Sustainability-Reliable Emergency Facility Location Determination with Consideration of Complex Polygonal Barriers and the Risk of Facility Disruption

Abstract

This paper presents a reliable emergency facility location optimization model that considers complex polygon barriers and the risk of facility disruption. From a sustainable perspective, with capacity, cost, and demand constraints, the model maximizes time satisfaction and minimizes cost as the objective function to determine the optimal facility location and allocation. The paper proposes the barrier path optimization algorithm and the Kepler optimization algorithm (KOA) to solve the model and validates the model and algorithm through simulation experiments of various scales. Finally, the paper conducts a sensitivity analysis of the disruption probability using the control variable method to explore the impact of parameter changes on the decision results and compare the advantages of considering a sustainable perspective versus not considering it. The results show that the model and algorithm designed in this paper can effectively optimize the barrier path and obtain the optimal location-allocation scheme. The research findings will provide mathematical models and methodological strategies for emergency facility location decision-making.

1. Introduction

Emergency facility location decision-making is the foundation of emergency rescue operations. In the context of global climate change and frequent natural disasters, the rational layout of emergency facilities has become a key issue concerning public safety and social stability worldwide. For example, in the 2011 Tohoku earthquake in Japan, the unreasonable distribution of emergency shelters led to delays in rescue work, resulting in more casualties; in the 2020 Australian bushfires, the shortage of emergency material storage points made it difficult to deliver relief supplies to affected areas in a timely manner. These international cases fully demonstrate that efficient facility location and rational resource allocation are crucial for the success of rescue operations following various types of emergencies.

Due to uncertainties in reality, traditional emergency facilities generally assume that once established, the facilities are completely reliable. However, over time, disruptions in facility operation can occur and have an impact on the efficient conduct of rescue work. In addition, emergency events often involve barriers such as collapsed buildings, sunken road surfaces, and falling rocks. To enhance environmental protection while ensuring efficient rescue operations from a sustainable perspective, it is important to consider the environmental pressures associated with transporting relief supplies. Therefore, researching reliable and efficient location-allocation schemes for sustainable emergency facilities under scenarios involving complex polygonal barriers holds significant importance.

2. Literature Review

In the existing emergency facility location literature, the facility-disruption scenarios mainly come from the facilities themselves, while reliability, as a safety evaluation of protecting the system from emergencies, needs to be analyzed from multiple angles to ensure that the selected emergency facilities can play the greatest role in disasters. The reliability facility location problem (RFLP) was first proposed by Snyder and Daskin [1] in 2005, and an optimal Lagrangian relaxation algorithm was designed to solve it. Subsequently, considering reliable facility location and material distribution, Mohammadi et al. [2] proposed a new multi-objective reliability emergency facility location optimization model for humanitarian decision-making. In addition, designing a reliable supply chain network is also an effective method for handling emergency facility disruptions and uncertain demand. Tolooie et al. [3] consider multi-period capable facility location and distribution scenarios and build a two-stage stochastic mixed integer model. A L-shaped stochastic linear programming method is then designed to solve the model. Haghjoo et al. [4] addressed the problem of disruption risk and demand uncertainty in emergency facilities in disaster scenarios, and built a dynamic supply chain network location-allocation model. For large-scale problems, they proposed two meta-heuristic algorithms, namely the self-adaptive imperialist competitive algorithm (ICA) and the invasive weed optimization (IWO), to solve the model. A robust and service-limited emergency facility location optimization model was proposed considering facility-disruption risk in [5]. The model was solved using the Non-dominated Sorting Genetic Algorithm II (NSGA-II) with an elite strategy to obtain a multi-objective Pareto solution set. More information on reliability-oriented emergency facility location can be found in [6,7,8,9] and the references therein.

Barriers are objectively present and vary in emergency facility location; they affect the efficiency of emergency response by affecting the transportation distance of rescue materials. When barrier factors are interwoven throughout the location decision-making process, how to scientifically and flexibly bypass barriers and optimize routes is the focus of this paper. After an emergency event occurs in the complex geographical environment with diverse barriers, Kate and Cooper [10] first considered the constraint of barriers in the Weber facility location model, and depicted the barriers with circular regions. They built a single-objective facility location optimization model and solved it. Amiri-Aref et al. [11] proposed an extended multi-weber facility location problem and designed an exact divide-and-conquer-based strategy (EDC) and an approximate divide-and-conquer-based strategy (ADC) to solve the model. The literature [12] proposed a multi-emergency facility location problem under convex barriers, considering that the barrier area cannot be selected, and designed a gray wolf optimization algorithm coupled with a visible convex point barrier path algorithm to solve the model. In addition, polygonal barriers often cause changes in the Voronoi diagram structure, and Byrne and Kalcsics [13] conducted a detailed study of the facility location problem with continuous demand and polygonal barriers. More research on facility location under more barrier constraints can be referred to [14,15,16].

Due to the urgency of emergencies, people often focus excessively on service efficiency, leading to large-scale waste and ecological pollution. This not only adds pressure to reconstruction efforts but also increases environmental burdens, making sustainable scenarios crucial for research on emergency facility location. Li et al. [17] reviewed models and optimization techniques for emergency response facility location and planning in the literature over the past few decades, noting a prevalence of single-objective models compared to multi-objective ones that comprehensively integrate various factors. In research on multi-objective problems, as early as 1980, Schilling [18] developed a dynamic, multi-period version of the Maximum Coverage Location Problem (MCLP) and adopted multi-objective methods to obtain near-optimal solutions. Malik et al. [19] explored multi-stage facility location decisions amid highly uncertain demand during crises (e.g., epidemics). They analyzed trade-offs between economic costs and carbon dioxide emissions from facility operations, proposing a multi-objective method based on the Non-dominated Sorting Genetic Algorithm II (NSGA-II). Carnero Quispe et al. [20] conducted a systematic literature review and found that most studies focus on constructing temporary facilities with little attention to their closure. Tatarczak et al. [21] proposed a framework for building alliances suitable for horizontal supply chain collaboration, balancing cost and efficiency. This framework enables multi-criteria joint planning and employs a multi-criteria group decision-making approach, achieving optimization by integrating methodologies such as the Analytic Hierarchy Process, Choquet Integral, and Shapley Value. Tatarczak and Anna [22] introduced an innovative mixed group decision-making method for alliance formation in multi-enterprise collaborative replenishment, integrating Multi-Criteria Decision-Making (MCDM) and Intuitive Fuzzy Set (IFS) theory to enhance operational efficiency. Ali et al. [23] considered the problem of scrap vehicle recycling from a sustainable perspective and proposed an efficient multi-criteria optimization method, N-type-2-ARAS, for a detailed study in practical situations. Ala et al. [24] constructed a dynamic capacity facility location model for mobile renewable energy charging stations, considering two-stage stochastic programming. To better balance vehicle working time and operating costs in sustainable facility location, Shi et al. [25] established a double objective mixed integer linear programming model. More research on facility location under sustainable conditions can be referred to [26,27,28,29].

In summary, existing research on reliable emergency facility location under barrier scenarios remains limited: most studies only account for convex barriers, failing to address the complex concave polygonal barriers prevalent in real-world environments. Moreover, while some works touch on aspects of rescue efficiency or environmental impact, few integrate these elements comprehensively in location decisions. This paper introduces a novel perspective by simultaneously incorporating diverse complex polygonal barriers—an often-overlooked feature in prior models—and adopting a holistic approach that balances sustainability (encompassing environmental pressures) and rescue efficiency from both demand area and resource allocation viewpoints. Unlike existing multi-objective models that typically focus on narrow criteria, our bi-objective optimization framework for sustainable emergency facility location under complex polygonal barriers offers a more comprehensive solution. By combining the Kepler optimization algorithm (KOA) with the convex hull method and path optimization algorithms to optimize barrier-avoidance paths, this research fills a critical gap in current literature. The resulting model and solution approach provide actionable insights for real-world emergency facility location decisions, enhancing responsiveness to sudden disasters.

3. Mathematical Model Description

3.1. Problem Formulation

Due to the suddenness of disasters and the impact of secondary disasters, some established emergency facilities may suffer varying degrees of damage, resulting in a loss of disaster relief capabilities. In such cases, the supply of materials at many relief points must be met by other emergency facilities, leading to a significant reduction in disaster relief efficiency. Therefore, potential facility-disruption scenarios need to be considered during the optimization process for emergency facility location. Furthermore, the specificity of environmental sustainability rescue also imposes higher requirements on emergency facility location decisions. In sustainable rescue operations, low-carbon rescue and service coverage need to be as extensive as possible to reach as many affected people as possible. Additionally, due to changes in terrain that often accompany disasters such as ground fractures, landslides, and mudslides, complex geographical environments create transportation challenges. Timely rescue efforts can greatly reduce the suffering and losses of affected populations; therefore, response speed and rational resource allocation under complex barrier scenarios are crucial. These requirements make decision-making for sustainable emergency facility location under complex polygonal barrier scenarios more complicated and challenging. This paper focuses on emergency facility location decisions and uses satisfaction with demand-side time and cost consumption from a sustainable perspective as objective functions. It constructs an objective evaluation system for decision-making based on these criteria before building a bi-objective optimization model for sustainable emergency facility location under complex polygonal barrier scenarios. The assumptions made by this model are:

(1)

Clustering is used to group emergency demand points, and the emergency demand areas are depicted using circular regions.

(2)

Various types of polygons are used to depict barrier areas, and transportation cannot pass through these areas.

(3)

Facility locations are not placed within barrier areas or the demand regions.

(4)

The shortest path optimization distance is the straight-line distance between the facility point and the centroid of the irregular demand region.

(5)

The feasible regions between facility points and demand regions are all accessible.

The mathematical notations that we will use hereafter are introduced in

Table 1. The notations.

Indexes
i	the facility location.
j	the demand region.
k	the barrier area.
Sets
M	the set of facility nodes, $\forall i\in M$.
N	the set of demand nodes, $\forall j\in N$.
K	the set of barrier nodes, $\forall k\in K$.
D	the set of circular demand regions, $\forall D_j\in D$.
F	the set of facility locations, $\forall F_i\in F$.
B	the set of barrier areas, $\forall B_k\in B$.
C	the set of facility construction costs, $\forall c_i\in C$.
P	the set of feasible areas, $P=R^2\setminus int\left(B\right)$.
Parameters
${\omega }_1$	the weighted satisfaction of the demand region without facility disruption.
${\omega }_2$	the weighted satisfaction of the demand region with the risk of facility disruption.
$c_i$	the per cost of building a facility location i.
$c_{ij}$	the per transportation cost between facility location i and demand region j,
	which is measured in dollar.
$t_{ij}$	the time between facility location i and demand region j.
$v_{ij}$	the average speed between facility location i and demand region j.
$d_{\mathit{ij}}$	the distance between facility location i and demand region j.
$U_j$	the upper bound supply of demand region.
$L_j$	the lower bound supply of demand region.
${\beta }_i$	the positive time sensitivity coefficient in the Sigmoid time satisfaction function.
$S_i$	the maximum amount of resources that facility location i can provide.
$L_i$	the resource safety stock (i.e., the minimum stock) of facility location i.
$w_{ij}$	the quantity of supply allocates facility location i to demand region j.
$w_j$	the quantity of demand resource in demand region j.
p	the total number of facilities constructed.
${\lambda }_i$	the increased cost of facility location i disruption.
u	the cost of returning an empty vehicle.
$V_c$	the loading quantity of the vehicle from facility location i to demand region j.
$EM$	the cost required per ton of carbon dioxide production.
${\delta }_i$	the carbon dioxide emissions generated by facility location i.
${\varphi }_j$	the amount of CO2 generated between facilities to demand region j unit distance,
	which is measured in kilometer.
$Y_{ij}$	It equals 1 if vehicle from facility location i to demand region j, 0 otherwise.
q	the probability of facility disruption.
Q	the budget limit.
$p_j(p_{x_j},p_{y_j})$	the center coordinate of circular demand region j.
$(x_{c_1},y_{c_1})$	the upper bound of map coordinates.
$(x_{c_0},y_{c_0})$	the lower bound of map coordinates.

.

3.2. Time Satisfaction Function

In emergency management, time is a critical factor as it directly relates to the efficiency and effectiveness of rescue operations. The time satisfaction function is used to measure the level of satisfaction with the time it takes for demand points to receive supplies or services from emergency facility points. The time satisfaction function is typically a decreasing function, indicating that as time increases, the satisfaction level at demand points decreases. This is because rapid response is crucial in emergency situations. If an emergency facility point is far from a demand point or if transportation times are long, then the satisfaction level at the demand point will decrease. By introducing the time satisfaction function, we can consider time factors in the location selection process and choose facility points that can provide services to demand points more quickly. This not only improves emergency response efficiency but also reduces disaster losses, thereby protecting people’s lives and property. The satisfaction function is established based on literature [30], set $t_{ij}$, $v_{ij}$, and $d_{ij}$ represent the time, average speed, and distance from the i th facility to the j th demand area. $L_j$ and $U_j$ are the upper and lower bounds on the waiting time for disaster relief materials to be obtained in the demand area, with $L_j\le U_j$, and introduce the Sigmoid time satisfaction function with a decreasing index as follows

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f\left(t_{ij}\right)=\left\{\begin{array}{cc}1,\hfill & if{d}_{ij}\le L_j{v}_{ij},\hfill \\ {\displaystyle \frac{2e^{-{\beta }_i(d_{ij}/v_{ij}-L_j)}}{1+e^{-{\beta }_i(d_{ij}/v_{ij}-L_j)}}},\hfill & if{d}_{ij}>L_j{v}_{ij},\hfill \end{array}\right.\end{array}\end{array}$$

where ${\beta }_i$ is a positive coefficient that indicates the level of sensitivity over time, with a larger value representing stronger sensitivity as shown in Figure 1. The figures represent the Sigmoid time satisfaction functions with decreasing indices for parameters ${\beta }_i$ of $0.02$, $0.03$, $0.04$, $0.05$, and $0.005$.

3.3. Model Formulation

In this section, we construct a bi-objective optimization model for the sustainable location of emergency facilities under complex polygonal barrier scenarios. The model considers cost consumption and demand-side time satisfaction from a sustainability perspective as the objective functions. The mathematical notations used are provided in Table 1.

Objective functions:

$$\begin{array}{c}\begin{array}{c}\hfill \min Z_1={\displaystyle \sum _{i\in M}}{\displaystyle \sum _{j\in N}}q[p(c_i+{\delta }_{i}EM)\end{array}\\ \begin{array}{c}\hfill \hspace{1em}\hspace{1em}\hspace{1em}+Y_{ij}({\mathfrak{c}}_{ij}{d}_{ij}{V}_c+d_{ij}{\varphi }_{j}EM+u)+{\lambda }_i],\end{array}\end{array}$$

(1)

$$\begin{array}{c}\max Z_2=\alpha {\omega }_1+(1-\alpha ){\omega }_2,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & {\omega }_1=\sum _{j=1}^n\sum _{i=1}^{m}f\left(t_{ij}\right)w_{ij},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\omega }_2=\sum _{j=1}^n\sum _{i=1}^m(1-q)f\left(t_{ij}\right)w_{ij},\hfill \end{array}$$

(4)

Constraints:

$$\begin{array}{cc}\hfill & F\in P,P=R^2\setminus int\left(B\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill {\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{Y}_{ij}\ge 1,\end{array}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill {\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{V}_c{Y}_{ij}\le S_i,i=1,2,\cdots ,m,\end{array}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill {\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{V}_c{Y}_{ij}\ge {\displaystyle \sum _{i=1}^m}{\displaystyle \sum _{j=1}^n}{w}_{ij},\end{array}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill (1-q)S_i-{\displaystyle \sum _{j=1}^n}{w}_{ij}\ge (1-q)L_i,i=1,2,\cdots ,m,\end{array}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & \sum _{i=1}^m{c}_i\le Q,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & d_{ij}=d(F_i,p_j),i=1,2,\cdots ,m,j=1,2,\cdots ,n,\hfill \end{array}$$

(11)

Decision variables:

$$\begin{array}{cc}\hfill & \begin{array}{c}\hfill Y_{ij}=\{0,1\},i=1,2,\cdots ,m,j=1,2,\cdots ,n,\end{array}\hfill \end{array}$$

(12)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & x_{c_0}\le x_i\le x_{c_1},i=1,2,\cdots ,m,\hfill \\ \hfill & y_{c_0}\le y_i\le y_{c_1},i=1,2,\cdots ,m,\hfill \\ \hfill & x_{c_0}\le p_{x_j}\le x_{c_1},j=1,2,\cdots ,n,\hfill \\ \hfill & y_{c_0}\le p_{y_j}\le y_{c_1},j=1,2,\cdots ,n.\hfill \end{array}\end{array}$$

(13)

In the above mathematical model, (1) indicates the objective is to minimize disruption cost under sustainable conditions, which is divided into facility fixed cost, facility operation carbon dioxide emission cost, material transportation cost and carbon dioxide emission cost, and penalty cost for facility disruption, where ${\lambda }_i=10w_{ij}$; (2) indicates the objective is to maximize the desired time weighted satisfaction of the demand region in the event of a facility disruption, where $\alpha$ is a weighting factor and there is $0\le \alpha \le 1$; (3) indicates the weighted satisfaction of the demand region in the case of normal operation of the facilities and without facility disruption based on Section 3.2; (4) indicates the weighted satisfaction of the demand region in the presence of the risk of facility disruption based on Section 3.2, where q is the probability of facility disruption, with $0\le q\le 1$ in the normal case; (5) indicates the feasible area for facility locations and is not allowed to be located in the area of the barrier; (6) indicates that each demand point is served at least once; (7) indicates the capacity constraint, means that the vehicle may be transported from facility location i to demand region j, more than once, but the total quantity of resource needs to be less than the maximum capacity at facility location i; (8) indicates that the material needs of each demand point are met; (9) indicates the safety stock constraint, ensuring that each facility has a minimum stock capacity limit in case of facility disruption; (10) indicates that the total cost of establishing facility locations does not exceed the total funding limit; (11) indicates the shortest bypass distance between the i th facility location and the j th demand region, expressed as the optimized distance between the center coordinates of the facility location and the center coordinates of the square demand region; (12) indicates that it equals 1 if vehicle from facility location i to demand region j, 0 otherwise. (13) indicates the geographical environment constraint for the demand region and the location area, requiring the demand region inscription and facility location to be within a specified square area.

The mathematical model constructed above is a nonlinear programming model with complex constraints, making it an NP-hard problem. However, when facing complex geographical environments such as barrier constraints, single solving algorithms may not provide efficient and convenient solutions for the model. Based on this model, a scenario of sustainable emergency facility location under complex polygonal barriers is proposed. In this context, a path optimization algorithm based on the convex hull algorithm is designed and combined with KOA, which has good optimization performance, to solve the model.

4. Solution Algorithms

4.1. Implementation of Convex Hull Algorithm

In this section, we introduce the application of the convex hull algorithm with barrier location information. The convex hull algorithm [31] is a method for computing the smallest convex polygon enclosing a given set of points in a plane. A convex polygon is defined as one where no line segment between any two vertices passes through the interior of the polygon. The concept of a convex hull is relative to a set of points, and for a given set, the convex hull consists of a subset of these points that form a convex polygon completely surrounding all the points in the set. A common analogy for understanding the concept of convex hull is to imagine all the points in the point set as nails, and then using an elastic band to enclose all these nails. The resulting shape formed by this stretched elastic band represents the convex hull, with each nail tightly held by it representing one vertex on this boundary. The typical steps involved in implementing convex hull algorithms are as follows:

Step 1: Find the point with minimum y-coordinate from within the point set and sort them based on their polar angles from this reference point. If two or more points have identical polar angles, they are sorted based on their distances from this reference point.

Step 2: Select the first two sorted points to construct an initial edge forming part of our convex hull.

Step 3: Iteratively selecting subsequent points; if they lie outside our current convex hull, we add them into it; otherwise, we remove them from it. This determination can be made by calculating cross products between vectors $AB$ and $AC$ (where A and B represent adjacent vertices already present in our convex hull while C represents a candidate new vertex), if positive, then C lies counterclockwise to $AB$, indicating its exterior position else interior.

Step 4: Repeat Step 3 until all inputted data has been processed, yielding our final convex hull. Both methods require sorting operations which result in time complexity $O\left(nlogn\right)$.

The convex hull algorithm is widely used to effectively find the smallest convex polygon that encloses all given points. This has broad applications in computer graphics, computational geometry, and robot path planning. In Figure 2, the point set $Q=\{p_0,p_1,\cdots ,p_{15}\}$ is enclosed by a polygon $P=[p_2;p_4;p_6;p_9;p_{11};p_{13};p_{14}]$ formed by line segments, which represents the convex hull of point set Q.

In the case where complex polygonal barriers exist, for the sake of convenience in calculation, all concave barriers are converted into convex barriers using the convex hull method. The conversion result is shown in Figure 3. The gray shaded area represents the original barrier shape, and the converted barrier shape is the blue-framed area. There is a total of 12 barriers.

4.2. Implementation of Path Optimization Algorithm

The path optimization algorithm is a method used to solve the problem of route selection in transportation, logistics, network flow, and other fields. The objective is to find one or multiple optimal paths within a given network in order to minimize certain costs or maximize certain benefits. Path optimization algorithms typically involve knowledge from various fields such as graph theory, optimization theory, and computer science. Reasonable planning of paths is crucial for the efficient distribution of emergency supplies between important nodes in emergency facility location and demand areas. Different forms of barriers present significant challenges that need to be overcome by a path optimization algorithm. Common path optimization algorithms include: 1. Shortest path algorithms: Used to find the shortest path from a starting point to an endpoint within a graph. Common shortest path algorithms include Dijkstra’s algorithm, Bellman-Ford algorithm, Floyd’s algorithm, etc., which can be applied in areas such as traffic navigation and logistics transportation. 2. Traveling salesman problem (TSP): This classic combinatorial optimization problem aims to find the shortest possible route that visits a series of cities and returns to the starting point. TSP has many variants, such as TSP with time windows, multiple traveling salesman problems, etc., with solving algorithms including genetic algorithms, simulated annealing algorithm, ant colony algorithm, among others. 3. Vehicle routing problem (VRP): An important problem in the field of logistics aimed at finding an optimal set of routes so that a certain number of vehicles can transport goods from warehouses to customers according to demand constraints like vehicle capacity, time windows, and transportation costs. Solving VRP involves heuristic algorithms, meta-heuristic methods, mixed integer programming, among others. However, the choice and adjustment of these algorithms depend on specific characteristics when applying them practically.

In order to address this issue, in this section, we will continue with the introduction of a path optimization algorithm based on the convex hull algorithm discussed in Section 4.1. We have established a “visual environment” and proposed a reasonable path optimization algorithm to achieve optimal route planning from emergency facility points to demand points. The algorithm includes several steps: First, based on the implementation of the convex hull algorithm in Section 4.1, identify the first barrier along the straight-line path between facility points and demand points. Next, determine an effective point sequence bypassing the previously identified barrier and use its endpoint as a new starting point for surveying the next barrier, sequentially expanding these effective point sequences. Finally, shorten the effective point sequences along their paths to reduce the number of convex vertices encountered when traversing barriers. Figure 4 provides a complete example process of barrier-avoidance optimization from emergency facility points to demand points.

With $(20,1)$ as the emergency facility point (i.e., the arrowhead marks the starting point, marked as the start point), and $(11,20)$ as the demand area center (i.e., the arrowhead marks the ending point, marked as the end point), using path optimization algorithms, the allocation path from the emergency facility point to the demand point is optimized. The optimization steps include the following:

(1)

Connect the emergency facility point and the demand point, as shown by the black dashed line in Figure 4, and find the first barrier between them, marked as barrier 1.

(2)

Determine the points to be bypassed around barrier 1 as $(20,1)(17,2)$, and take the endpoint of the sequence $(17,2)$ as the new starting point (marked as start point 1), i.e., considering the new connection between the emergency facility point and the demand point for calculation, as shown by the orange solid line in Figure 4, find the first barrier under the new starting point, marked as barrier 2, and determine the effective points to be bypassed around barrier 2 as $(16,17)$.

(3)

Continue to take the endpoint of Step 2 $(16,17)$ as the new starting point for the next barrier search (marked as start point 2), and there is no next barrier on this path, proceed to the next calculation. At this time, the effective points between the emergency facility point and the demand area can be obtained as $(20,1)(17,2)(16,17)(11,20)$, shown by the blue dashed line in Figure 4.

(4)

Determine whether there are any barriers that have been penetrated in the obtained valid point sequence. There is only one barrier between $(17,2)$ and $(16,17)$, marked as barrier 3, and the sequence for bypassing barrier 3 is $(15,14)$. The resulting extended sequence is $(20,1)(17,2)(15,14)(16,17)(11,20)$, marked in purple lines in the figure. At this point, the point sequences in the figure no longer have any barriers.

(5)

Determine whether there are any deletable point sequences. If there are any deletable point sequences, delete them to obtain the deletion sequence. If not, then directly obtain the optimal path as $(20,1)(17,2)(15,14)(16,17)(11,20)$.

(6)

The deletion sequence obtained in Step 5 is the optimal path provided by the barrier-avoidance optimization algorithm, marked in purple lines in Figure 4.

4.3. Model Solving Based on KOA

The Kepler optimization algorithm [32] is an intelligent optimization algorithm based on the principles of planetary motion. The algorithm primarily derives from the study of planetary motion laws, particularly Kepler’s laws, which describe the orbital motion of planets around the sun. KOA applies these laws to optimization problems by simulating the motion of planets to find the optimal solution. The main advantages of the algorithm include: 1. Global search capability: KOA can perform a global search in the search space by simulating the motion of planets, thus avoiding becoming trapped in a local optimal solution. 2. Diversity maintenance: The multiple planet candidate solutions in the algorithm help maintain the diversity of the population, which helps avoid premature convergence and enhances the algorithm’s global optimization ability. 3. Flexible parameter adjustment: The parameter of KOA adjustment is relatively flexible, and can be adjusted according to the specific characteristics of the problem to achieve better optimization results. In this paper, KOA is combined with the convex hull method and path optimization algorithm to solve the location model and resource allocation problem.

KOA is a novel optimization algorithm inspired by Kepler’s laws of planetary motion, where all planet positions are treated as emergency facility locations in the universe. The sun (the optimal solution) and planets (candidate solutions) that orbit around it in an elliptical orbit form the search space. At different times, planets will be in different positions on their orbits, which is equivalent to a process of gradually optimizing. This strategy effectively implements the exploration and development stage. The gravitational attraction between the sun and planets, as well as their rotation speed, also determines the proximity of planets to the sun. Additionally, to adapt this algorithm, the term “time” is more suitable for the iteration word in the model solving, and the calculation result of the fitness of each planet based on its position is the model’s objective function. KOA, inspired by Kepler’s laws of planetary motion, offers distinct advantages over other popular meta-heuristic algorithms such as the Genetic Algorithm (GA) and Simulated Annealing (SA). KOA stands out for its strong global exploration capabilities, effectively avoiding premature convergence to local optima due to its orbital-based search strategy, making it especially suitable for complex multimodal problems. Additionally, KOA strikes a good balance between exploration and exploitation, enabling both broad search and localized optimization. Its simple and intuitive mechanism, similar to genetic algorithms, facilitates easy implementation and performs well in multi-objective optimization tasks. However, KOA is not without its limitations. It tends to have a slower convergence rate, particularly in high-dimensional or complex optimization problems. The emphasis on exploration can also result in longer search times and may require more iterations to fine-tune the solution. Here is the specific process of KOA solving the sustainable emergency facility location model with a complex polygon barrier scenario:

Step 1: Initialization: Model-related parameter setting: set the position information of the demand region, and randomly generate the demand $w_j$ of the demand region j within a reasonable range. To ensure that the total amount of facilities is greater than the total demand. Set up the fixed cost required for the establishment of the emergency facilities $c_i$, total cost Q, unit transportation cost $c_{ij}$, the loading capacity of the vehicle $V_c$, the cost of an empty vehicle return u, disruption probability q of emergency facility i, increased cost ${\lambda }_i$ entry of emergency facility i disruption, time-sensitive coefficient ${\beta }_i$, total amount of relief supplies $kk$, the average transportation speed $v_{ij}$, the upper boundary of the waiting time to obtain relief supplies is $U_j$, the lower boundary is $L_j$, the maximum capacity at the emergency facility point i is $S_i$, safety stock (i.e., the minimum capacity) is $L_i$, the total number of emergency facilities points built is p.

The algorithm-related parameter setting: initialize the population individual, which is the location of the randomly generated facility point and the demand region allocation amount, ensure that the location of each initial facility point is not repeated, and each individual in the population represents a candidate solution of the problem. The initialize planet position formula is

$$\begin{array}{c}\hfill X_i^j=X_{i,low}^j+ran{d}_{[0,1]}\times \left(X_{i,up}^j-X_{i,low}^j\right),\left\{\begin{array}{c}i=1,2,\dots ,N,\hfill \\ j=1,2,\dots ,d,\hfill \end{array}\right.\end{array}$$

(14)

where $X_i$ represents the i th planet in the search space (candidate emergency facility point), $X_{i,up}^j$ represents the upper bound of the j th decision variable, $X_{i,low}^j$ represents the lower bound of the j th decision variable, N represents the population size which is the number of candidate solutions in the search space, d represents the population size, the position of the planet will be randomly distributed in two-dimensional space which represents the decision variable of the optimization model, $ran{d}_{[0,1]}$ represents a random between 0 and 1, initializing the orbital eccentricity $e_i$ have

$$\begin{array}{c}\hfill e_i=ran{d}_{[0,1]},i=1,...,N,\end{array}$$

(15)

where $ran{d}_{[0,1]}$ represents a random between 0 and 1, initializing the orbital period $T_i$ have

$$\begin{array}{c}\hfill T_{\mathrm{i}}=\left|\begin{array}{c}r\end{array}\right|,i=1,...,N,\end{array}$$

(16)

where r is the random number generated by a normal distribution, setting the maximum number of iterations to $Ite{r}_{max}$, the current number of iterations is 1.

Step 2. Definition of gravity: The main reason for the movement of planets around the sun is the sun’s gravity, which constantly changes its direction during the movement of planets, making the planets move around the sun in an elliptical trajectory. The size of the gravity of each planet is related to its own mass, and the change in the planet’s gravity also determines the speed of the planet. The closer the planet is to the sun, the greater the orbital speed will be. Its motion obeys the law of gravitation

$$\begin{array}{c}\hfill F_{g_i}\left(t\right)=e_i\times \mu \left(t\right)\times {\displaystyle \frac{{\overline{M}}_s\times {\overline{m}}_i}{{\overline{R}}_i^2+\epsilon }}+r_1,\end{array}$$

(17)

where ${\overline{M}}_s$ and ${\overline{m}}_i$ represent $M_s$ and $m_i$ normalized values, representing the mass of $M_s$ and $m_i$. The mass formula is (19) and (20), $\epsilon$ is a minimum value, and $\mu$ is the general gravitational coefficient formula as shown in (23), $e_i$ is the eccentricity of the planetary orbit, indicated by Formula (15), $r_1$ represents the random number generated between 0 and 1, ${\overline{R}}_i$ is the normalized values of $R_i$, represents the Euclidean distance between $X_s$ and $X_i$, and $R_i$ is defined as

$$\begin{array}{c}\hfill R_i\left(t\right)={\Vert X_s\left(t\right)-X_i\left(t\right)\Vert }_2=\sqrt{\sum _{j=1}^d{\left(X_{sj}\left(t\right)-X_{ij}\left(t\right)\right)}^2},\end{array}$$

(18)

where $\Vert X_s\left(t\right)-X_i{\left(t\right)\Vert }_2$ represents $X_S$ (optimal emergency facility point) and $X_i$ represents Euclidean distance at time t, $X_S$ and $X_i$ the mass formula of is defined as follows

$$\begin{array}{c}\hfill M_s=r_2{\displaystyle \frac{fi{t}_s\left(t\right)-worst\left(t\right)}{{\sum }_{k=1}^N\left(fi{t}_k\left(t\right)-worst\left(t\right)\right)}},\end{array}$$

(19)

$$\begin{array}{c}\hfill m_i={\displaystyle \frac{fi{t}_i\left(t\right)-worst\left(t\right)}{{\sum }_{k=1}^N\left(fi{t}_k\left(t\right)-worst\left(t\right)\right)}},\end{array}$$

(20)

where $r_2$ represents a random between 0 and 1, $X_S$ and $X_i$ the mass at time t is determined by the fitness function value at time t, considering that the minimized objective function has a fitness formula

$$\begin{array}{c}\hfill fi{t}_s\left(t\right)=best\left(t\right)=\underset{k\in 1,2,...,N}{min}fi{t}_k\left(t\right),\end{array}$$

(21)

$$\begin{array}{c}\hfill worst\left(t\right)=\underset{k\in 1,2,...,N}{max}fi{t}_k\left(t\right).\end{array}$$

(22)

The gravitational coefficient $\mu$ is a function of presenting an exponential decrease with time t, with equation

$$\begin{array}{c}\hfill \mu \left(t\right)={\mu }_0\times \exp (-\gamma {\displaystyle \frac{t}{T_{max}}}),\end{array}$$

(23)

where $\gamma$ is a constant, ${\mu }_0$ is the initial values, t and $T_{max}$ is the number of current and maximum iterations.

Step 3. Speed calculation:

When the planet is changed by the sun’s force, its speed will also change. The closer to the sun, the larger the orbit speed, and the smaller the farther away from the sun, there is a formula that the speed changes with time t

$$\begin{array}{cc}\hfill V_i\left(t\right)& =\left\{\begin{array}{c}\mathit{\ell }\times \left(2r_4{\overrightarrow{X}}_i-{\overrightarrow{X}}_{\mathbf{b}}\right)+\ddot{\mathit{\ell }}\times \left({\overrightarrow{X}}_a-{\overrightarrow{X}}_b\right)+\left(1-R_{i-norm}\left(t\right)\right)\hfill \\ \times \mathcal{F}\times {\overrightarrow{U}}_1\times {\overrightarrow{r}}_5\times \left({\overrightarrow{X}}_{i,up}-{\overrightarrow{X}}_{i,low}\right),\hfill & if R_{i-norm}\left(t\right)\le 0.5,\hfill \\ r_4\times \mathcal{L}\times \left({\overrightarrow{X}}_a-{\overrightarrow{X}}_i\right)+\left(1-R_{i-norm}\left(t\right)\right)\hfill \\ \times \mathcal{F}\times U_2\times {\overrightarrow{r}}_5\times \left(r_3{\overrightarrow{X}}_{i,up}-{\overrightarrow{X}}_{i,low}\right),\hfill & else,\hfill \end{array}\right.\hfill \end{array}$$

(24)

where $V_i\left(t\right)$ denotes the t time ${\overline{X}}_i$, the velocity of planet i, $r_3$ and $r_4$ represents a 0 to 1 between the resulting random numbers, and ${\overrightarrow{r}}_5$ represent a random vector generated by numerical values between 0 and 1, ${\overrightarrow{X}}_a$ and ${\overrightarrow{X}}_b$ represent two vectors indicate from the population. The two solutions selected have other parameter formulas as follows:

$$\begin{array}{c}\hfill \mathit{\ell }=\overrightarrow{U}\times \mathcal{M}\times \mathcal{L},\end{array}$$

(25)

$$\begin{array}{c}\hfill \mathcal{L}={\left[\mu \left(t\right)\times \left(M_S+m_i\right)\left|{\displaystyle \frac{2}{R_i\left(t\right)+\epsilon }}-{\displaystyle \frac{1}{a_{i\left(t\right)}+\epsilon }}\right|\right]}^{{\displaystyle \frac{1}{2}}},\end{array}$$

(26)

$$\begin{array}{c}\hfill \mathcal{M}=(r_3\times (1-r_4)+r_4),\end{array}$$

(27)

$$\begin{array}{c}\hfill \overrightarrow{U}=\left\{\begin{array}{cc}0,\hfill & \overrightarrow{r_5}\le \overrightarrow{r_6},\hfill \\ 1,\hfill & else,\hfill \end{array}\right.\end{array}$$

(28)

$$\begin{array}{c}\hfill \mathcal{F}=\left\{\begin{array}{cc}1,\hfill & if{r}_4\le 0.5,\hfill \\ -1,\hfill & else,\hfill \end{array}\right.\end{array}$$

(29)

$$\begin{array}{c}\hfill \ddot{\mathit{\ell }}=\left(1-\overrightarrow{U}\right)\times \overrightarrow{\mathcal{M}}\times \mathcal{L},\end{array}$$

(30)

$$\begin{array}{c}\hfill \overrightarrow{\mathcal{M}}=(r_3\times (1-\overrightarrow{r_5})+\overrightarrow{r_5}),\end{array}$$

(31)

$$\begin{array}{c}\hfill \overrightarrow{U_1}=\left\{\begin{array}{cc}0,\hfill & \overrightarrow{r_5}\le r_4,\hfill \\ 1,\hfill & else,\hfill \end{array}\right.\end{array}$$

(32)

$$\begin{array}{c}\hfill U_2=\left\{\begin{array}{cc}0,\hfill & r_3\le r_4,\hfill \\ 1,\hfill & else,\hfill \end{array}\right.\end{array}$$

(33)

where ${\overrightarrow{r}}_6$ represents a random vector generated by numerical values between 0 and 1, $a_i$ represents the time t of the object i in an elliptic orbit. The half-main axis is derived from Kepler’s third law; the specific content is as follows

$$\begin{array}{c}\hfill a_i\left(t\right)=r_3\times {\left[T_i^2\times {\displaystyle \frac{\mu \left(t\right)\times (M_s+m_i)}{4{\pi }^2}}\right]}^{{\displaystyle \frac{1}{3}}},\end{array}$$

(34)

where $a_i$ will gradually decrease with the increasing iteration, and the solution will gradually approach the global optimal. $R_{i-norm}$ denotes time t to $X_s$ and $X_i$ the Euclidean distance of is normalized and defined as follows:

$$\begin{array}{c}\hfill R_{i-norm}\left(t\right)={\displaystyle \frac{R_i\left(t\right)-min\left(R\left(t\right)\right)}{max\left(R\right(t\left)\right)-min\left(R\right(t\left)\right)}},\end{array}$$

(35)

If $R_{i-norm\left(t\right)}\le 0.5$, the planet will gradually approach the sun and gradually increase its movement speed. Otherwise, the planet will move at slower speeds.

Step 4. Planetary position update:

The planets move around the sun in their respective orbits, away from the sun for a period during the rotation, and then closer towards the sun. The KOA divides the process into two stages: exploration and exploitation. When the planet is moving away from the sun to find a new solution, it is called the exploration stage, and when the planet moves near the sun to find a better solution, it becomes the development stage

$$\begin{array}{c}\hfill {\overrightarrow{X}}_i\left(\begin{array}{c}t+1\end{array}\right)={\overrightarrow{X}}_i\left(\begin{array}{c}t\end{array}\right)+\mathcal{F}\times {\overrightarrow{V}}_i\left(\begin{array}{c}t\end{array}\right)+\left(\begin{array}{c}{F}_{g_i}\left(\begin{array}{c}t\end{array}\right)+\left|\begin{array}{c}r\end{array}\right|\end{array}\right)\times \overrightarrow{U}\times \left(\begin{array}{c}{\overrightarrow{X}}_S\left(\begin{array}{c}t\end{array}\right)-{\overrightarrow{X}}_i\left(\begin{array}{c}t\end{array}\right)\end{array}\right),\end{array}$$

(36)

where ${\overrightarrow{X}}_i(t+1)$ is the position of the planet at time $t+1$, ${\overrightarrow{V}}_i\left(t\right)$ is the speed at which the planet needs to reach the next position, ${\overrightarrow{X}}_S\left(t\right)$ is the optimal solution, the position of the sun, $\mathcal{F}$ is used to control the search direction.

Step 5. Update the distance from the sun: In order to further improve the exploration and exploitation stage of the planet, KOA further optimizes the operator h to optimize the exploration and exploitation stage. When the planet is close to the sun, KOA will focus on the optimization of the operator, and when the planet is away from the sun, KOA will focus on optimizing the exploration operator. The position update formula of the planet is as follows

$$\begin{array}{c}{\overrightarrow{X}}_{\mathrm{i}}(t+1)\begin{array}{c}\hfill ={\overrightarrow{X}}_{\mathrm{i}}\left(t\right)\times {\overrightarrow{U}}_1+\left(1-{\overrightarrow{U}}_1\right)\times \left({\displaystyle \frac{{\overrightarrow{X}}_{\mathrm{i}}\left(t\right)+{\overrightarrow{X}}_{\mathrm{s}}+{\overrightarrow{X}}_{\mathrm{a}}\left(t\right)}{3.0}}\right)\end{array}\\ \hfill \begin{array}{c}\hfill +h\times \left({\displaystyle \frac{{\overrightarrow{X}}_{\mathrm{i}}\left(t\right)+{\overrightarrow{X}}_{\mathrm{s}}+{\overrightarrow{X}}_{\mathrm{a}}\left(t\right)}{3.0}}-{\overrightarrow{X}}_{\mathrm{b}}\left(t\right)\right)),\end{array}\end{array}$$

(37)

where h is the adaptive factor at time t controlling the distance between the sun and the current planet, defined as

$$\begin{array}{c}\hfill h={\displaystyle \frac{1}{e^{\eta r}}},\end{array}$$

(38)

where r is a number randomly generated according to a normal distribution, but rather a linear decreasing factor ranging from 1 to $-2$, defined as

$$\begin{array}{c}\hfill \eta =(a_2-1)\times r_4+1,\end{array}$$

(39)

where $a_2$ is the cyclic control parameters, throughout the optimization process $\overline{T}$ the cycle gradually decreases from $-1$ to $-2$, defined as

$$\begin{array}{c}\hfill a_2=-1-1\times \left({\displaystyle \frac{t\%\frac{T_{max}}{\overline{T}}}{\frac{T_{max}}{\overline{T}}}}\right).\end{array}$$

(40)

Step 6. Elite strategy:

To further filter out the optimal solution and ensure the optimal position of the planets and the sun, using the equation   

$$\begin{array}{c}\hfill {\overrightarrow{X}}_{i,new}(t+1)=\left\{\begin{array}{cc}{\overrightarrow{X}}_i(t+1),\hfill & iff\left({\overrightarrow{X}}_i(t+1)\right)\le f\left({\overrightarrow{X}}_i\left(t\right)\right),\hfill \\ {\overrightarrow{X}}_i\left(t\right),\hfill & else.\hfill \end{array}\right.\end{array}$$

(41)

KOA implements the reliability emergency facility location model for the polygon barrier scenario in Algorithm 1. The pseudo-code for the KOA is described in Algorithm 2.

Algorithm 1 KOA for solving sustainability, reliable emergency facility location determination with consideration of complex polygonal barriers and the risk of facility disruption

Step 1. Initialization: Model correlation setting: input polygon barrier position information and convert it into barrier matrix, depict the location information of demand region, randomly generate the demand of demand regions within a reasonable range, to ensure that the sum of facilities exceeds the sum of demand based on safety inventory. Algorithm parameter setting: initialize the planet population by randomly generating the positions of emergency facility points, ensuring that each initialization position is not repeated and does not appear in barriers. Calculate the fitness of each individual planet as the objective function of the model, and record the discovered optimal position as $X_{best}$, which is the current position of the sun. Traverse the planet with a scale of N, the maximum number of iterations $T_{max}$, the constant $\overline{T}$ in $a_2$, and the constants ${\mu }_0$ and $\gamma$ in $\mu$. Step 2. Perform a loop iteration and substitute the initialized planetary position, eccentricity, and orbital period. Step 3. Calculate the fitness value of the planet (model objective function value) and search for the position of the sun (global optimal solution). Step 4. Determine the Euclidean distance between the sun and planets through calculation. Step 5. Calculate the solar gravitational force on each planet. Step 6. Obtain the velocity value of the planet’s orbit around the sun. Step 7. Generate two random numbers. If the random number $rd>r$, update the position information of the planet itself; otherwise, update the distance between the sun and the planet. Step 8. Recalculate the fitness value (model objective function value) of the planet through elite strategy, update the sun position $X_{best}$ (global optimal solution), and discard inferior solutions. Step 9. If the termination iteration count condition is met, terminate. If the termination condition is not met, continue iterating from Step 2 with $X_{best}$.

Algorithm 2 KOA solves the sustainability-reliable emergency facility location determination with consideration of complex polygonal barriers and the risk of facility disruption

Input:  • $B_j=$ randomly initialize polygonal barriers, $D_j=$ circular demand region. $U_j=$ upper bound of time satisfaction, $L_j=$ lower bound of time satisfaction. ${\beta }_i=$ time sensitivity coefficient, $v_{ij}=$ delivery speed. $S_i=$ maximum facility capacity, $L_i=$ facility safety stock. $w_j=$ demand quantity, $q=$ probability of disruption of each facility. $Q=$ total cost limit, $c_i=$ fixed cost of establishing emergency facilities. $c_{ij}=$ unit transportation cost, $V_c=$ vehicle loading capacity. $u=$ cost of returning an empty vehicle, ${\lambda }_i=$ increased cost of disrupting emergency facility. $kk=$ total amount of rescue supplies, $p=$ total number of emergency facility points. $N=$ traversing organism size, $T_{max}=$ maximum iteration. $\overline{T}$ constant of $a_2$, ${\mu }_0=$ constant of $\mu$, $\gamma =$ constant of $\mu$. Output:  $Z_1=$ cost of each solution, $Z_2=$ satisfaction of each solution, $X_{best}=$ the best solution found so far. Initialize emergency facility locations population with random positions, orbital eccentricities, and orbital periods using (14), (15), and (16), respectively. Evaluate objective function values for the initial population. Determine the global best solution ($X_s$) as the sun.   1: while  $t<T_{max}$  do   2:    Update $e_i$$(i=1,2,...,N)$, $best\left(t\right)$, $worst\left(t\right)$ and $\mu \left(t\right)$, using (21), (22) and (23), respectively.   3:    for $i=1:N$ do   4:      Calculate the Euclidean distance between the global best solution and the candidate solution i using (18).   5:      Calculate the gravitational force between the global best solution and the candidate solution i using (17).   6:      Calculate the velocity of the candidate solution $X_i$ using (24).   7:      Generate two random numbers r, between 0 and 1.   8:      if $r>r_1$ then   9:         Update the position of the candidate solution. 10:         Update the candidate solution position using (36). 11:      else 12:         Update the distance between the candidate solution and the global best solution. $$ 13:         Update the candidate solution position using (37). 14:      end if 15:      Apply an elitist strategy to select the best position of the global best solution and the candidate solutions, using (41). 16:      $t=t+1$ 17:    end for 18: end while 19: return $X_{best}$, $F_{best}$.

5. Numerical Study and Analysis

In this section, the model and algorithm of this section will be used to solve and analyze the problems proposed in the design. In order to verify the feasibility and applicability of the constructed model and algorithm, a total of 5 groups of experiments are designed. KOA, combined with the convex hull method and path optimization algorithm, was used to solve the model, and a comparative analysis was carried out. Using MATLAB 2020b as the experimental platform, the algorithm is executed on Intel (R), Core (TM) i7-6600UCPU, 2.80 GHz, 16.00 GB memory, Windows 10 operating system, and complex polygon barrier data and demand area location information are customized. The convex hull algorithm and path optimization algorithm designed in this paper are used to solve the location model in combination with the KOA, and the performance of the algorithm proposed in this paper and the optimized facility location situation are verified by experiments with a variety of different scale examples.

In this part, hypothetical data are used instead of real-world data mainly due to the difficulty in obtaining comprehensive, accurate data on complex polygonal barriers—their boundaries, shapes, and distributions are often restricted by privacy, security, or ownership issues and prone to incompleteness—as well as challenges in acquiring reliable demand area and resource demand data, with dynamic, uncertain real-world factors and scattered existing facility data requiring extensive coordination; additionally, real-world uncontrollable variables like weather, traffic, and delays can interfere with model validation, while hypothetical data enables controlled testing to demonstrate the model and algorithm’s effectiveness, making it feasible for validation.

The coordinates of the vertices for the polygonal barriers are set as:

$B1=[1,22;3,24;6,22;4,21]$	$B2=[10,21;10,23;15,23;15,21]$
$B3=[19,17;21,19;21,21;23,19;22,15]$	$B4=[9,16;11,19;16,17]$
$B5=[4,14;3,15;4,16;5,15]$	$B6=[12,11;13,12;14,12;15,11;14,10;13,10]$
$B7=[2,7;2,9;4,8;7,8;8,7]$	$B8=[2,1;1,4;5,4]$
$B9=[11,1;11,5;13,4;14,2]$	$B10=[18,7;20,8;20,11;22,11;22,8;24,7]$
$B11=[17,2;24,5;24,2]$	$B12=[22,22;23,22;23,23]$

Example 1. 

Simulation experiment of a single-facility emergency facility location model.

Cluster the demand points to obtain 5 demand regions, and determine the location of a single facility point under 12 barrier constraints. We set the facility point $F_1$ with the facility capacity randomly generated in the range $[150,250]$. The demand volume is 40. The total cost of facility point construction is set at 800, and the shipping speed is 10. Additionally, the safety stock is set at $0.5$, while the search traversal area is defined as a square area of $[25,25]$. The disruption risk q is $0.1$. The empty vehicle cost u is 20, the vehicle loading capacity $V_c$ is 10, the cost of carbon dioxide per ton $EM$ is 10, the carbon dioxide emissions from each emergency facility point ${\delta }_i$ is $0.8$, the carbon dioxide emissions per unit distance of vehicle transportation materials ${\varphi }_j$ is $0.0002$, the initial population size of KOA is 50, the number of iterations is 500, the constant $\overline{T}$ is 3, the ${\mu }_0$ is $0.1$, and the $\gamma$ is 15. The coordinates of the center of the circle demand point and its radiation radius are as follows:

$D1=(6,20,0.8),D2=(9,13,0.5),$

$D3=(16,5,1.3),D4=(17,15,1),$

$D5=(18,22,0.9).$

Using KOA, combine the convex hull algorithm with the path optimization algorithm, we optimize the objective and results of single-facility reliability emergency facility location. The single-facility location assignment scheme is obtained as shown in Figure 5, where $F_1$ indicates the optimal location of the determined single facility, and the blue dashed line represents the optimized final path, the red star indicate the final facility location, the red circle illustrates the demand regions, and the gray polygon outlines the barriers. In addition, we obtain the optimal location coordinates of the facility location as $(12.9874,14.2661)$. The convergence speed of the model is relatively fast, and the optimal target value $Z_1$ is $139.8554$, $Z_2$ is $139.8554$.

Example 2. 

Simulation experiments of multi-facility emergency facility location model.

The capacity of the $F_1,F_2,F_3,F_4$ facility is randomly generated within the range of 80 to 100. The total cost of constructing facility points can reach up to 3200, while all other parameters remain unchanged as Example 1. The coordinates of the center of the square demand point and its radiation radius are as follows:

$D1=(6,20,0.8),D2=(9,13,0.5),$

$D3=(16,5,1.3),D4=(17,15,1),$

$D5=(18,22,0.9).$

By using KOA, combining the convex hull algorithm with the path optimization algorithm, we optimize the objective and results of multi-facility reliability emergency facility locations. In Figure 6a, we have derived multi-facility location-allocation schemes, where $F_1,F_2,F_3$ indicate the optimal locations of the identified facilities. The blue dashed line represents the optimized final path, the red stars indicate the final facility locations, the red circle illustrates the demand regions, and the gray polygon outlines the barriers. We also determined the 3 optimal coordinates of the facility points as $(6.9498,19.3005)$, $(15.2239,9.8919)$ and $(18.8154,20.1569)$, resulting in the optimal target value $Z_1$ is $281.4054$, $Z_2$ is $172.9496$.

In Figure 6b, according to the current conditions, we have derived multi-facility location-allocation schemes, where $F1,F2,F3,F4$ indicate the optimal locations of the identified facilities. In addition, we determined the 4 optimal location coordinates of the facility point as $(9.6224,14.2917)$, $(4.9021,18.3339)$, $(17.3403,20.0076)$ and $(13.5458,9.3376)$, resulting in the optimal target value $Z_1$ is $348.8629$, $Z_2$ is $167.9143$. It can be seen that the allocation of facility points to demand points is not limited only to a single facility point, but there may be a scenario where multiple facility points serve a demand point at the same time.

Example 3. 

Simulation experiment on a large-scale emergency facility location model with multiple facility reliability.

The 15 demand areas are set to be depicted by circular areas in Figure 7. Under the constraints of capacity, cost, and 12 barriers, the time satisfaction of the demand side and cost consumption from the perspective of sustainability are taken as the objective function, and other parameters remain unchanged as compared with the previous section, so as to determine the location and distribution plan of 6 facility points. Set the demand of the demand area as a random number between 30 and 50. The center coordinate of the circular demand point and its coverage radius are as follows:

$D1=(6,20,0.8),D2=(9,13,0.5),$	$D3=(16,5,1.3),D4=(17,15,1)$,
$D5=(18,22,0.9),D6=(2,10,0.3),$	$D7=(7,5,0.3),D8=(10,10,0.4)$,
$D9=(20,5,0.5),D10=(24,15,0.6),$	$D11=(15,20,0.5),D12=(8,24,0.7)$,
$D13=(22,12,0.9),D14=(12,20,0.5),$	$D15=(17,10,0.5)$.

A combination of the convex hull algorithm, path optimization algorithm, and KOA was used to solve the model, and the reliability location-allocation topology of large-scale multi-facility emergency facilities was obtained, as shown in Figure 8. The blue dashed line represents the optimized final path, the red star represents the final facility location, the red circle represents the demand area, and the gray polygon represents the barrier, resulting in the optimal target value $Z_1$ is $561.9877$, $Z_2$ is $458.0966$.

The facility capacities and locations are detailed in

Table 2. Facility capacity.

No.	Capacity	Location
1	$102.366$	$(16.7169,8.3535)$
2	$130.1312$	$(15.434,17.8079)$
3	$134.9242$	$(9.084,14.9432)$
4	$92.15094$	$(9.7188,9.0651)$
5	$102.8357$	$(6.4747,24.5589)$
6	$131.2465$	$(6.1115,3.4726)$

, while

Table 3. Facility capacity boundary.

No.	Boundary	Capacity
1	Upper boundary	100
	Lower boundary	120
2	Upper boundary	120
	Lower boundary	140
3	Upper boundary	130
	Lower boundary	140
4	Upper boundary	90
	Lower boundary	100
5	Upper boundary	100
	Lower boundary	120
6	Upper boundary	120
	Lower boundary	140

outlines the specific range of facility capacities, with a total storage volume of approximately 700. Random generation within the range of $[30,50]$ is employed to determine the demand volume for the demand region, as depicted in

Table 4. Demand point information.

No.	Demand Volume	Center Coordinates	Radius
1	31.49764	(6, 20)	0.8
2	38.98215	(9, 13)	0.5
3	46.95067	(16, 5)	1.3
4	31.8134	(17, 15)	1
5	41.27111	(18, 22)	0.9
6	39.13247	(2, 10)	0.3
7	37.70701	(7, 5)	0.3
8	33.97671	(10, 10)	0.4
9	41.7648	(20, 5)	0.5
10	35.73031	(24, 15)	0.6
11	41.89461	(15, 20)	0.5
12	41.87234	(8, 24)	0.7
13	43.04936	(22, 12)	0.9
14	49.44994	(12, 20)	0.5
15	31.34577	(17,10)	0.5

. Construction costs for each facility location are also randomly generated within the interval $[500,800]$, with a total cost ceiling of 4800. Additionally, shipping speed is randomized within $[8,10]$, the safety stock is set at $0.5$, the disruption risk q is $0.1$, the empty vehicle cost u is 20, the vehicle loading capacity $V_c$ is 10, the cost of carbon dioxide per ton $EM$ is 10, the carbon dioxide emissions from each emergency facility point ${\delta }_i$ is $0.8$, the carbon dioxide emissions per unit distance of vehicle transportation materials ${\varphi }_j$ is $0.0002$, the initial population size of KOA is 50, the number of iterations is 500, the constant $\overline{T}$ is 3, the ${\mu }_0$ is $0.1$, and the $\gamma$ is 15. The search traversal area covers the square region of $[25,25]$, as illustrated in

Table 5. Search region.

Name	Boundary	Region
x	Upper boundary	0
	Lower boundary	25
y	Upper boundary	0
	Lower boundary	25

.

From the allocation results in Table 2, it can be seen that the optimal facility point number is set to 6, the number of demand regions is 15, and the safety stock of the facility point is $0.5$ when the optimal allocation of emergency facilities to demand regions is obtained by combining the number of emergency facilities and the demand of the demand regions. The effective allocation of emergency facilities to demand regions is 20 times. Briefly describing the first three effective allocations: $\left(1\right)$ The allocation of emergency facility 2 to demand region 14 is $49.4499$, the distance between the emergency facility and the demand region is $4.5405$, and the time satisfaction is $0.90224$. $\left(2\right)$ The allocation of emergency facility 1 to demand region 3 is $46.9507$, the distance between the emergency facility and the demand region is $4.5966$, and the time satisfaction is $0.92658$. $\left(3\right)$ The allocation of emergency facility 1 to demand region 13 is $12.1056$, the distance between the emergency facility and the demand region is $7.2406$, and the time satisfaction is $0.83086$.

In

Table 6. The result of facility locations and allocation.

No.	Facility Location	Demand Location	Allocation Volume	Distance	Time Satisfaction
1	2	14	49.4499	4.5405	0.90224
2	1	3	46.9507	4.5966	0.92658
3	1	13	12.1056	7.2406	0.83086
4	2	13	30.9438	9.4629	0.77745
5	2	11	41.8946	2.6593	0.99335
6	5	12	41.8723	2.3230	0.97266
7	1	9	41.7648	5.0731	0.91734
8	5	5	41.2711	12.711	0.69714
9	4	6	23.7238	8.0746	0.89459
10	6	6	3.9584	8.7092	0.87892
11	3	6	11.4502	8.9232	0.87366
12	3	2	38.9821	2.4209	0.95171
13	6	7	37.7070	2.0584	0.97968
14	1	10	35.7303	10.1809	0.77894
15	4	8	29.8162	1.3630	0.96603
16	6	8	4.1605	7.9414	0.80412
17	4	4	31.8134	10.3027	0.79450
18	5	1	31.4976	5.3567	0.94826
19	1	15	3.2960	2.1009	1.00000
20	4	15	4.9318	7.8377	0.92740

, it can be observed that emergency facility point 1 is allocated to 5 demand regions, namely demand regions $3,13,9,10$, and 15; Emergency facility point 2 is allocated to 3 demand regions, namely demand regions $14,13$, and 11; Emergency facility point 3 is allocated to 2 demand regions, namely demand regions 6 and 2; Emergency facility point 4 is allocated to 4 demand regions, namely demand regions $6,8,4,$ and 15; Emergency facility point 5 is allocated to 3 demand regions, namely demand regions $12,5,$ and 1; Emergency facility point 6 is allocated to 3 demand regions, namely demand regions $6,7,$ and 8. The last column indicates the time satisfaction of demand regions, and the results show that the satisfaction of demand regions is generally high. Among them, after statistical analysis, there is only one allocation with a time satisfaction of $1,11$ items greater than $0.9,16$ items greater than $0.8$, and no extremely low time satisfaction, reflecting the robustness and reliability of the emergency facility reliability location model.

Example 4. 

Sensitivity analysis.

In this section, the sensitivity analysis of different disruption risks is considered, and the impact of different disrupt probabilities on emergency facility disruptions is studied. Different decision results can be obtained by adjusting the disrupt probability parameters to solve the example, and then a comparative observation is carried out. In addition, in order to further explore the feasibility of the sustainability perspective proposed in the paper, this section will conduct comparative experiments between models that consider sustainability and those that do not. The initial population size of KOA is 50, the number of iterations is 300, and other parameters were unchanged compared with those in Section 2. The solution results are analyzed. By redefining the objective function ${\overline{Z}}_1$, we obtain the objective function without considering sustainability as

$$min{\overline{Z}}_1={\displaystyle \sum _{i\in M}}{\displaystyle \sum _{j\in N}}q[c_{i}p+Y_{ij}(c_{ij}{d}_{ij}{V}_c+u)+{\lambda }_i].$$

(42)

The control variable method is used to adjust the disrupt probability of the sustainable model and the non-sustainable model, respectively, and the optimized results are shown in

Table 7. Sensitivity analysis of different disruption probabilities.

Probability of Disruption of Emergency Facilities		0.1	0.2	0.3	0.4
$Z_1$	Considering sustainability	280.181	546.6332	814.5979	1075.353
	Unconsidering sustainability	284.955	540.2239	804.4287	1036.107
$Z_2$	Considering sustainability	172.5315	142.7806	112.9142	88.69013
	Unconsidering sustainability	168.0872	145.2802	103.7401	84.76184

.

With the increase of the probability of disruption, the emergency facilities are faced with a greater risk of damage after the establishment, and it is still necessary to pay attention to cost and demand-side satisfaction. The cost increases with the increase in the probability of disruption. Considering the location of sustainable emergency facilities, when the probability of disruption increases from $0.1$ to $0.4$, the cost increases gradually from $280.181$ to $1075.3528$, and the weighted satisfaction decreases gradually from $172.5315$ to $88.690129$. Without considering the location of sustainable emergency facilities, when the probability of disruption increases from $0.1$ to $0.4$, the cost gradually increases from $284.955$ to $1036.1065$, and the weighted satisfaction gradually decreases from $168.0872$ to $84.761843$. Furthermore, different interrupt probabilities of emergency facilities are visualized. Figure 9 shows the topological structure of the location-allocation network with different disrupt probabilities without considering sustainability.

It is observed that when the disruption probability of emergency facilities is $q=0.1$, the difference of the objective function $Z_1$ between the two strategies is about $0.0168$, and the difference of the objective function $Z_2$ is about $0.0264$. When the emergency facility-disruption probability is $q=0.2$, the difference of the objective function $Z_1$ between the two strategies is about $0.0119$, and the difference of the objective function $Z_2$ is about $0.0172$. When the disrupt probability of emergency facilities is $q=0.3$, the difference of objective function $Z_1$ between the two strategies is about $0.0126$, and the difference of objective function $Z_2$ is about $0.0884$. When the emergency facility-disruption probability $q=0.4$, the difference of the objective function $Z_1$ of the two strategies is about $0.0379$, and the difference of the objective function $Z_2$ is about $0.0463$, as shown in Figure 10 after visualization. According to statistics, the impact of the two strategies on cost and satisfaction is less than $0.1$, and the location of emergency facilities is also reasonable. Considering the location of emergency facilities from the perspective of reducing environmental pressure, the strategy proposed in this paper is more reasonable, which further proves the feasibility of the sustainable location of emergency facilities proposed in this paper.

In order to explore the influence of facility capacity and demand capacity on the model, we conducted the sensitivity analysis on the single-facility location experiment. While all other parameters remain unchanged as Example 1. After several experiments, the results are as follows:

In

Table 8. Sensitivity analysis of different facility capacity.

Facility Capacity	250	300	400
$Z_1$	127.4354	140.2133	139.6051
$Z_2$	148.2843	156.3298	159.7906

and

Table 9. Sensitivity analysis of different sums of the demand capacity.

Demand Capacity	150	200	250
$Z_1$	127.4149	139.6918	151.8591
$Z_2$	120.0962	156.4354	196.472

, it can be observed that the two objective functions of the model gradually increase with the increase in the amount of facility capacity and the sum of the demand capacity.

Example 5. 

Expand experiments.

In order to more accurately reflect the unpredictable nature of real-world emergencies, considering the impact of parameter changes on the experiment, a dynamic interruption scenario is introduced, and the probability of facility disruption q is defined as a random number in the range of $[0,1]$. Since the parameters of the KOA algorithm itself will also have an impact on the solution efficiency, we set the gravitational coefficient (23) to an adaptive parameter as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mu }_0={\mu }_{0_{min}}+({\mu }_{0_{max}}-{\mu }_{0_{min}})\times (1-{\displaystyle \frac{t}{T_{max}}}),\end{array}\end{array}$$

(43)

where ${\mu }_{0_{min}}=0.1$, ${\mu }_{0_{max}}=0.3$,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \gamma ={\gamma }_{max}-({\gamma }_{max}-{\gamma }_{min})\times (1-{\displaystyle \frac{t}{T_{max}}}),\end{array}\end{array}$$

(44)

where ${\gamma }_{min}=15$, ${\gamma }_{max}=30$, while all other parameters remain unchanged as Example 1.

The initial value of the gravitational coefficient $\mu$ starts from large and gradually decreases: We can start with a larger $\mu$ to increase the breadth of our search. With the increase of the number of iterations, $\mu$ is gradually decreased to enhance the local attraction between individuals and help the algorithm converge to the optimal solution faster. Using KOA, combine the convex hull algorithm with the path optimization algorithm, we optimize the objective and results of single-facility reliability emergency facility location. The single-facility location assignment scheme is obtained as shown in Figure 11, where $F_1$ indicates the optimal location of the determined single facility, and the blue dashed line represents the optimized final path, the red star indicate the final facility location, the red circle illustrates the demand regions, and the gray polygon outlines the barriers. In addition, we obtain the optimal location coordinates of the facility location as $(8.7778,9.715)$. The convergence speed of the model is relatively fast, and the optimal target value $Z_1$ is $157.4758$, $Z_2$ is $160.4823$.

By using KOA, combine the convex hull algorithm with the path optimization algorithm, we optimize the objective and results of multi-facility reliability emergency facility locations. In Figure 12, we have derived multi-facility location-allocation schemes, where $F_1,F_2,F_3$ indicate the optimal locations of the identified facilities. The blue dashed line represents the optimized final path, the red stars indicate the final facility locations, the red circle illustrates the demand regions, and the gray polygon outlines the barriers. We also determined the 3 optimal coordinates of the facility points as $(18.021,13.574)$, $(2.7478,14.0868)$ and $(8.0103,21.5236)$, resulting in the optimal target value $Z_1$ is $280.7582$, $Z_2$ is $181.244$.

6. Discussion

The proposed bi-objective model effectively balances time satisfaction and cost minimization for emergency facility location under complex polygonal barriers, integrating sustainability, capacity, time, and economic factors—an improvement over single-objective existing studies. The combined convex hull, path optimization, and KOA algorithms efficiently handle barriers and outperform traditional methods like NSGA-II in large-scale scenarios.

Limitations include static barrier assumptions (unlike dynamic real disaster scenarios) and reliance on hypothetical data, lacking real-case verification. While some existing studies use specific cases for stronger practical guidance, they often lack universality.

7. Conclusions

This paper develops a bi-objective model for sustainable emergency facility location under complex polygonal barriers, considering disruption risk and multiple factors. Solved via combined algorithms, simulations show it achieves high satisfaction, minimizes costs, eases environmental pressure, and aids rescues, with the algorithm framework efficient for large-scale applications. Future work will address dynamic barriers and real cases to boost practicality, offering insights for complex environment challenges.