A Mathematical Model for Integrated Disaster Relief Operations in Early-Stage Flood Scenarios

Abstract

When a flood strikes, the two most critical tasks are evacuation and relief distribution. It is essential to integrate these tasks, particularly before the floodwater reaches the vulnerable area, to minimize loss and damage. This paper presents a mathematical model of vehicle routing problems to optimize an integrated disaster relief operation. The model addresses routing for both the evacuation and relief distribution tasks in the early stages of a flood, aiming to identify a minimal number of vehicles required with their corresponding routes to transport vulnerable individuals and simultaneously distribute emergency relief. The new model incorporates several features, including vehicle reuse, multi-trip and split delivery scenarios for evacuees and emergency relief items, uncertainty in evacuation demands, and closing time windows at evacuation points. Due to the complexity of vehicle routing problems, particularly in large-scale scenarios, the exact approach for obtaining optimal solutions is time-consuming. Therefore, we propose the use of a metaheuristic algorithm, specifically a modified genetic algorithm, to find an approximate solution for the proposed model. We apply the developed model and modified algorithm to various simulated flood scenarios and a real-life case study from Indonesia. The experimental results demonstrate that our approach requires fewer vehicles compared to standard models for similar scenarios. Moreover, while the exact approach fails to find optimal solutions within a reasonable timeframe for large-scale scenarios, our new approach provides near-optimal solutions in a much shorter time. In smaller simulated scenarios, the modified genetic algorithm obtains optimal or near-optimal solutions approximately 92.5% faster than the exact approach.

1. Introduction

A natural disaster is an event or phenomenon that causes death, injury, other health effects, property damage, social and economic disruption, or environmental damage. According to data from the International Disaster Database, floods are the most common and most expensive natural disaster worldwide [1].

Several regions worldwide experience frequent flooding due to various factors such as climate, geography, and human activities. For instance, Southeast Asian countries like Indonesia face frequent flooding. Due to its geographic position near the equator, the region has a tropical climate with long rainy seasons and high rainfall levels. In addition, steep slopes and abundant rivers can overflow during heavy rains, making flooding the most dangerous type of natural disaster in the country. Other countries, such as Bangladesh and India, experience frequent flooding during the monsoon season, while the varied topography of European countries like Germany and the Netherlands, including mountain ranges and low-lying areas, also leads to floods.

Most flood deaths are caused by flash floods, with approximately half of these caused by people attempting to cross swollen streams or flooded roads. Victims frequently underestimate the power of water and drive into flooded areas when the water level is already high. Aside from taking human lives, a flood has economic consequences. In the first half of 2017, there were 149 events in 73 countries, which resulted in 3162 deaths and affected more than 80.6 million people, with the economic damages estimated to be US$32.4 billion [1].

When a flood strikes, the two highest priorities—evacuation and relief distribution—are considered. According to Emergency Management Australia (EMA), an evacuation is “a risk management strategy which may be used as a means of mitigating the effects of an emergency or disaster on a community”. It is an act of moving people directly and quickly from areas at risk to safer locations. This involves not only transferring victims to shelters once a flood strikes but also transporting vulnerable people from prone areas before the water strikes.

Since conditions in a disaster-affected area are generally unfavourable, disaster management (DM) plays a critical role in meeting demands [2]. Allocating evacuees to available vehicles, selecting the most efficient routes for evacuation, and distributing sufficient initial emergency relief are all time-sensitive decisions. A proper selection of temporary shelters in safer places can also facilitate the efficient flow of relief supplies. However, the infrastructure in the affected areas may be destroyed due to the impact of floodwater [3]. Disruptions of the transportation network and the uncertainty of the number of evacuees make the tasks even harder.

To date, both real-life practice and present research have tended to focus on disaster relief operations after a disaster happens or reaches the prone areas [3,4]. There are also studies that emphasize minimizing the operation costs or times, for instance, [5,6]. However, evacuating people before a disaster, i.e., floodwater, reaches them is crucial at this stage to avoid more significant losses and damage. Still, in a real-life situation, it is unlikely that all evacuees can reach safer areas without help. Increasing water volume may affect a vehicle’s ability to travel. Water only needs to be about 12 inches deep to wash away a typical vehicle [7]. Based on the historical data, the elderly are the most common primary victims in every disaster [8]. The amount of time needed to transfer evacuees into a vehicle and drop them off has a significant impact on human life. Additionally, receiving emergency relief to support the evacuees’ lives for the first few hours can also reduce further anxiety and illness. The need to find better strategies to reduce losses is obvious, and one approach is to design an efficient routing model focused on humanity values in the disaster preparedness stage.

In this research, we develop a mathematical model to integrate evacuation and relief distribution at the early stages of a flood, prioritizing humanitarian values. It incorporates a unified response plan, offering a comprehensive overview of the flood management scenario. In the proposed model, the distribution vehicles are scheduled to be dispatched after the evacuation begins for a specified period. A modified genetic algorithm has been developed to provide optimal or near-optimal solutions for the minimum number of vehicles and their corresponding best routes within a reasonable computational time. These results will be communicated to relevant stakeholders for effective and timely evacuation and relief distribution.

The remainder of this paper is organized as follows. In Section 2, we provide a review of the related work. Section 3 discusses the research gaps and outlines our main contributions. Section 4 describes the proposed mathematical formulation. Section 5 presents the proposed solution approach using the modified genetic algorithm. Section 6 reports the numerical results for synthetic datasets and a case study. Finally, Section 7 presents future research directions and concluding remarks.

2. Related Work

According to the United States Federal Emergency Management Agency and the Red Cross, the early stage of a flood refers to the critical initial phase, typically from the issuance of the initial flood warning to the first few hours of active flooding. Several studies, such as [9,10], emphasize the crucial importance of timely rapid response, coordination, and effective communication during this stage to minimize the impact of flooding.

An evacuation model is typically categorized as either no-notice or short-notice. During a short-notice evacuation, a warning is issued a few hours before a disaster strikes, prompting people to evacuate before the floodwater reaches the area [11]. While such warnings save lives, the willingness of all residents to evacuate remains uncertain. The study of mass evacuation using public transportation for vulnerable individuals gained prominence after the Hurricane Katrina event in the United States in 2005. Comprehensive surveys on evacuation operations were provided in [12,13].

Some studies have explored evacuation solely or in conjunction with location-allocation decisions for resources [14,15,16]. Others have focused exclusively on developing optimal plans and solutions for relief distribution, such as those discussed in [17] and the references therein. There are also studies that integrated relief distribution with facility location decisions [18], dynamic allocation models with prior stock positioning [19], and multi-depot location-routing models for determining depot and distribution center locations [20].

To manage large demands or loads exceeding vehicle capacities, some studies have employed strategies like split delivery routing, known as the split delivery vehicle routing problem (SDVRP). This approach involves serving an affected area multiple times to reduce operational costs [21,22,23]. Furthermore, some researchers have addressed the inefficiencies of single-trip vehicle assumptions by proposing multi-trip vehicle routing problems (MTVRP), where vehicles can be dispatched from depots more than once within a planning period [5,24]. This flexibility is particularly advantageous in emergency scenarios.

In real-life flood situations, quick decisions are crucial for disaster management (DM) authorities, especially when numerous people and resources are involved. While exact methods can provide optimal solutions, their high computational requirements may delay decision-making. Consequently, many studies have turned to metaheuristic approaches to approximate optimal solutions efficiently [25]. According to [26], the genetic algorithm (GA) outperforms other metaheuristic algorithms such as ant colony systems, particle swarm optimization, tabu search, and simulated annealing in terms of solution quality. The GA’s ability to quickly search and find approximate or near-optimal solutions within acceptable computational timeframes has made it widely applicable in various DM domains, including relief distribution or humanitarian aid [21,27], evacuation planning [11], and integrated problems like relief distribution combined with facility location-allocation [6,28]. However, disaster relief scenarios are often dynamic and unpredictable, leading to the varying performance of metaheuristic algorithms based on specific problem characteristics and operational complexities. Therefore, the continuous adaptation and fine-tuning of these approaches are essential to leverage their strengths effectively.

3. Research Gaps and Our Contribution

As previously mentioned, coordinating relief vehicles during natural disasters poses significant challenges. Comprehensive information on the number and locations of evacuees, availability of supplies and vehicles, capacities of vehicles, and distribution centers’ (DCs) and shelters’ locations must be meticulously managed. Therefore, efficient plans are essential for planning safe evacuations and effective relief distribution.

In our previous study [29], a mathematical model that integrates split delivery and multi-trip routing strategies for emergency evacuations was introduced. The model considers closing time windows for each evacuation point (EP) and dependent service times for evacuees. In this paper, we present a mathematical model that combines emergency evacuation strategies from [29] with relief distribution to optimize routes and to identify the required minimum number of vehicles for both evacuation and relief distribution.

The main contributions of this paper are as follows:

• Extending the developed model in [29] by

  (a)

  Integrating the evacuation plan with relief distribution, ensuring emergency relief is available for evacuees upon arrival at shelters;

  (b)

  Determining the minimum number of vehicles required simultaneously for both evacuation and relief distribution during the critical initial hours after arrival at shelters;

  (c)

  Considering vehicle reuse by implementing multi-trip and split delivery strategies, uncertainties in evacuation demands, specific loading times, closing time windows at evacuation points, and unloading times at shelters based on the number of evacuees served.

• Extending the genetic algorithm (GA) to be utilized in our new model to obtain an approximate solution within a reasonable computation time by

  (a)

  Generating the initial population using a constructive heuristic method to lay the foundation for the genetic algorithm’s optimization process;

  (b)

  Implementing a modified best cost route crossover operator during genetic algorithm execution to enhance the quality of solution paths;

  (c)

  Applying a series of local searches for the mutation to further refine and improve offspring solutions derived from the genetic algorithm;

  (d)

  Applying the developed model and solution approach to address simulated flood scenarios and a specific case study based in Indonesia.

4. The Proposed Mathematical Model

Historically, the elderly, young children, and other vulnerable individuals have been unlikely to reach safe areas without evacuation support. Therefore, we consider a mass evacuation scenario where these groups are transported using vehicles provided by government agencies. To account for people’s reactions to evacuation warnings, we allow for a 1-hour (1-h) period for evacuees to make decisions, prepare, and walk to the nearest designated EP for pickup. For safety reasons, each EP has a closing time window; once this window passes, the floodwater reaches the site, preventing vehicle passage. A 2-hour (2-h) clearance time is set based on the longest time window, requiring all evacuation vehicles to complete their missions within this time frame.

According to the Indonesia Bureau of Statistics, vulnerable individuals include young children aged 0–10 years (9%), female adults (36%), the homeless, and the elderly (defined as over 60 years old, comprising 17%), making up approximately 60% of the impacted population [30]. Consequently, our study assumes that 60% of the population will rely on vehicles provided by authorities to move to safer locations during the initial stages of a flood.

In the initial hours following an evacuation, there are often shortages of essential relief supplies. To address this problem, we develop an integrated disaster relief operations model that simultaneously handles both evacuation and relief distribution. The proposed model utilises two fleets of vehicles: one dedicated to evacuation and the other to relief distribution. Each fleet operates from its own depot. Relief supplies are received and stored at DCs, which have limited storage capacities and predetermined locations. These distribution depots and DCs are situated in safe areas close to the shelters to ensure the quick and efficient delivery of supplies.

The amount of relief supplies delivered to each shelter is based on the number of evacuees present. As outlined in [29], a distribution vehicle is dispatched from the depot to deliver these supplies. The vehicle visits one or more DC to load the relief supplies before transporting them to temporary shelters where evacuees have been accommodated. Figure 1 illustrates a relief disaster operation that encompasses both evacuation and relief distribution, demonstrating the application of split delivery and multi-trip strategies. The figure shows evacuation routes followed by an evacuation vehicle, from its depot, leading to evacuation points to a shelter. In a similar manner, a distribution vehicle takes a designated route, starting from its depot, picking up supplies from distribution centers and dropping them off at shelters where evacuees await. Different arrows indicate a different route for each vehicle. Additionally, the figure shows how the same vehicle makes multiple trips between shelters and evacuation points, minimizing vehicle usage. A split delivery strategy is also shown, both for evacuation and for relief distribution. It shows how evacuees are divided into batches and how supplies are distributed in batches, allowing for partial deliveries to multiple shelters within different routes.

The split delivery strategy in our model is similar to the one described in [23]. However, our model differs by incorporating a multi-trip approach and closing time windows alongside the split delivery. This means that once an evacuation vehicle picks up evacuees and drops them off at a shelter, it can make additional trips to transport evacuees from other EPs to the same or different shelters until all evacuees have been evacuated. Similarly, once a distribution vehicle picks up relief supplies from a DC and delivers them to one or more shelters, it can make additional trips until all evacuees have received relief packages or the time limit is reached. Our goal is to accomplish these tasks using the minimum number of vehicles.

4.1. Assumptions and Notations

We consider the following assumptions:

• Evacuees

  –

  An estimated number of vulnerable people willing to evacuate is known at each EP.

  –

  Each evacuee conceives the evacuation procedure and immediately heads to their designated EP when an alert is issued.

• Relief distributions

  –

  Each evacuee will receive one relief package regardless of any individual differences (age, gender, special needs, etc.).

  –

  Each relief package is assumed to be sufficient for one person. It includes all of the essentials required for several hours of survival—water, food, a blanket, and medicine.

  –

  The number of relief packages required at each shelter is determined by the number of evacuees at that location.

  –

  Each DC has an adequate supply of relief packages.

• Vehicles and road network

  –

  A number of identical vehicles are always on hand at the evacuation vehicle depot as soon as an evacuation warning is issued.

  –

  A fleet of distribution vehicles is stationed at this depot and can leave simultaneously to their designated shelters.

  –

  The travel time between nodes in the network is identified and symmetric.

  –

  An EP or shelter can be served by multiple appropriate vehicles (i.e., split delivery).

  –

  A DC can be visited by more than one distribution vehicle.

  –

  A vehicle may be assigned to multiple trips.

  –

  For safety reasons, a vehicle ends its route at the visited node of its last trip.

• Time

  –

  The evacuation begins after an evacuation warning is issued.

  –

  Since there is an hour allocated to prepare and reach an EP, there is no waiting time at any EP.

  –

  Each EP has a limited time to be served, i.e., a closing time window ($b_i$). All of the vehicles can only operate until this time window expires.

  –

  Each EP has a loading time ($lt$), the total time it takes for each vehicle to load its assigned evacuees, and an unloading time ($ult$), the amount of time for each vehicle it takes to deliver evacuees to a shelter. These service times depend on the number of evacuees to be loaded or unloaded at each visit.

  –

  As the flood is expected to hit all areas within a specific time frame, all vehicles should complete the evacuation before the clearance time.

  –

  The distribution of relief begins at $\alpha$ time unit after the evacuation begins.

• Temporary shelters

  –

  There is a known number of temporary shelters located in a safe place.

  –

  The shelter capacity varies based on its size and utilities.

  –

  There is no time window for shelters as they are located outside flood-prone areas.

  –

  Shelters receive evacuees from any evacuation point.

• Distribution centers (DCs)

  –

  The locations and capacities of DCs are predetermined and known.

  –

  A DC has a service time, i.e., the time to load the relief supplies onto a vehicle.

  –

  DCs have different relief capacities depending on their size.

In addition, we define some notations that we use to formulate our model. Some notations are similar to those defined in [23,29]. To differentiate the decision variables of the evacuation task, we use “′” for variables related to the relief distribution task. The depots for a fleet of evacuation vehicles and a fleet of relief distribution vehicles are denoted by 1 and $1^{\prime }$, respectively.

• Sets

C	set of evacuation points
S	set of temporary shelters
V	set of all evacuation nodes ($1\cup C\cup S$)
K	set of evacuation vehicles
R	set of evacuation trips
$\tilde{R}$	$=\left\{1,2,\dots ,\left|R\right|-1\right\}$
DC	set of distribution centers
$V^{\prime }$	set of all distribution nodes ($1^{\prime }\cup DC\cup S$)
$K^{\prime }$	set of distribution vehicles
$R^{\prime }$	set of distribution trips
$\widetilde{R^{\prime }}$	$=\left\{1,2,\dots ,|R^{\prime }|-1\right\}$

• Parameters

$t_{ij}$	traveling time associated with arc $(i,j)$, where $i,j\in V,i\ne j$
$d_i$	demand (number of evacuees) at evacuation point $i\in C$
$d_i^{\prime }$	supply at distribution center $i\in DC$
Q	capacity of evacuation vehicles
$Q^{\prime }$	capacity of distribution vehicles
$L_i$	capacity of shelter $i\in S$ for evacuees
$L_i^{\prime }$	capacity of shelter $i\in S$ for relief
$lt$	loading time per person at evacuation point $i\in C$
$ult$	unloading time per person at shelter $i\in S$
$b_i$	closing time window at evacuation point $i\in C$
$s{t}^{\prime }$	distribution service time at each $DC\cup S$
$\alpha$	minimum time required to dispatch distribution vehicle after
	the evacuation starts
M	a big number

• Decision variables

$x_{ijkr}$	= 1 if vehicle $k\in K$ in trip $r\in R$ travels directly from node $i\in V$
	to node $j\in V$; 0 otherwise
$x_{ijkr}^{\prime }$	= 1 if vehicle $k\in K^{\prime }$ in trip $r\in R^{\prime }$ travels directly from node $i\in V^{\prime }$
	to node $j\in V^{\prime }$; 0 otherwise
$z_k$	= 1 if vehicle $k\in K$ is dispatched; 0 otherwise
$z_k^{\prime }$	= 1 if vehicle $k\in K^{\prime }$ is dispatched; 0 otherwise
$y_{ikr}$	= 1 if vehicle $k\in K$ in trip $r\in R$ is allocated to node $i\in C\cup S$
$y_{ikr}^{\prime }$	= 1 if vehicle $k\in K^{\prime }$ in trip $r\in R^{\prime }$ is allocated to node $i\in DC\cup S$
$f_{kr}$	= 1 if vehicle $k\in K$ is assigned to trip $r\in R$; 0 otherwise
$f_{kr}^{\prime }$	= 1 if vehicle $k\in K^{\prime }$ is assigned to trip $r\in R^{\prime }$; 0 otherwise
$p_{ikr}$	number of evacuees (demand) picked up from evacuation point $i\in C$
	by vehicle $k\in K$ in trip r
$q_{ikr}$	number of evacuees (demand) dropped off at shelter $i\in S$
	by vehicle $k\in K$ in trip r
$p_{ikr}^{\prime }$	number of relief packages picked up at $i\in DC$ by vehicle $k\in K^{\prime }$
	in trip $r\in R^{\prime }$
$q_{ikr}^{\prime }$	number of relief packages dropped to shelter $i\in S$ by vehicle $k\in K^{\prime }$
	in trip $r\in R^{\prime }$
$a_{ikr}$	arrival time of vehicle $k\in K$ in trip $r\in R$ at node $i\in V$
$a_{ikr}^{\prime }$	arrival time of vehicle $k\in K^{\prime }$ in trip $r\in R^{\prime }$ at node $i\in V^{\prime }$

Based on the assumptions and notations above, we formulate our integrated evacuation-relief distribution model.

4.2. Mathematical Formulation

When a flood begins, the most important thing, after the number of evacuees is calculated, is to determine the number of vehicles required to rescue all the people and provide emergency relief. Thus, our objective is to minimize the number of operational vehicles needed to transport all evacuees and deliver relief packages. This is defined as follows:

$$\mathrm{minimize}\left(\sum _{j\in C}\sum _{k\in K}\sum _{r\in R}{x}_{1jkr}+\sum _{j\in DC}\sum _{k\in K^{\prime }}\sum _{r\in R^{\prime }}{x}_{1^{\prime }jkr}\right)$$

(1)

The above objective is subject to some constraints. The first set of constraints is related to vehicle routing, defined in constraints (2)–(13), and described next, respectively. The evacuation and relief distribution vehicles must start their first trip from their designated depots; a vehicle must be allocated to the first trip if used; all subsequent trips must end at shelters; no arc will be passed if the vehicle is not in use; and any evacuation or distribution vehicle entering an EP or a shelter should use only one route.

$$\begin{array}{}\mathrm{(2)}& \hfill \sum _{j\in C}{x}_{1jk1}& =z_k\hfill & \hfill & \forall k\in K\hfill \mathrm{(3)}& \hfill \sum _{i\in DC}\sum _{j\in S}{x}_{ijk1}& =z_k^{\prime }\hfill & \hfill & \forall k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(4)}& \hfill f_{k1}& =z_k\hfill & \hfill & \forall k\in K\hfill \end{array}$$

$$\begin{array}{}\mathrm{(5)}& \hfill f_{k1}^{\prime }& =z_k\hfill & \hfill & \forall k\in K^{\prime }\hfill \mathrm{(6)}& \hfill \sum _{i\in C}\sum _{j\in S}{x}_{ijkr}& \le f_{kr}\hfill & \hfill & \forall r\in R,k\in K\hfill \mathrm{(7)}& \hfill \sum _{i\in DC}\sum _{j\in S}{x}_{ijkr}^{\prime }& \le f_{kr}^{\prime }\hfill & \hfill & \forall r\in R^{\prime },k\in K^{\prime }\hfill \mathrm{(8)}& \hfill \sum _{i\in V}\sum _{j\in V}\sum _{r\in R}{x}_{ijkr}& \le M·z_k\hfill & \hfill & \forall k\in K\hfill \mathrm{(9)}& \hfill \sum _{i\in V^{\prime }}\sum _{j\in V^{\prime }}\sum _{r\in R^{\prime }}{x}_{ijkr}^{\prime }& \le M·z_k^{\prime }\hfill & \hfill & \forall k\in K^{\prime }\hfill \mathrm{(10)}& \hfill \sum _{i\in 1\cup C,i\ne j}{x}_{ijkr}& =y_{jkr}\hfill & \hfill & \forall j\in C,k\in K,r\in R\hfill \mathrm{(11)}& \hfill \sum _{i\in C}{x}_{ijkr}& =y_{jkr}\hfill & \hfill & \forall j\in S,k\in K,r\in R\hfill \mathrm{(12)}& \hfill \sum _{i\in 1^{\prime }\cup DC,i\ne j}{x}_{ijkr}^{\prime }& =y_{jkr}^{\prime }\hfill & \hfill & \forall j\in DC,k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(13)}& \hfill \sum _{i\in DC}{x}_{ijkr}^{\prime }& =y_{jkr}^{\prime }\hfill & \hfill & \forall j\in S,k\in K^{\prime },r\in R^{\prime }\hfill \end{array}$$

To guarantee route continuity, the second set of constraints, (14)–(17), are applied. An evacuation vehicle leaves an EP after picking up evacuees; a distribution vehicle leaves a DC after taking some relief packages on board; and if an evacuation vehicle ends a trip at a shelter, it departs from the same shelter on the next trip unless it is the route’s final trip, and likewise for the relief distribution vehicles.

$$\begin{array}{}\mathrm{(14)}& \hfill \sum _{i\in V,i\ne h}{x}_{ihkr}& \le \sum _{j\in C\cup S,h\ne j}{x}_{hjkr}\hfill & \hfill & \forall h\in C,k\in K,r\in R\hfill \mathrm{(15)}& \hfill \sum _{i\in V^{\prime },i\ne h}{x}_{ihkr}^{\prime }& \le \sum _{j\in DC\cup S,h\ne j}{x}_{hjkr}^{\prime }\hfill & \hfill & \forall h\in DC,k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(16)}& \hfill \sum _{j\in C}{x}_{hjk(r+1)}& \le \sum _{i\in C}{x}_{ihkr}\hfill & \hfill & \forall j\in S,k\in K,r\in \tilde{R}\hfill \mathrm{(17)}& \hfill \sum _{i\in DC}{x}_{hjk(r+1)}& \le \sum _{i\in DC}{x}_{ihkr}\hfill & \hfill & \forall j\in S,k\in K^{\prime },r\in \widetilde{R^{\prime }}\hfill \end{array}$$

Next, we define the third set of constraints, (18)–(23), to manage the capacity limitations of resources. A vehicle cannot be assigned more evacuees than its capacity in a trip; the total number of relief packages delivered on each trip cannot exceed the vehicle’s capacity; and, finally, the relationship between each allocated vehicle, the evacuees, and the relief must be maintained.

$$\begin{array}{}\mathrm{(18)}& \hfill p_{ikr}& \le Q·z_k\hfill & \hfill & \forall i\in Ck\in K,r\in R\hfill \mathrm{(19)}& \hfill p_{ikr}^{\prime }& \le Q^{\prime }·z_k^{\prime }\hfill & \hfill & \forall i\in DC,k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(20)}& \hfill p_{ikr}& \le Q·y_{ikr}\hfill & \hfill & \forall i\in C,k\in K,r\in R\hfill \mathrm{(21)}& \hfill q_{ikr}& \le Q·y_{ikr}\hfill & \hfill & \forall i\in S,k\in K,r\in R\hfill \mathrm{(22)}& \hfill p_{ikr}^{\prime }& \le Q^{\prime }·y_{ikr}^{\prime }\hfill & \hfill & \forall i\in C^{\prime },k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(23)}& \hfill q_{ikr}^{\prime }& \le Q·y_{ikr}^{\prime }\hfill & \hfill & \forall i\in S,k\in K^{\prime },r\in R^{\prime }\hfill \end{array}$$

The fourth set of constraints, (24)–(30), handles demands allocations and distributions as follows: the number of relief packages delivered must be equal to the number of evacuees in each shelter; the number of evacuees collected from EPs must be equal to those dropped off at shelters; there should be an equal number of relief packages collected from DCs and delivered to shelters; all allocated evacuees are safely transported to shelters; all allocated relief at the DC is transported to shelters; and all shelters cannot accept more people or relief than can be accommodated.

$$\begin{array}{}\mathrm{(24)}& \hfill \sum _{k\in K}\sum _{r\in R}{q}_{ikr}& =\sum _{k\in K^{\prime }}\sum _{r\in R^{\prime }}{q}_{ikr}^{\prime }\hfill & \hfill & \forall i\in S\hfill \mathrm{(25)}& \hfill \sum _{i\in C}{p}_{ikr}& =\sum _{i\in S}{q}_{ikr}\hfill & \hfill & \forall k\in K,r\in R\hfill \mathrm{(26)}& \hfill \sum _{i\in DC}{p}_{ikr}^{\prime }& =\sum _{i\in S}{q}_{ikr}^{\prime }\hfill & \hfill & \forall k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(27)}& \hfill \sum _{k\in K}\sum _{r\in R}{p}_{ikr}& =d_i\hfill & \hfill & \forall i\in C\hfill \mathrm{(28)}& \hfill \sum _{k\in K^{\prime }}\sum _{r\in R^{\prime }}{p}_{ikr}^{\prime }& =d_i^{\prime }\hfill & \hfill & \forall i\in DC\hfill \mathrm{(29)}& \hfill \sum _{k\in K}\sum _{r\in R}{p}_{ikr}& \le L\hfill & \hfill & \forall i\in S\hfill \mathrm{(30)}& \hfill \sum _{k\in K^{\prime }}\sum _{r\in R^{\prime }}{p}_{ikr}^{\prime }& \le L_i^{\prime }\hfill & \hfill & \forall i\in S\hfill \end{array}$$

To maintain the time schedule, the fifth set of constraints (31)–(36) is employed as follows: an evacuation vehicle must arrive at an EP before the closing time window; distribution vehicles must start at a minimum of $\alpha$ on their first trip; and the arrival time must be kept on track when a vehicle visits a particular stop.

$$\begin{array}{}\mathrm{(31)}& \hfill a_{ikr}& \le b_i\hfill & \hfill & \forall i\in C,k\in K,r\in R\hfill \mathrm{(32)}& \hfill \alpha ·\sum _{j\in DC}{x}_{1jk1}^{\prime }& \le a_{1k1}^{\prime }\hfill & \hfill & \forall k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(33)}& \hfill a_{ikr}+(lt·p_{ikr})+t_{ij}-M·\left(1-x_{ijkr}\right)& \le a_{jkr}\hfill & \hfill & \forall i\in C,j\in C\cup S,k\in K,r\in R\hfill \mathrm{(34)}& \hfill a_{ikr}^{\prime }+s{t}^{\prime }+t_{ij}-M·(1-x_{ijkr}^{\prime })& \le a_{jkr}^{\prime }\hfill & \hfill & \forall j\in DC,i\in 1^{\prime }\cup S,k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(35)}& \hfill a_{ikr}+(ult·q_{ikr})+t_{ij}-M·\left(1-x_{ijk(r+1)}\right)& \le a_{jk(r+1)}\hfill & \hfill & \forall i\in S,j\in C,k\in K,r\in \tilde{R}\hfill \mathrm{(36)}& \hfill a_{ikr}^{\prime }+s{t}^{\prime }+t_{ij}-M·(1-x_{ijk(r+1)}^{\prime })& \le a_{jk(r+1)}^{\prime }\hfill & \hfill & \forall j\in S,i\in DC\cup S,k\in K^{\prime },r\in \widetilde{R^{\prime }}\hfill \end{array}$$

Finally, constraints (37)–(50) impose the domain of decision variables.

$$\begin{array}{}\mathrm{(37)}& \hfill x_{ijkr}& \in \{0,1\}\hfill & \hfill & \forall i,j\in V,k\in K,\forall r\in R\hfill \mathrm{(38)}& \hfill x_{ijkr}^{\prime }& \in \{0,1\}\hfill & \hfill & \forall i,j\in V^{\prime },k\in K^{\prime },r\in R^{\prime }\hfill \end{array}$$

$$\begin{array}{}\mathrm{(39)}& \hfill z_k& \in \{0,1\}\hfill & \hfill & \forall k\in K\hfill \mathrm{(40)}& \hfill z_k^{\prime }& \in \{0,1\}\hfill & \hfill & \forall k\in K^{\prime }\hfill \mathrm{(41)}& \hfill y_{ikr}& \in \{0,1\}\hfill & \hfill & \forall i,j\in C\cup S,k\in K,r\in R\hfill \mathrm{(42)}& \hfill y_{ikr}^{\prime }& \in \{0,1\}\hfill & \hfill & \forall i,j\in DC\cup S,k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(43)}& \hfill f_{kr}& \in \{0,1\}\hfill & \hfill & \forall k\in K,r\in R\hfill \mathrm{(44)}& \hfill f_{kr}^{\prime }& \in \{0,1\}\hfill & \hfill & \forall k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(45)}& \hfill p_{ikr}& \ge 0\hfill & \hfill & \forall i\in C,k\in K,r\in R\hfill \mathrm{(46)}& \hfill p_{ikr}^{\prime }& \ge 0\hfill & \hfill & \forall i\in DC,k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(47)}& \hfill q_{ikr}& \ge 0\hfill & \hfill & \forall i\in C,k\in K,r\in R\hfill \mathrm{(48)}& \hfill q_{ikr}^{\prime }& \ge 0\hfill & \hfill & \forall i\in DC,k\in K^{\prime },r\in R^{\prime }\hfill \mathrm{(49)}& \hfill a_{ikr}& \ge 0\hfill & \hfill & \forall i\in V,k\in K,r\in R\hfill \mathrm{(50)}& \hfill a_{ikr}^{\prime }& \ge 0\hfill & \hfill & \forall i\in V^{\prime },k\in K^{\prime },r\in R^{\prime }\hfill \end{array}$$

5. The Solution Approach

This section describes the MGA used in our integrated model to find approximate solutions within a reasonable time frame. Some modifications are inspired by the works of [31,32].

5.1. Solution Procedure

The procedure for the MGA is to:

• Generate N number of chromosomes as an initial population by employing a constructive heuristic approach;
• Evaluate the fitness of each chromosome in the population;
• Select the two fittest chromosomes to become parents;
• Apply the modified best cost route crossover operator to obtain their offspring;
• Improve the chromosomes using a set of mutation operators to guide the search towards a promising solution area;
• Apply the constructive heuristic approach to repair any infeasible chromosomes;
• Evaluate the fitness of each chromosome in the newly generated population;
• Replace the less-fit chromosomes with fitter offspring to maintain the size of the new population. The best solution (i.e., the fittest chromosome) obtained during this process will be recorded;
• Terminate if the stopping criterion is met (i.e., the maximum allowable number of generations is reached).

Algorithm 1 presents the pseudo-code for the MGA, and its flowchart is depicted in Figure 2. The following notations are used:

• Population size (N): the maximum number of chromosomes in one generation;
• Maximum generation ($maxGen$): the maximum number of times a new population can be generated;
• Stopping criteria ($\beta$): a particular value used to terminate the algorithm when the best solution remains unchanged;
• Crossover rate or probability ($P_c$): a ratio of how many couples will be picked for mating;
• Mutation rate ($P_m$): the probability that mutation will occur at a particular mating.

Algorithm 1 Modified genetic algorithm (MGA)

Require:  Numbers $N,maxGen,\beta ,P_c$ and $P_m$ Ensure:  Solution ($P^{*}$) 1: $S_0=P_1,\dots ,P_N\leftarrow Initial-Solutions$ ▹ /*Generate N feasible solutions as an initial population using the constructive heuristic*/ 2: Set $t=1$ ▹ /*Set an initial iteration counter*/ 3: $f_j,j=P_1,\dots ,P_N\leftarrow minVehicles$ ▹ /*Compute the fitness value of each chromosome 4: Determine the best fitness value at iteration t = 1, $f_{j|t}^{*}$ 5: while ($t<maxGen$) or ($f_{j|t-1}^{*}!=f_{j|t-\beta }^{*}$) do 6:   Randomly select fittest chromosomes $P_i$ and $P_j$, where $1\le i,j\le N,i\ne j$ 7:   $O_c,O_d\leftarrow P_i\oplus P_j=Best-Cost-Route-Crossover(P_i,P_j)$ ▹ /*Generate offspring $O_c$ and $O_d$ using crossover operator with $P_c$*/ 8:   $O_c,O_d\leftarrow Modified-Local-Search(O_c,O_d)$ ▹ /*Improve $O_c,O_d$ using mutation operator with $P_m$*/ 9:   $O_c,O_d\leftarrow Constructive-Heuristic(O_c,O_d)$ ▹ /* Repair $O_c,O_d$*/ 10:   if $O_c$ or $O_d$ is better than $P^{*}$ then 11:    $P^{*}\leftarrow O_c\vee O_d$ 12:   end if 13:   Update the best fitness value $f_{j|t}^{*}$ 14:   Set $t=t+1$ 15: end while 16: return ($P^{*}$)

Remark 1.

The algorithm stops if there is no further improvement on the best fitness value, when the best solution remains unchanged for β generations, or when the maximum generation maxGen is reached.

5.2. Initial Solutions

The MGA begins by generating an initial population, consisting of a set of N solutions. We propose the following constructive heuristic approach, which emphasizes diversifying the solutions, allowing their quality to be enhanced through the MGA:

• Create an empty set of tours;
• Create an empty tour for a vehicle;
• While the vehicle capacity and closing time window are not violated, carry out the following:

  (a)

  create a new trip;

  (b)

  randomly choose an unfulfilled $E{P}_i$ that still has demand (i.e., evacuees to be picked up) and add it to the current trip;

  (c)

  select another unfulfilled EP that is close to the current EP & insert it into the current trip;

• Find any possible shelter that still has available capacity to receive the evacuees. Add the shelter to the current trip;
• If there is still unsatisfied demand at some EPs, the vehicle will start another trip from the current shelter to serve those EPs (repeat Step 3). Otherwise, the tour will end at the last visited shelter. Add the constructed trip to the current tour;
• If there is still unfulfilled demand but it is infeasible because the vehicle cannot reach those EPs within its closing time window, then another vehicle will be selected to fulfil the remaining demand with a new tour. The new tour will start with the first trip from the depot. A set of new trips is then constructed by repeating Steps 2–6.

This procedure continues until the demand from all the EPs is met. Based on the number of evacuees dropped at each shelter, we then generate a set of tours for relief distribution using the same heuristic approach. Finally, all the constructed routes are concatenated to form a feasible solution or chromosome.

5.3. Fitness Calculation

The fitness value of each chromosome $P_i,i\in \{1,2,\dots ,N\}$ reflects the quality of the solution and is produced by a fitness function. In this research, the value is determined by the number of vehicles required to complete the entire operation and the total travel time of all of the used vehicles. The fitness function is composed of two parts. The first part is the minimum number of vehicles required to carry out the evacuation and distribution (1), and the second part is the contribution of the total traveling time to the system. The smaller the number of vehicles used and the total traveling time obtained, the fitter the chromosome is. Therefore, the fitness function is calculated as

$$\begin{array}{cc}\hfill f& =\sum _{j\in C}\sum _{k\in K}\sum _{r\in R}{x}_{1jkr}+\sum _{j\in DC}\sum _{k\in K^{\prime }}\sum _{r\in R^{\prime }}{x}_{1^{\prime }jkr}+\frac{\gamma }{1-{\displaystyle \sum _{i,j\in V,i\ne j}}{\displaystyle \sum _{k\in K}}{\displaystyle \sum _{r\in R}}{t}_{ij}{x}_{ijkr}},\hfill \end{array}$$

(51)

where $\gamma$ is a coefficient in [0, 1]. We set $\gamma =0.2$ in our experiments. The objective is to achieve the fittest chromosome as the best final solution to the problem.

5.4. Selection and Crossover

The selection of the fittest chromosomes from the current population is described next. The roulette wheel selection (RWS) method is used to scale the fitness values of the chromosomes in the population. For each chromosome, its probability of being selected is given by

$$p_i=\frac{f_i}{{\sum }_{j\in N}{f}_j},$$

(52)

where $f_i$ is the fitness value of a chromosome or individual in the population and N is the total number of chromosomes in the population. Inspired by the best cost route crossover method in [32], we use the following approach:

• Generate random number $R_c\in [0,1]$, for comparison with crossover probability $P_c$
• Set a value for $R_c$ by randomly generating a number between 0 and 1
• While $P_c\le R_c$, carry out the following:

  (a)

  randomly select two cut points on $P_i$ and $P_j$ for creating a sub-string.

  (b)

  from each parent, choose a sub-string randomly.

  (c)

  delete the genes present in the sub-string of one parent from the sub-string of the other parent to create offspring.

  (d)

  place the genes of the selected string of one parent in the offspring corresponding to the other parent. The method will place the genes of the selected string in the most feasible location in the corresponding offspring.

Remark 2.

To find the best possible location for each gene, we look for a location that satisfies both the vehicle capacity and the time horizon limitations.

5.5. Mutation and Repair

The local search procedure, referred to as the modified local search, plays a key role in the MGA as the mutation operator to improve the solutions. We have combined the moves in both intra-route and inter-route operations. Intra-route moves involve adjustments made within the same route, whereas inter-route moves involve operations between two distinct routes. Specifically, for the evacuation segment within the chromosome, the intra-route moves are detailed as follows:

• Intra-node insertion: an EP or a shelter is relocated within the same route;
• Intra-node swap: two EPs that are in the same route are swapped or exchanged their positions;
• Intra-arc insertion: an arc of two nodes in a particular route, which could be two consecutive EPs or one EP and one shelter, is relocated within the same route;
• Intra-arc swap: two arcs of consecutive nodes (either two EPs or one EP and one shelter) are swapped within the same route.

In inter-route moves, the operator works between the routes. The inter-route moves (for evacuation string) are as follows:

• Inter-node insertion: an EP or a shelter from one route is selected and relocated into a different route within the same chromosome;
• Inter-node swap: two nodes (EPs or shelters) from different routes are swapped, or their positions are exchanged;
• Inter-arc insertion: an arc of two nodes in a particular route, which could be two consecutive EPs or one EP and one shelter, is relocated to a different route;
• Inter-arc swap: two arcs of consecutive nodes (either two EPs or one EP and one shelter) in different routes are swapped.

Remark 3.

To maintain the feasibility of each chromosome after mutation, we employed a repair operator using our constructive heuristic. The chromosome is feasible if the time windows and the vehicle and shelter capacity are satisfied. Similar moves are applied to the relief distribution string in the chromosome.

6. Numerical Experiments and Discussion

We first demonstrate the performance of the proposed integrated model and solution approach using synthetic datasets. Then, we apply this model and approach to a case study in the Bontoala district of Indonesia. Our numerical experiments were conducted on a computer equipped with an Intel(R) Core(TM) i5-6500 CPU @3.20 GHz and 8.00 GB of RAM, utilizing Python 3.7 and the CPLEX solver 12.10.

The selection of parameter values for the MGA is problem-dependent; it depends on the problem search space. Therefore, we determined the most suitable parameter combination through some experiments. The parameters employed in this research are specified as follows: $N=100,maxGen=100,\beta =20,P_c=0.8$, and $P_m=0.1$.

6.1. Results Using Synthetic Datasets

We have generated several synthetic datasets with different numbers of EPs, shelters, and DCs. The traveling time and network detail for one of the datasets are presented in Figure 3. In this figure, there are 11 nodes. Node 1 represents the evacuation vehicle depot, where eight identical vehicles, each with a capacity of 20 people, are available. Six evacuation points, $E{P}_2,E{P}_3,E{P}_4,E{P}_5,E{P}_6$, have a demand of $d_2=d_3=30,d_4=20,d_5=d_6=10$, respectively. The temporary shelters with a capacity of 60 people are located at nodes $S_7$ and $S_8$. It is assumed that it will take 10 min to load and unload evacuees at a shelter, respectively. There are two predetermined DCs ($D{C}_9$ and $D{C}_{10}$) with a capacity of $L_9^{\prime }=L_{10}^{\prime }=100$ and a service time of 10 min. Four identical distribution vehicles with a capacity of 50 are available at the distribution vehicle depot $1^{\prime }$ (node 11). It is assumed that the flood will hit the predicted area within 2-h. The distribution of relief packages will start 1-h after the start of the evacuation. We assume that the completion time for each distribution vehicle is 150 min.

To demonstrate the effectiveness of our integrated model, we compare it with the standard hierarchical model. In the hierarchical model, the minimum number of vehicles required for the evacuation task is calculated first, followed by optimizing the relief distribution task. The comparative results are presented in Figure 4 and

Table 1. Evacuation and relief distribution routing plans derived from the integrated and hierarchical models.

Integrated Model
Route/Vehicle	Trip of Vehicle	No. of Evacuees Picked up at Node	Arrival time at node
1	1–2–7	0–20–0	0–20–40
	7–2–7	0–10–0	50–60–80
2	1–3–7	0–20–0	0–30–80
3	1–4–8	0–20-0	0–10–60
	8–6–5-8	0–10–0	70–80–95–120
4	1–3–8	0–20—0	0–30–80
Total evacuation vehicles: 4
Route/vehicle	Trip of vehicle	Relief packages picked up at node	Arrival time at node
1	11–9–7–10–8	0–50–0–40–0	60–70–100–125–150
Total distribution vehicles: 1
Hierarchical Model
Route/vehicle	Trip of vehicle	No. of evacuees picked up at node	Arrival time at node
1	1–7	0–20–0	0–20–40
	7–2–7	0–10–0	50–60–80
2	1–3–7	0–20–0	0–30–80
3	1–4–8	0–20–0	0–10–60
	8–6–5–8	0–10–0	70–80–95–120
4	1–3–7	0–20–0	0–30–80
Total evacuation vehicles: 4
Route/vehicle	Trip of vehicle	Relief packages picked up at node	Arrival time at node
1	11–9–7–10–8	0–50–0–40–0	60–70–100–125–150
2	11–10–7	0–10–0	60–80–105
Total distribution vehicles: 2

. Our findings indicate that our integrated model outperforms the hierarchical approach for both the evacuation and relief distribution tasks.

Specifically, while the hierarchical model necessitates two distribution vehicles and four evacuation vehicles, our integrated model achieves the same task with just one distribution vehicle and four evacuation vehicles. This improvement is primarily attributed to the hierarchical model’s allocation of 60 evacuees to shelter $S_7$ and 40 evacuees to shelter $S_8$, which results in the need for an additional distribution vehicle to adequately serve shelter $S_7$ as each distribution vehicle can only accommodate up to 50 relief packages. In contrast, our integrated model optimally allocates 50 evacuees to both shelter $S_7$ and shelter $S_8$. Although vehicle 4’s arrival time remains consistent across both models, the integrated model’s choice of shelter $S_8$ as the final stop enables all of the required relief supplies to be delivered by a single distribution vehicle.

As an additional analysis, we conducted computational experiments using both an exact approach and the MGA on six simulated flood problems/instances (named $PI1$–$PI6$). Each instance comprises a network of nodes representing the positions and distances between depots, EPs, shelters, and DCs. The loading and unloading times vary based on the number of evacuees. Given that the MGA is an approximate method, we executed ten independent runs for each instance and computed the average solutions. The results are summarized in

Table 2. Results obtained by the exact approach and the MGA on synthetic flood problems.

Instances	Size $\left(\right|\mathit{C}|/|\mathit{S}|/|\mathbf{DC}\left|\right)$	#Vehicles (EV/RD)				CPU Time (s)
		Exact	MGA	sdv	Exact	MGA	sdt	Time Reduction
$PI1$	5/2/2	4/1	4/1	0.49	341.7	40.1	2.30	88.3 (%)
$PI2$	6/3/2	5/1	5/1	0.64	1454.7	46.3	6.09	96.8 (%)
$PI3$	8/4/3	-/-	8/2	0.49	-	47.0	1.10	-
$PI4$	10/5/3	-/-	10/2	0.46	-	57.4	1.05	-
$PI5$	13/7/3	-/-	12/2	0.49	-	61.2	5.37	-
$PI6$	15 /8/3	-/-	15/3	0.51	-	59.5	2.94	-

. The column ‘$\left(\right|C|/|S|/|DC\left|\right)$’ indicates the number of EPs, shelters, and DCs in each instance. The column ‘#vehicles (EV/RD)’ shows the total number of vehicles required for evacuation and relief distribution, obtained using both the exact (‘Exact’) and the MGA (MGA’). For example, ‘4/1’ means that four evacuation vehicles and one relief distribution vehicle are necessary for the integrated operation.

The computational time (in seconds) is reported in the column labeled ‘CPU time ($sec$)’. The column ‘${\mathrm{sd}}_v$’ presents the standard deviation of the solutions for the total minimum vehicles (evacuation + relief distribution) across the ten experiments conducted using the MGA. Meanwhile, column ‘${\mathrm{sd}}_t$’ lists the standard deviation of the corresponding CPU time. The final column, ‘Time reduction (%)’, indicates the percentage reduction in computational time achieved by the MGA compared to the exact approach. Due to the exact approach’s significant increase in computational time with larger problem sizes, we limited its runtime to one hour.

The results from this table show that the exact approach was able to produce optimal solutions only for $PI1$ and $PI2$. However, it failed to produce any solution for $PI3$–$PI6$ within the given time limit (denoted by ‘-’), whereas the MGA obtained optimal or near-optimal solutions for all instances within a reasonable time. When comparing the CPU time required by the exact approach and the MGA for $PI1$ and $PI2$, it is evident that the MGA completed the computations notably faster, achieving a time reduction of 92.5%.

6.2. Results on the Case Study from Indonesia

Indonesia experiences a significant number of floods annually, resulting in substantial human and economic losses, making it one of the most flood-prone countries. According to the 2020 report by the Indonesian National Board for Disaster Management (BNBP) [33], South Sulawesi province, where the Bontoala district is situated, has low capacity and effectiveness in handling major floods.

On 23 January 2019, the entire district experienced a devastating flash flood. Unfortunately, evacuation efforts commenced only after floodwaters had already inundated the area. The limited availability of vehicles posed a major challenge in evacuating and rescuing victims. The incident resulted in at least 45 fatalities, 25 missing persons, 47 injuries, and the evacuation of 2121 individuals across the district.

Given these challenges, we applied our proposed model to this region as a case study to develop more effective strategies for future evacuation routing and scheduling, as well as relief distribution.

According to data from the Indonesia Bureau of Statistics and the National Disaster Management Agency (https://bnpb.go.id (accessed on 6 May 2021)), we estimated the demand at each evacuation point (Figure 5) and their respective closing time windows (in minutes).

The total expected number of evacuees was 2296, as detailed in [29]. These individuals were to be transported by the provided buses to designated temporary shelters located in safer areas within the region (Figure 6). Each bus has a capacity of 50 people, and specific shelter capacities are outlined in [29]. For further details on the evacuation data, please refer to the cited paper. It is assumed that the flood would reach the predicted area within 3.5 h.

Since the proposed model involves relief supply distribution, we ensured that each evacuee received a package containing essential relief items such as water, food, blankets, and medical supplies. In the Bontoala region, five locations were designated as distribution centers (DCs), as detailed in

Table 3. Location and capacity of distribution centers (DCs).

DC	Location	Capacity
$D{C}_1$	Kantor Desa Tamanyeleng	1000
$D{C}_2$	SMKN 1 Pallangga	1000
$D{C}_3$	Gudang BPPD Sulawesi Selatan	1000
$D{C}_4$	Gudang Pajak Gowa	1000
$D{C}_5$	Gudang Kementerian Dalam Negeri	1000

. Trucks with a capacity of 300 packages each were used to distribute the relief supplies.

Due to the large number of nodes in the problem, solving the mathematical model exactly within a reasonable computational time became challenging, as illustrated in Section 6.1. Therefore, to enhance the solution efficiency while preserving near-optimal quality, we employed the MGA. We conducted ten experiments and averaged the obtained solutions. The results are summarized in

Table 4. Results of evacuation operations using the MGA.

Vehicle	Trip of Vehicle	Evacuees	Arrival Time (min)
1	0–$E{P}_{15}$–$E{P}_{16}$–$S_7$	0–19–31–50	0–9–29–67
	$S_7$–$E{P}_{11}$–$S_7$	0–50–50	117–120–173
2	0–$E{P}_{13}$–$S_7$	0–50–50	0–8–63
	$S_7$–$E{P}_{11}$–$S_7$	0– 50–50	113–116–169
3	0–$E{P}_{21}$–$S_2$	0–50–50	0–7–63
	$S_2$–$E{P}_{11}$–$S_1$	0–17–17	113–125–133
4	0–$E{P}_{18}$–$S_1$	0–40–40	0–8–55
	$S_1$–$E{P}_{16}$–$S_1$	0–32–32	105–111–149
5	0–$E{P}_{15}$–$S_7$	0–50–50	0–9–67
	$S_7$–$E{P}_{14}$–$S_7$	0–28–28	117–122–155
6	0–$E{P}_{20}$–$S_1$	0–50–50	0–6–66
	$S_1$–$E{P}_{20}$–$E{P}_{27}$–$S_1$	0–4–26–30	116–125–132–171
7	0–$E{P}_2$–$E{P}_{22}$–$S_6$	0–34–14–48	0–8–47–70
	$S_6$–$E{P}_{12}$–$S_6$	0–50–50	118–123–178
8	0–$E{P}_8$–$S_5$	0–50–50	0–4–64
	$S_5$–$E{P}_{12}$–$S_6$–$S_1$	0–50–50	114–123–178
9	0–$E{P}_8$–$S_6$	0–50–50	0–4–61
	$S_5$–$E{P}_{17}$–$S_5$	0–50–50	111–119–182
10	0–$E{P}_{26}$–$S_1$	0–50–50	0–9–70
	$S_1$–$E{P}_{25}$–$S_1$–$S_1$	0–50–50	120–130–190
11	0–$E{P}_2$–$S_7$	0–50–50	0–8–64
	$S_7$–$E{P}_{27}$–$S_7$–$S_1$	0–50–50	94–105–166
12	0–$E{P}_3$–$S_8$	0–50–50	0–6–61
	$S_8$–$E{P}_9$–$S_5$–$S_1$	0–50–50	111–116–176
13	0–$E{P}_{17}$–$E{P}_{15}$–$S_5$	0–13–37–50	0–8–23–72
	$S_5$–$E{P}_{24}$–$S_2$–$S_1$	0–46–46	112–126–187
14	0–$E{P}_5$–$S_8$	0–50–50	0–4–58
	$S_8$–$E{P}_7$–$E{P}_{11}$–$S_7$	0–33–17–50	108–112–146–166
15	0–$E{P}_{10}$–$E{P}_{21}$–$S_7$	0–20–28–48	0–6–32–69
16	0–$E{P}_{11}$–$S_7$	0–50–50	0–6–59
	$S_7$–$E{P}_{10}$–$S_7$–$S_1$	0–50–50	109–112–165
17	0–$E{P}_{19}$–$S_2$	0–50–50	0–11–13
	$S_2$–$E{P}_{22}$–$S_2$–$S_1$	0–50–50	63–75–137
18	0–$E{P}_9$–$E{P}_{13}$–$S_5$	0–30–16–46	0–3–36–62
	$S_5$–$E{P}_{14}$–$S_5$–$S_1$	0–22–22	108–118–150
19	0–$E{P}_{25}$–$E{P}_{26}$–$S_2$	0–10–40–50	0–9–23–78
	$S_2$–$E{P}_{19}$–$S_2$–$S_1$	0–50–50	128–130–182
20	0–$E{P}_{18}$–$S_2$	0–50–50	0–8–70
	$S_2$–$E{P}_{19}$–$S_2$–$S_1$	0–14–14	120–122–138
21	0–$E{P}_{20}$–$S_7$	0–50–50	0–6–62
	$S_7$–$E{P}_{14}$–$S_7$	0–38–38	112–117–160
22	0–$E{P}_{24}$–$S_1$	0–46–46	0–10–65
	$S_1$–$E{P}_{23}$–$S_1$	0–50–50	111–120–179
23	0–$E{P}_1$–$E{P}_3$–$S_8$	0–28–22–50	0–8–39–66
	$S_8$–$E{P}_2$–$E{P}_{25}$–$S_1$	0–27–23–50	116–123–156–190
24	0–$E{P}_5$–$S_8$	0–50–50	0–4–58
	$S_8$–$E{P}_4$–$S_8$	0–22–22	108–113–140
25	0–$E{P}_{26}$–$S_8$	0–50–50	0–2–54
	$S_8$–$E{P}_6$–$E{P}_7$–$S_7$	0–3–47–50	104–108–113–166
26	0–$E{P}_{24}$–$S_8$	0–50–50	0–5–60
	$S_8$–$E{P}_4$–$E{P}_9$–$S_7$	0–24–18–42	110–115–142–165
27	0–$E{P}_1$–$S_8$	0–50–50	0–8–65
	$S_8$–$E{P}_5$–$S_8$	0–16–16	115–119–139

and

Table 5. Results obtained by the MGA on the Bontoala case study.

Run No.	EV Vehicles	RD Vehicles	Total Vehicles	CPU Time (s)
1	28	10	38	69.8
2	27	10	37	61.0
3	28	11	39	68.6
4	27	10	37	62.8
5	27	11	40	61.9
6	27	10	37	60.3
7	28	12	40	60.9
8	27	11	38	63.4
9	28	10	38	62.8
10	27	11	40	69.4
Mean	27	11	38	66.5
sd	0.75	0.66	1.20	3.18
Best	27	10	37	61.9

.

Table 4 provides a detailed overview of the evacuation operation solution obtained from a single computational run using the MGA. The table includes information such as the route (‘Trip of vehicle’) for each vehicle, encompassing the depot, EP(s), and shelters. The column labeled ‘Evacuees’ indicates the number of evacuees picked up and dropped off at each sequence node. Additionally, the arrival time of the vehicle at consecutive visited nodes is presented in the column labeled ‘Arrival time (min)’.

In Table 5, the columns labeled ‘Run No’, ‘EV vehicles’, and ‘RD vehicles’ denote the sequence of computational runs, the optimal solution for the number of evacuation vehicles required, and the optimal solution for the total number of relief distribution vehicles, respectively. The column ‘Total vehicles’ presents the total number of vehicles needed for both the evacuation and relief distribution tasks. The computational runtime (in seconds) for solving each instance is recorded in the CPU time (s)’ column. Additionally, the table includes average results (‘Mean’), standard deviations (‘sd’), and the best results achieved (‘Best’).

The results demonstrate the MGA’s effectiveness, yielding an average objective value of 38. This signifies that 27 buses are minimally required for evacuating vulnerable individuals from flood-prone areas. Additionally, to ensure timely initial emergency relief for these evacuees upon their arrival at designated shelters, 11 trucks are needed for relief distribution. The small standard deviations observed in the results (0.75 for evacuation vehicles and 0.66 for distribution vehicles) indicate consistent performance across different scenarios. This consistency enhances the reliability of our findings and underscores the MGA’s robust performance.

In summary, our empirical results demonstrate that the proposed integrated model, along with the MGA, can assist disaster relief authorities at the early phase of a flood to determine minimum fleet sizes and their corresponding optimal routes for both evacuation and relief distribution within reasonable computational time limits.

Furthermore, the MGA’s effectiveness in finding high-quality solutions is underscored by its robust convergence properties, ensuring it consistently converges to near-optimal solutions within a reasonable timeframe. Table 2 and Table 5 illustrate small standard deviations across multiple runs, indicating the reliable convergence of the MGA. In our case study, the MGA consistently achieved an average near-optimal objective value of 38 across multiple executions, with a completed time of up to two minutes. This reliability is attributed to the fine-tuned exploration mechanisms in the MGA, including selection, crossover, and mutation, which optimize convergence speed while maintaining solution quality. Consequently, the MGA’s convergence behaviour ensures its capacity to deliver robust and timely solutions in intricate disaster relief scenarios.

7. Conclusions and Future Directions

In practical disaster scenarios, the swift coordination of evacuation and relief distribution tasks, optimized with minimal vehicle usage, is crucial for effective disaster relief operations.

This paper introduces a vehicle routing model designed to efficiently manage simultaneous evacuation and relief distribution tasks during the initial stages of a flood. The model integrates multi-trip and split delivery routing strategies, accommodates uncertainties in evacuation demands, and incorporates closing time windows at evacuation points. However, achieving optimal solutions for such routing challenges, especially in large-scale scenarios, can be time-intensive when using exact approaches. To overcome this challenge, we have adapted and applied a modified genetic algorithm to solve the integrated model.

Our proposed model and solution approach were validated through testing on various simulated flood scenarios and a real-life case study in Indonesia. The experimental results demonstrate that our integrated model requires fewer vehicles for evacuation and relief distribution tasks compared to standard models. Furthermore, the computational time was significantly reduced using the modified genetic algorithm to obtain approximate solutions. For instance, in larger scenarios $PI3$–$PI6$, where exact methods struggled, our approach delivered near-optimal solutions promptly. In smaller cases $PI1$ and $PI2$, our method achieved optimal or near-optimal solutions with a computational time reduction of up to 92.5% compared to exact approaches. These findings underscore the effectiveness of our model and solution approach as a robust tool for optimizing evacuation and relief distribution plans under resource constraints within reasonable timeframes.

In real-world disaster-prone areas, various social, health, demographic, and geographical factors influence operational dynamics. Despite these unique conditions, principles and strategies derived from our study can offer transferable insights to other regions. The proposed methodologies, problem-solving approaches, and algorithm are applicable to other flood-impacted areas, with adjustments tailored to local conditions.

During life-threatening situations, such as rescue operations under high-stress conditions, unforeseen adjustments like altering the number of stops for vehicles may become imperative but were not explicitly covered in our model. Future extensions of our model could consider these dynamic scenarios. Moreover, integrating real-time electronic databases could bolster the robustness of the evacuation-relief distribution model. Furthermore, one can explore additional task integration and alternative solution approaches to enhance the reliability of the proposed model.