equal to the number of people sent to this shelter. Equation (5) establishes that the number of humanitarian aid kits that must be sent from distribution center $k$ to shelter $j$ is less than the capacity of the distribution center. Equation (6) allows that the costs of evacuating people and the distribution of humanitarian aid are less than the budget allocated to manage a flood disaster. Finally, Equation (7) establishes that the variable $y_j$ is 1 if the shelter opens and 0 otherwise.

3.3. Model Solution

To address the ILP model for the evacuation described in Equations (1)–(7), we employed Lingo 17.0 software, a tool specifically designed for constructing and solving mathematical optimization models.

To conduct sensitivity analysis, the number of people to be evacuated was used as an input parameter in the Lingo model. After obtaining the optimal response with the original data, the model was solved with a 10% variation in demand (both increasing and decreasing). This was performed to examine how the total cost of evacuation and humanitarian aid delivery behaves under different demand scenarios.

3.4. Distribution of Humanitarian Aid

In this research, a vehicle routing problem (VRP) was used for determining the routes of the collection centers. The VRP can be represented as the following graph theory problem: $let G=(V,A)$ be a complete graph, where $V=\{0,1,...,n\}$ is the set of vertices and A is the set of arches. The vertices $j=1,...,n$ correspond to customers, each with a known non-negative demand, while vertex 0 corresponds to the deposit. A non-negative cost is associated with each arc and represents the cost of traveling from vertex $i$ to vertex $j.$ The VRP consists of finding a set $k$ of simple circuits, each one corresponding to a vehicular route with a minimum cost defined as the sum of the costs of the arcs of the circuits, in such a way that (a) each circuit visits vertex 0, (b) each vertex $j$ is visited by exactly one circuit, and (c) the sum of the demands of the vertices visited by a circuit does not exceed the capacity of the vehicle [42].

The VRP that we used considered the allocation of a vehicle at each distribution center, specifically at the State of Hidalgo Government, DIF Hidalgo, and the Presidency of Tula. The investigation encompassed 34 shelters characterized by normal demand. To obtain real coordinates and estimated travel times, while accounting for traffic conditions, Bing Maps data were utilized. The analysis assumed that each humanitarian aid kit weighed approximately 1 kg and that articulated vehicles with a capacity of 25,000 kg were employed, allowing each vehicle to transport 25,000 kits. These vehicles were assumed to possess a fuel efficiency of 100 km per 35 L of diesel [43], resulting in a cost of MXN 8.28 per kilometer traveled. Additionally, a delivery cost of MXN 500 was incorporated for each humanitarian aid drop-off. The VRP model was implemented with a single vehicle at each distribution center. Subsequently, the number of vans was determined through sensitivity analysis. To estimate the hours required for evacuation, the research considered the deployment of 5, 10, 15, and 20 vans. Furthermore, a 10% variation in demand was considered, encompassing both decreases and increases.

This methodology provides a well-structured guide to address the identified problem in a systematic and effective way. It should be noted that this methodology was designed for quick responses in pre- and post-disaster evacuation and aid distribution, with few temporary shelters and a small population.

4. Results

4.1. Evacuation Model

To solve the ILP for the evacuation model proposed in Equations (1)–(7), Lingo 17.0 software was used. Solving the model through the branch-and-bound solver included in Lingo, the optimal allocation was obtained from each of the affected municipalities (concentration centers for victims) to the different shelters, and at the same time the optimal distribution of aid was determined. It should be noted that a sensitivity analysis was carried out by modifying the demand parameters, with a variation of 10% above and below, and these were used as input parameters in Lingo. The objective function with the initial demand was MXN 2,969,813.00, with decreasing demand by 10% it was MXN 2,635,202.00, and with increasing demand by 10% it was MXN 3,250,149.00. It can be seen that with a small modification of the demand, the cost of evacuation and distribution of aid increases by 9% (MXN 280,336.00) with respect to the initial demand because, to meet the demand, three more shelters must be opened (see Figure 4).

The solutions for the three different scenarios of demand are shown in the repository [40]; only 34 shelters were opened, and the shelter information can be found in

Table 4. Shelter information.

Shelters	Address	Latitude	Longitude
Unidad Deportiva de Tula	Osa Mayor 15, El Cielito, 42803 Tula.	20.0526796	−99.3315105
Auditorio Municipal de Tula	Centro, Tula de Allende, Hgo.	20.0559197	−99.3428116
Auditorio Infonavit San Marcos	San Marcos, 42831 San Marcos, Hgo.	20.0293121	−99.3379669
Escuela Primaria Franciso Sarabia, Pueblo Nuevo	Pueblo Nuevo, 42845 San Miguel Vindho, Hgo.	19.9939747	−99.3316803
Catedral de Tula	5 de Mayo #5, Centro, 42800 Tula de Allende, Hgo.	20.0543701	−99.3441306
Escuela Primaria Benito Juárez	C. Sabino 37, Barrio Alto, 42800 Tula de Allende, Hgo.	20.0471437	−99.3560835
Escuela Primaria Miguel Hidalgo y Costilla	C. Francisco Zarco 529, Barrio Alto 2da Secc, 42807 Tula de Allende, Hgo.	20.0536890	−99.3493398
Escueal Secundaria General Niños Héroes	Emiliano Zapata 1, Centro, 42760 Tezontepec de Aldama, Hgo.	20.1424996	−99.2413238
Escuela Primaria María Angeles	Rep Mexicana 40, Noxtongo 1ra, 42850 Tepeji del Rio de Ocampo, Hgo.	19.9047249	−99.3394578
Escuela Melchor Ocampo	Juan Rulfo, Tlaxinacalpan, 42855 Tepeji del Rio de Ocampo, Hgo.	19.9232103	−99.3418594
Escuela Primaria Lázaro Cárdenas	42853, Av. Melchor Ocampo 37, San Mateo 1ra, Tepeji del Rio de Ocampo, Hgo.	19.9004832	−99.3418973
Colegio Simón Bolivar	C. Guillermo Prieto 17, San Francisco 1ra, 42854 Tepeji del Rio de Ocampo, Hgo.	19.9028700	−99.3446700
Iglesia Cristiana Maranatha	San Mateo 1ra, 42853 Tepeji del Rio de Ocampo, Hgo.	19.9072718	−99.3313870
Escuela Primaria General 1ro de mayo	Bóvedas, 42982 Atotonilco de Tula, Hgo.	20.0224432	−99.2202657
Escuela Secundaria General Felipe Ángeles	16 de Enero S/N, Centro, 42980 Atotonilco de Tula, Hgo.	20.0078759	−99.2150485
Escuela Secundaria Tecnica No.58	Av. del Trabajo SN, Boxfi, 42980 Atotonilco de Tula, Hgo.	19.9996695	−99.2095964
Colegio de Bachillerato del Estado de Hidalgo	C. República del Salvador SN, Bóvedas, 42980 Atotonilco de Tula, Hgo.	20.0064429	−99.2210060
Escuela Primaria Francisco Sarabia	Iturbide, Dendho, 42970 Cardonal, Hgo.	20.0572434	−99.2279053
Prepa Fray Diego de Rodríguez	Insurgentes 77, El Tablón, 42970 Atitalaquia, Hgo.	20.0541176	−99.2162695
Escuela Primaria Adolfo López Mateos	Veracruz 4, El Tablón, 42970 Atitalaquia, Hgo.	20.0496058	−99.2193827
Escuela Primaria Urbana General Ma. Anaya	Niños Héroes 88, 3ra Demarcación, 42700 Mixquiahuala de Juárez, Hgo.	20.2370703	−99.2043504
Escuela Primaria Amado Nervo	Gabino Barreda 1, Centro, 1ra Demarcación Poniente, 42700 Mixquiahuala de Juárez, Hgo.	20.2563827	−99.1903234
Escuela Primaria Pedro Ma. Anaya	Miguel Hidalgo 6, Educación, 42952 Tlaxcoapan, Hgo.	20.0912805	−99.2242954
Escuela Primaria Urbana General Ford 72	Fco Villa 20, Rancho Viejo, 42780 Tlahuelilpan, Hgo.	20.1353506	−99.2340138
Escuela Primaria Miguel Hidalgo Loma México	42325 Ixmiquilpan, Hgo.	20.4835415	−99.2191544
Escuela Primaria General Ignacio Zaragoza	San Pedro Capula, Ixmiquilpan, Hgo.	20.5124985	−99.1572957
Escuela Primaria Francisco I.Madero	Progreso, 42302 Ixmiquilpan, Hgo.	20.4898208	−99.2222736
Escuela Secundaria General Libertadores de América	Libertadores de América 12, El Fithzi, 42300 Ixmiquilpan, Hgo.	20.4658258	−99.2092965
Escuela Primaria Teniente Juan de la Barrera	Fco I Madero Sn, El Fithzi, 42300 Ixmiquilpan, Hgo.	20.4836712	−99.2192688
Escuela Primaria Lázaro Cárdenas del Rio	C. Allende 1, El Mirador, 55660 Apaxco de Ocampo, Méx.	19.9763946	−99.1715745
Escuela Primaria Club de Leones	Del Trabajo y, Santa Anita, 42337 Zimapán, Hgo.	20.7356244	−99.3867279
Escuela Secundaria General Josefa Bustamante	Heroico Colegio Militar 11, Centro, 42330 Zimapán, Hgo.	20.7365736	−99.3834011
Escuela Primaria Alvaro Obregón	Seguro Social 2, Calvario Bajo, 42381 Tasquillo, Hgo.	20.5387707	−99.3689880
Bachillerato Alfajayucan TES	Colonia Centro, 42390 Alfajayucan, Hgo.	20.4080620	−99.3495331

. Considering the solution in the repository for the original demand, the people who need to be evacuated from a central location can be determined; for example, the Tula Presidency moves 4500 people (p) to the Tula Sports Unit, 4000 p to the Tula Municipal Auditorium, 4000 p to the Infonavit San Marcos Auditorium, 2000 p to the Cathedral of Tula, 1342 p to the Benito Juárez Elementary School, and 2000 p to the Miguel Hidalgo y Costilla Elementary School, giving a total of 17,842 p.

4.2. Evacuation Time and Vans Required

Based on the best results in the provided repository [40], the required evacuation time was calculated, considering variations in the number of vans for the transfer of people; it was considered that each van has a transfer capacity of 15 people. In order to know the number of hours required for evacuation, 5, 10, 15, and 20 vans were considered, as well as considering a 10% variation in the demand (decrease and increase). Since the municipalities of Tula, Tepeji, and Ixmiquilpan are the ones with the largest numbers of people to evacuate, they are also the ones that require the longest time. For example, if 5 vans are assigned to the Tepeji Presidency, considering the average demand, 62.04 h is required to evacuate, while with 10 vans 31.0244 h is required, and with 20 vans 15.51 h is required, changing proportionally. The total time varies between 8.65% and 11.38% less when the demand goes down and increases by between 25.48% and 29.34% when the demand increases (see

Table 5. Time for evacuation, original demand.

Presidency	5 a Vans	10 Vans	15 Vans	20 Vans	3 b h	4 h	8 h
Tula	52.50 c	25.13	17.50	13.13	88 d	53	33
Tezontepec	27.56	13.78	9.19	6.89	46	28	18
Tepeji	62.05	31.02	20.68	15.51	104	63	39
Atotonilco de Tula	30.93	15.47	10.31	7.73	52	31	20
Atitalaquia	26.02	13.01	8.67	6.50	44	27	17
Mixquiahuala	44.10	22.05	14.70	10.24	74	45	28
Tlaxcoapan	21.84	10.92	7.28	5.46	37	22	14
Tlahuelilpan	9.07	4.53	3.02	1.71	16	10	6
Ixmiquilpan	120.93	60.46	40.31	29.58	202	121	76
Apaxco	17.65	8.83	5.88	3.34	30	18	12
Zimapán	0.00	0.00	0.00	0.00	0	0	0
Tasquillo	0.00	0.00	0.00	0.00	0	0	0
Alfajayucan	0.00	0.00	0.00	0.00	0	0	0
Total vans					693	418	263

^a Five vans assigned to every presidency; ^b hours needed for evacuation; ^c time in hours; ^d vans required.

).

To provide an adequate response to the evacuation problem in the face of a natural disaster, time is a key factor. In this sense, the numbers of vans required to evacuate in smaller timeframes (3, 5, and 8 h) were calculated. Table 5 shows that when demand decreases, the number of trucks needed decreases by between 10.65% and 11.11%, while if demand increases, the number of trucks needed increases by between 24.71% and 25.12%. The number of vans required to evacuate the population using the average demand would be 418 for a time of 5 h, while 3 h would require 693 vans, and for 8 h 263 vans would be needed; for more information, consider

Table A5. Routes and approximate times for delivery of kits by vehicle 2—first strategy.

Vehicle:	V2	Stops:	14	Net Profit:	−7652.10
Stop Count	Location Name	Distance Travelled	Driving Time	Arrival Time	Departure Time	Working Time	Profit Collected	Load
0	DIF Hidalgo	0.00	0:00		08:00	0:00	0	24,927
1	Escuela Primaria Amado Nervo	63.41	1:15	09:15	09:15	1:15	0	22,927
2	Escuela Primaria Urbana General Ma. Anaya	67.67	1:25	09:25	09:25	1:25	0	20,927
3	Escuela Secundaria General Niños Héroes	79.84	1:45	09:45	09:45	1:45	0	18,927
4	Escuela Primaria Urbana General Ford 72	81.76	1:51	09:51	09:51	1:51	0	16,927
5	Bachillerato Alfajayucan TES	128.09	3:01	11:01	11:01	3:01	0	14,995
6	Escuela Primaria Álvaro Obregón	144.88	3:36	11:36	11:36	3:36	0	12,995
7	Escuela Secundaria General Josefa Bustamante	177.57	4:20	12:20	12:20	4:20	0	10,995
8	Escuela Primaria Club de Leones	178.06	4:22	12:22	12:22	4:22	0	10,000
9	Escuela Primaria Francisco I. Madero	226.92	5:28	13:28	13:28	5:28	0	8000
10	Escuela Primaria Teniente Juan de la Barrera	228.03	5:31	13:31	13:31	5:31	0	6000
11	Escuela Primaria Miguel Hidalgo Loma México	228.05	5:31	13:31	13:31	5:31	0	4000
12	Escuela Secundaria General Libertadores de América	231.17	5:40	13:40	13:40	5:40	0	2000
13	Escuela Primaria General Ignacio Zaragoza	240.45	6:03	14:03	14:03	6:03	0	0
14	DIF Hidalgo	320.30	7:40	15:40		7:40	0	0

.

Concerning the strategy evaluation for using different distribution centers using the VRP, the VRP Spreadsheet Solver v3.8 was used. Two different strategies were evaluated. The first strategy made use of three collection centers: two in Pachuca (DIF Hidalgo and State of Hidalgo Government) and the third center in the Tula Presidency, each with one articulated vehicle with a capacity of 25,000 kits; the solution is shown in

Table A6. Routes and approximate times for delivery of kits by vehicle 3—first strategy.

Vehicle:	V3	Stops:	11	Net Profit:	−5386.58
Stop Count	Location Name	Distance Travelled	Driving Time	Arrival Time	Departure Time	Working Time	Profit Collected	Load
0	Presidencia de Tula	0.00	0:00		08:00	0:00	0	23,478
1	Auditorio Municipal de Tula	1.45	0:05	08:05	08:05	0:05	0	19,478
2	Escuela Primaria Miguel Hidalgo y Costilla	2.46	0:09	08:09	08:09	0:09	0	17,478
3	Escuela Primaria Benito Juárez	3.85	0:14	08:14	08:14	0:14	0	15,478
4	Colegio Simón Bolivar	22.16	0:49	08:49	08:49	0:49	0	13,478
5	Escuela Primaria Lázaro Cárdenas	22.90	0:52	08:52	08:52	0:52	0	11,478
6	Escuela Primaria María Ángeles	23.56	0:54	08:54	08:54	0:54	0	9478
7	Iglesia Cristiana Maranatha	25.66	1:00	09:00	09:00	1:00	0	7478
8	Escuela Melchor Ocampo	28.36	1:08	09:08	09:08	1:08	0	5478
9	Escuela Primaria Francisco Sarabia, Pueblo Nuevo	37.69	1:24	09:24	09:24	1:24	0	4000
10	Auditorio Infonavit San Marcos	43.51	1:39	09:39	09:39	1:39	0	0
11	Presidencia de Tula	46.69	1:49	09:49		1:49	0	0

,

Table A7. Routes and approximate times for delivery of kits by vehicle 1—second strategy.

Vehicle:	V1	Stops:	14	Net Profit:	−7652.10
Stop Count	Location Name	Distance Travelled	Driving Time	Arrival Time	Departure Time	Working Time	Profit Collected	Load
0	DIF Hidalgo	0.00	0:00		08:00	0:00	0	24,927
1	Escuela Primaria Amado Nervo	63.41	1:15	9:15	09:15	1:15	0	22,927
2	Escuela Primaria Urbana General Ma. Anaya	67.67	1:25	9:25	09:25	1:25	0	20,927
3	Escuela Secundaria General Niños Héroes	79.84	1:45	09:45	09:45	1:45	0	18,927
4	Escuela Primaria Urbana General Ford 72	81.76	1:51	09:51	09:51	1:51	0	16,927
5	Bachillerato Alfajayucan TES	128.09	3:01	11:01	11:01	3:01	0	14,995
6	Escuela Primaria Álvaro Obregón	144.88	3:36	11:36	11:36	3:36	0	12,995
7	Escuela Secundaria General Josefa Bustamante	177.57	4:20	12:20	12:20	4:20	0	10,995
8	Escuela Primaria Club de Leones	178.06	4:22	12:22	12:22	4:22	0	10,000
9	Escuela Primaria Francisco I. Madero	226.92	5:28	13:28	13:28	5:28	0	8000
10	Escuela Primaria Teniente Juan de la Barrera	228.03	5:31	13:31	13:31	5:31	0	6000
11	Escuela Primaria Miguel Hidalgo Loma México	228.05	5:31	13:31	13:31	5:31	0	4000
12	Escuela Secundaria General Libertadores de América	231.17	5:40	13:40	13:40	5:40	0	2000
13	Escuela Primaria General Ignacio Zaragoza	240.45	6:03	14:03	14:03	6:03	0	0
14	DIF Hidalgo	320.30	7:40	15:40		7:40	0	0

and

Table A8. Routes and approximate times for delivery of kits by vehicle 2—second strategy.

Vehicle:	V2	Stops:	11	Net Profit:	−5399.32
Stop Count	Location Name	Distance Travelled	Driving Time	Arrival Time	Departure Time	Working Time	Profit Collected	Load
0	Presidencia de Tula	0.00	0:00		08:00	0:00	0	23,978
1	Auditorio Infonavit San Marcos	3.08	0:09	08:09	08:09	0:09	0	19,978
2	Escuela Primaria Francisco Sarabia, Pueblo Nuevo	8.82	0:25	08:25	08:25	0:25	0	18,500
3	Colegio Simón Bolivar	20.78	0:46	08:46	08:46	0:46	0	16,500
4	Escuela Primaria Lázaro Cárdenas	21.52	0:49	08:49	08:49	0:49	0	14,500
5	Escuela Primaria María Ángeles	22.17	0:51	08:51	08:51	0:51	0	12,500
6	Iglesia Cristiana Maranatha	24.28	0:57	08:57	08:57	0:57	0	10,500
7	Escuela Melchor Ocampo	26.98	1:05	09:05	09:05	1:05	0	8500
8	Escuela Primaria Benito Juárez	43.43	1:37	09:37	09:37	1:37	0	6500
9	Escuela Primaria Miguel Hidalgo y Costilla	45.13	1:42	09:42	09:42	1:42	0	4500
10	Unidad Deportiva de Tula	47.45	1:50	09:50	09:50	1:50	0	0
11	Presidencia de Tula	48.23	1:54	09:54		1:54	0	0

in Appendix A, where the routes to be followed, approximate time, and total cost each time that the aid is delivered (MXN 19,646) are shown. The second strategy made use of only two collection centers: DIF Hidalgo, with one vehicle, and a second center in the Tula Presidency with two vehicles. The routes are shown in Table A7, Table A8 and

Table A9. Routes and approximate times for delivery of kits by vehicle 3—second strategy.

Vehicle:	V3	Stops:	12	Net Profit:	−5560.40
Stop Count	Location Name	Distance Travelled	Driving Time	Arrival Time	Departure Time	Working Time	Profit Collected	Load
0	Presidencia de Tula	0.00	0:00		08:00	0:00	0	24,000
1	Auditorio Municipal de Tula	1.45	0:05	08:05	08:05	0:05	0	20,000
2	Catedral de Tula	1.75	0:06	08:06	08:06	0:06	0	18,000
3	Escuela Primaria Pedro Ma. Anaya	17.52	0:32	08:32	08:32	0:32	0	16,000
4	Prepa Fray Diego de Rodríguez	22.32	0:44	08:44	08:44	0:44	0	14,000
5	Escuela Primaria Adolfo López Mateos	23.34	0:47	08:47	08:47	0:47	0	12,000
6	Escuela Primaria Francisco Sarabia	25.68	0:54	08:54	08:54	0:54	0	10,000
7	Escuela Primaria General 1ro de Mayo	31.29	1:08	09:08	09:08	1:08	0	8000
8	Colegio de Bachilleres del Estado de Hidalgo	33.33	1:14	09:14	09:14	1:14	0	6000
9	Escuela Secundaria Técnica no.58	35.06	1:19	09:19	09:19	1:19	0	4000
10	Escuela Primaria Lázaro Cárdenas del Rio	42.08	1:35	09:35	09:35	1:35	0	2000
11	Escuela Secundaria General Felipe Ángeles	51.18	1:55	09:55	09:55	1:55	0	0
12	Presidencia de Tula	67.68	2:19	10:19		2:19	0	0

. The second strategy was the best, with a cost of MXN 18,611 each time that the kits are delivered.

5. Conclusions

Currently, hazard events represent a great threat to humanity. A further increase in their incidence is expected due to climate change. This is why the HSC is gaining more and more interest in academia, government, and society. Some relevant problems are the location of shelters, the distribution of aid, inventories, and the mass evacuation of people.

In the literature reviewed here, each author adapted the model and solution strategies to the specific case that they were trying to solve. In this article, shelters are selected, people are evacuated to the shelters and, finally, strategies to distribute aid are explored considering demographic and real geographic data for the evacuation of people with mobility difficulties and disabilities.

The flood that occurred on 6 September 2021 was considered to be an extreme event, and year after year there are flooding problems. The main idea of this study was to provide strategies for municipal and state governments to make decisions about how many shelters to open and how to distribute the aid. The methodology consisted of calculating the population to be evacuated, identifying the possible shelters that could be set up, deciding where it would be convenient to locate the distribution centers, calculating the distance between locations and the expenses incurred to adapt a temporary shelter, and formulating the optimization model to decide which shelters would be opened and how people would be evacuated. Finally, a VRP with different strategies to distribute humanitarian aid was presented.

An analysis was conducted with three types of demand—the original, increased by 10%, and decreased by 10%—to evaluate the adequacy of the planned shelters. Furthermore, we calculated the number of vans needed to execute the evacuation. Our hope, as natives of this region, is that this research serves as a foundation for decision-making and can be employed to conduct similar analyses in other communities during extreme events.

Some of the limitations of this work include the reliance on Google Maps distances for the integer linear programming (ILP), which do not account for real-time traffic conditions. Future research may involve considering the evacuation of other vulnerable individuals in the region, such as those in hospitals, prisons, etc., in the event of other extreme events. Additionally, it could explore how aerial evacuations would be conducted in situations where all routes are obstructed and strategize on the organization of brigades within shelters and/or evacuation points.