Autonomous Coverage Path Planning Model for Maritime Search and Rescue with UAV Application

Abstract

Maritime transport is vital to the global economy, yet the frequency of natural disasters at sea continues to rise, resulting in more persons falling overboard. Therefore, effective maritime search and rescue (SAR) hinges on accurately predicting the probable distribution of drifting victims and on rapidly devising an optimal search plan. Conventional SAR operations either rely on rigid, pre-defined patterns or employ reinforcement-learning techniques that yield non-unique solutions and incur excessive computational time. To overcome these shortcomings, we propose an adaptive SAR framework that integrates three modules: (i) the AP98 maritime-drift model, (ii) Monte Carlo particle simulation, and (iii) a mixed-integer linear programming (MILP) model. First, Monte Carlo particles are propagated through the AP98 model to generate a probability density map of the victim’s location. Subsequently, the MILP model maximizes the cumulative probability of rescue success while minimizing a composite cost index, producing optimal UAV search trajectories solved via Gurobi. Experimental results on a 10 km × 10 km scenario with five UAVs show that, compared with traditional parallel-line search, the proposed MILP approach increases cumulative success probability by 12.4% within the first twelve search steps, eliminates path overlap entirely, and converges in 9.5 s with an optimality gap of 0.79%, thereby demonstrating both efficiency and real-time viability. When MIPFocus (a solver setting in Gurobi that controls the emphasis of the Mixed Integer Programming solver) aims at the optimal solution and uses the parallel solution method at the same time, the best result is achieved.

1. Introduction

With the continuous progress of the global economy, maritime transportation has become the primary mode of international cargo transportation. At the same time, as human exploration and production activities on the ocean increase, maritime operations are becoming more and more common.

However, the marine environment is extremely complex and changeable. Natural disasters such as strong winds, huge waves, storms, lightning strikes, and tsunamis may occur at any time. This also leads to frequent maritime accidents, which often result in people falling into the water and causing casualties. Therefore, formulating timely and efficient solutions is extremely crucial. Koopman [1] built upon the geometric and kinematic relationships between the observer and target established in the first paper; the present contribution formulates a probabilistic framework that quantifies the intrinsic uncertainties of detection under specific contact conditions, while underscoring that the resulting methods remain contingent upon the discrete and independent nature of visual glimpses. In the past, various departments often used helicopters for search. With the rapid development of UAVs, we can now try to control UAVs to assist in conducting maritime search work.

Generally, maritime rescue comprises two fundamental components, namely search and rescue, and search must be conducted prior to rescue. Given operational constraints and the critical time window in real missions, improving search efficiency is both effective in raising the overall probability of success and practical in terms of on-scene implementability. Given the vulnerability of people falling into water in the marine environment, rescue personnel must search for and locate them as soon as possible. Therefore, the search for people in the water encompasses three main tasks: (1) accurately and swiftly predicting the drift trajectory of people in the water (Brushett et al. [2]; Chen et al. [3,4]); (2) determining the optimal search area to ensure full coverage of the possible distribution range; (3) planning the search path of search and rescue (SAR) and optimizing the cumulative probability of success ($POS$) of the entire search process (Brown [5]; Kratzke et al. [6]; Mou et al. [7]).

When the rescue unit reaches the starting point of maritime search and rescue, due to the joint impacts of ocean currents, waves, and wind, the person in the water will keep drifting. The marine environment is extremely intricate, and there are numerous uncertain factors affecting the drifting process, which significantly heightens the difficulty of locating the person in the water. Furthermore, this complexity further augments the complexity of search path planning, ultimately making the entire search process more arduous. The Leeway model was initially put forward by Allen and Plourde [8]. In their papers “Review of Leeway” and “Leeway Divergence” [8], a quite comprehensive analysis of the wind pressure drift model was carried out. This model is mainly employed to quantify the drift of objects and has been widely applied in aspects like national search planning. Sea-based drift experiments are generally regarded as one of the most commonly used and reliable means for determining the leeway coefficient (Breivik et al. [9]; Kasyk et al. [10]; Meng et al. [11]; Sutherland et al. [12]; Tu et al. [13]; Wu et al. [14]). And Morin et al. [15] showed superior performance to greedy and MILP benchmarks and enabling implementation in operational SAR decision–support systems.

In terms of path planning, Zhang et al. [16] integrated heuristic crossover, real-time path straightening, and cubic B-spline smoothing into the basic SAR algorithm, yielding markedly faster convergence and safer, smoother 2-D/3-D trajectories than SAR, DE, ALO, and SSA counterparts. Du Y. [17] integrated an enhanced A* with a cooperative task-allocation scheme on grid-based 2D/3D maps, the proposed method markedly reduces computation time and memory. SARCPPF unites drift prediction, probability map and RL-guided coverage, Wu et al. [18] focused vessels on high-probability zones to outperform preset patterns. Zhan et al. [19] formulated a graph-based MSARPPP-MRC model and introduced a GA-RL algorithm that combined heuristic initialization with Q-learning-driven population management; experiments confirmed that GA-RL outperformed existing methods in optimizing multi-center UAV deployments for maritime SAR. Gramajo and Shankar [20] proposed a UAV flight algorithm that can maximize the use of the reserve energy of UAVs to cover the search area and optimize the turning method of UAVs to further shorten the search time without affecting the search coverage area.

For the full-coverage search path planning algorithm, the general goal is to minimize the repetition rate while maximizing the coverage area. Tan et al. [21] concluded open research problems and made suggestions for future research directions to address optimization problems related to coverage path planning (CPP) and improve the usability of robot applications. Cai et al. [22] focused on the Multi-robot Coverage Path Planning problem for maritime SAR missions using multiple UAVs. Hu and Mo [23] proposed a heuristic-crossover-enhanced search and rescue optimization algorithm (HC-SAR) that integrates real-time path adjustment and cubic B-spline smoothing, demonstrating significantly superior convergence speed and path-planning performance over SAR, DE, ALO, and SSA in both 2D and 3D threat environments. Li et al. [24] studied the path planning of multiple UAVs for sweep coverage in wireless sensor networks, considering the min-time max-coverage issue in forest fire early warning and monitoring. Dai et al. [25] proposed a novel method for multi-robot coverage analysis of large and complex structures with prior information. Li et al. [26] proposed an efficient, fault-tolerant multi-UAV coverage algorithm validated in simulations and field tests, outperforming existing methods in speed and robustness, which consists of three modules and uses deep reinforcement learning and a reward function. Pehlivanoglu and Pehlivanoğlu [27] solved the path planning problem of an autonomous UAV in target coverage by using artificial intelligence methods and proposed initial population enhancement methods in GA. Miao et al. [28] proposed IAACO, an adaptive ant-colony algorithm that uses angle guidance, obstacle avoidance, and multi-objective tuning to achieve faster, globally optimal indoor robot paths. Julius et al. [29] used a heuristic algorithm to optimize paths in smart mobile logistics applications in various environments. Gallego et al. [30] trained a convolutional neural network for body detection. A range-free and distributed method by using the application of the salp swarm algorithm for moving target node in the network for maritime rescue is proposed by Rani et al. [31]. Wen et al. [32] considered the scenarios where search and rescue targets (SRTs) may be mobile, and proposed a route planning method for UAVs applicable to the maritime SAR problem. An improved NSGA-II based on multi-task optimization (INSGA-II-MTO) is proposed to plan paths for multi-UAVs in the MSR tasks. In the INSGA-II-MTO, a novel population initialization method is proposed to improve the diversity of an initial population by Ma et al. [33]. Ai et al. [34] planned a search path with the shortest time-consuming and priority coverage of high-probability areas, considering complete coverage of maritime SAR areas and avoiding maritime obstacles. Cho et al. [35] proposed a two-phase method for solving the coverage path planning (CPP) problem of multiple-UAV areas in maritime SAR. An intelligent ship scheduling and path planning method is proposed for a practical application scenario wherein the emergency rescue center receives rescue messages and dispatches emergency rescue ships to the incident area. Chen et al. [36] developed a framework to develop and investigate trajectory planning algorithms for maritime SAR scenarios employing UAVs. Wang et al. [37] developed a fast, safe UAV 3D surface inspection path planner that cuts path length, time, and defects by over half in both simulations and real flights. Zhang et al. [38] proposed a path expansion strategy to prevent drones from becoming trapped in local optimal solutions.

Some algorithms have problems in complex environments, including not handling multiple constraints well (e.g., marine factors and search resources), leading to suboptimal paths and inefficiency; and some intelligent algorithms have slow convergence, taking too long to obtain a solution and not meeting real-time needs in search and rescue.

In light of the preceding review, four distinct yet inter-related shortcomings pervade the current literature. Static search theories [1,5] ignore oceanic drift and thus misstate the target’s probable location; swarm meta-heuristics [16,17,18] sacrifice timeliness, routinely requiring minutes to converge and breaching the golden-time imperative; geometric sweep planners [20,25,37] generate inherently redundant trajectories that squander limited battery capacity; and extant mathematical programs [28] recast critical endurance and sea-state limits as soft penalties, yielding routes that become infeasible once real wind–current loads are imposed. The proposed AP98-coupled MILP framework was expressly designed to close these gaps by embedding dynamic drift, enforcing hard real-time and endurance constraints, and guaranteeing overlap-free coverage, thereby offering a unified remedy to the deficiencies identified in prior work.

The framework of this paper is divided into three parts. Firstly, the AP98 drift model is adopted to simulate the drift trajectory of people in the water (PIW). A local probability distribution map is obtained through the Monte Carlo method within the final predicted area of the people in the water. Then, the MILP model aiming at minimizing time and maximizing the cumulative success probability is established. Constraints, such as integer constraints on decision variables, node flow constraints, maximum node carrying capacity constraints, and endurance constraints, are introduced to comprehensively solve the optimal search and rescue path. Finally, the advantages and disadvantages of the MILP method compared with the traditional parallel method are analyzed, and the optimal search parameters in the search and rescue process are adjusted.

The contributions of this paper are as follows:

(1)

The paper integrates search–resource constraints and the spatial probability distribution of targets. It explicitly models the UAV endurance limit and progressively re-weights the cumulative success–probability term in the objective function across solver iterations, which enhances both search efficiency and overall success rate.

(2)

The paper regards the UAV path as a flow and introduces realistic endurance constraints, enabling the UAV to automatically go to the supply point when the endurance is insufficient.

(3)

The proposed method converts the paths of all UAVs into multi-dimensional binary variable problems, and by solving, makes the model possess the dual functions of UAV scheduling and path planning simultaneously.

The remainder of the paper is structured to further elaborate on the proposed methods and their validation. Section 2 defines key probabilities in maritime search, including Probability of Containment ($POC$) and Probability of Detection ($POD$). Section 3 focuses on simulating the maritime SAR environment, using the AP98 model to predict drift trajectories and a Monte Carlo simulation. Section 4 details the mixed integer linear programming (MILP) model for coverage path planning, covering objective function design, various constraints, and model establishment. Section 5 experimentally verifies the effectiveness of the model. Finally, Section 6 concludes the study and suggests areas for future research.

To provide a clear benchmark for model validation, this paper formulates three explicit hypotheses:

• Persons-in-water (PIWs) exhibit no voluntary swimming motion or have exhausted their physical capacity.
• The endurance and cruising speed of each UAV remain constant during the entire mission.
• Oceanic environmental parameters (wind field and surface current) remain static within the operational time window.

2. The Probability Definitions in Maritime Search

The achievement of a maritime search operation hinges on comprehensive considerations in two respects. Firstly, the searchers have to carry out a search in the proper location. Secondly, the searchers at that location need to possess the capability to detect the distressed target. Since the actual position of a maritime distress target is influenced by wind and ocean currents, an appropriate location denotes a sea area that encompasses the distress target to a certain probable extent. Searchers can look for a specific distress target in this sea area via specific search equipment, and their ability to detect the target can also be gauged by a specific probable extent.

Comprehensive consideration of these two possible extents represents the possibility of detecting the target in this search operation, which is expressed by $POS$. The two possible extents are, respectively, expressed as: $POC$ and $POD$. The relationship among the three is shown in Equation (1).

$$POS=POC\times POD$$

(1)

According to the formula, to achieve a 100% search success rate, it is necessary to ensure that both the detection probability and the containment probability reach 100%. However, in the actual search and rescue process, due to human or environmental factors, ideal conditions cannot be achieved. Both probabilities fluctuate between 0% and 100%. Therefore, the success probability of a search operation is a random event. This also leads to the fact that in a search and rescue mission, the possibility of finding the target in the first search is not high. The search process often needs to be adjusted and repeated. Therefore, the concept of cumulative search success rate (Cumulative probability of success, denoted as $CPOS$) is born, which is used to measure the probability of finding a distress target in the entire search and rescue mission. Its calculation formula is shown in Equation (2).

$$CPOS={\sum }_{n=1}^{N}PO{S}_n$$

(2)

In the formula, $N$ represents the total number of searches, and ${POS}_n$ represents the success rate at the $n$-th search. $CPOS$ represents the cumulative probability of success over $N$ searches, computed as the running sum of $POS$, hence $CPOS$ $\in [0,1]$, providing a normalized measure of overall mission progress. Compared with $POS$, $CPOS$ can better represent the completion of the entire search area task and is also an important indicator for judging whether to abandon the search.

2.1. POC

Inclusion probability denotes the probability of encompassing a distress target within a specific search area and serves as a crucial foundation for formulating search plans. It is typically measured in percentages to signify the likelihood of a target being included in a particular area. As per the definition, provided that all areas that might potentially contain the distress target are incorporated into the category of the search area, the inclusion probability can attain 100%. Nevertheless, in actual search procedures, owing to constraints in manpower and resources, it is infeasible to search all sea areas that might potentially hold the distress target. To ensure search efficiency, search and rescue coordinators typically divide the entire search area that may potentially contain the distress target and prioritize searching the sub-areas with a relatively higher inclusion probability.

2.2. POD

$POD$ stands for Probability of Detection, which represents the possibility that a search unit can detect a SAR target and is a crucial indicator. It evaluates the effectiveness of SAR detectors in the search area (Abi-Zeid and Frost [39]). The calculation of $POD$ involves two important concepts, namely scan width and coverage. Scan width refers to the effective distance within a given search area at which a detector can locate a search target, and the specific parameters correspond to those in

Table 1. Weather correction coefficient for general search targets.

Winds (km/h) or Currents (m/s)	${\mathit{C}}_{\mathit{w}}$ for PIW, Life, Raft, or Ship < 10 m (33 ft)	${\mathit{C}}_{\mathit{w}}$ for Others
10–28 km/h or 0–1 m	1.0	1.0
28–46 km/h or 1–1.5 m	0.5	0.9
>46 km/h or >1.5 m	0.25	0.9

. In this study, it serves as an indicator of a UAV’s search capability. The numeric entries in the center and right columns are the weather correction coefficients ($C_w$) applied to the nominal scan width to obtain the environment-adjusted effective sweep width.

3. Maritime SAR Environment Simulation

Marine floating objects mainly move under the combined action of wind, waves, and currents when drifting. In this paper, a PIW is selected as a simulated specimen of the drifting target. According to the AP98 model, the influence of waves on its drifting process can be ignored. Therefore, when establishing the drifting model of maritime distress targets in this paper, only the influences of wind and currents are analyzed. In the entire SAR environment modeling, the overall process is shown in Figure 1. According to the real marine scene, the distribution probability map of people falling into the water is predicted. Then, MILP is used for resource scheduling and path planning of UAV search.

3.1. Principle of AP98 Model

For the calculation of the current drift vector, the Lagrangian integration method for sea surface flow velocity can generally be used by Prince [40]. However, it is more difficult to estimate the wind pressure drift vector. The construction of drift models has been developed to the third generation so far (the first-generation GDOC model [8] and the second-generation CASP model [8]), the AP98 wind pressure drift model (hereinafter referred to as the AP98 model) [8].

The principle of the AP98 model is that when the wind acts on a maritime drifting target, the actual wind pressure drift vector (Leeway) of the floating object can be decomposed into two parts: the downwind drift vector (Downwind Leeway, $DWL$) and the crosswind drift vector (Crosswind Leeway, $CWL$) [8], as shown in Figure 2.

In Figure 2, $W_{10\mathrm{m}}$ is the wind speed vector at a height of 10 m; Leeway is the wind pressure drift vector; $DWL$ is the downwind drift vector; $CWL$ is the crosswind drift vector. It is worth noting that when actually using the model, the probability of $CWL$ appearing on both sides of $DWL$ with an offset is uncertain and generally approximately equal. Allen classified different maritime drifting targets according to different wind pressure characteristics, summarized the proposed methods for the wind pressure divergence angle of the AP98 model for 63 common search and rescue targets, and established an initial AP98 model search and rescue database.

The process of calculating the drift of a specific maritime drifting target on the sea surface under the influence of wind force using the AP98 model is as follows: First, nine AP98 wind pressure drift parameters of this type of target and the wind field information of the sea state at the time of the incident need to be input. Then, according to the simulation, three directions (downwind is $DWL$, right-offset crosswind is denoted as $+CWL$, and left-offset crosswind is denoted as $-CWL$) of wind pressure vectors are synthesized to obtain the actual wind-induced drift vector. These nine parameters are fitted based on a large amount of drift data of this type of floating object. The slope term, intercept, and regression deviation of the linear regression equation were established with the actual wind speed in each direction.

$$DWL=a_d\times W_{10}+b_d+{\epsilon }_d$$

(3)

$$+CWL=a_{+c}\times W_{10}+b_{+c}+{\epsilon }_{+c}$$

(4)

$$-CWL=a_{-c}\times W_{10}+b_{-c}+{\epsilon }_{-c}$$

(5)

In Equations (3)–(5), $W_{10}$ is the wind speed vector at a height of 10 m, a represents the slope of each regression equation; $b$ represents the intercept of each regression equation; and $\epsilon$ is the regression deviation of each equation. And $d$ and $c$ explicitly distinguish the regression coefficients for the downwind and crosswind leeway speed components. $\epsilon =norm\ast S_{y/x}$ and $norm$ are random numbers selected from the standard normal distribution function, and $S_{y/x}$ is the regression standard deviation of each equation. For example, if the regression standard deviation of the crosswind component is 0.12 m/s, then $S_{y/x}=0.12$, and the random error term $\epsilon ~N(0,{0.12}^2)$.

Table 2. AP98 wind–pressure drift parameters (for a Person-in-Water, PIW).

Parameter Name	Physical Meaning (Units)	Typical Value for PIW
$a_d$	Downwind leeway slope (% per m s−1)	1.2 ± 0.3
$b_d$	Downwind leeway intercept (%)	0.5 ± 0.2
${\epsilon }_d$	Downwind leeway standard deviation (%)	0.4
$a_{+c}$	Crosswind leeway slope, right offset (% per m s−1)	0.3 ± 0.1
$b_{+c}$	Crosswind leeway intercept, right offset (%)	0.1 ± 0.05
${\epsilon }_{+c}$	Crosswind leeway standard deviation, right offset (%)	0.2
$a_{-c}$	Crosswind leeway slope, left offset (% per m s−1)	0.3 ± 0.1
$b_{-c}$	Crosswind leeway intercept, left offset (%)	0.1 ± 0.05
${\epsilon }_{-c}$	Crosswind leeway standard deviation, left offset (%)	0.2

lists the nine essential AP98 wind–pressure drift parameters for a PIW: three coefficients (slope, intercept, standard deviation) for the downwind leeway and three identical sets for both right- and left-offset crosswind leeway, fully defining how wind speed at 10 m height translates into PIW drift velocity and its statistical uncertainty.

3.2. Velocity Calculation of AP98 Model

In the practical use of the AP98 model to calculate the wind pressure velocity of marine drifting targets, it is necessary to project its velocity vector onto the east–west and north–south longitudinal coordinates, denoted by $u$ and $v$ respectively. The AP98 model does not directly give the calculation formula for its velocity. Denote the direction of wind-induced drift as $\alpha$, and its value interval is [0°, 90°), [90°, 180°), [180°, 270°), [270°, 360°). Since CWL can take two cases of $+CWL$ and $-CWL$, it forms wind-induced drift vectors $u_{wind}$ and $v_{wind}$ in eight environments. Among them, all four cases when taking $+CWL$ can be represented by Formulas (3)–(5), and all four cases when taking −CWL can be represented by Equations (6) and (7). In the absolute value sign in the formula, ± only represents the direction.

$$u_{wind}=DWL\times \mathit{sin}a\pm (\left|\pm CWL\right|\times \mathit{cos}a)$$

(6)

$$v_{wind}=DWL\times \mathit{cos}a\pm (\left|\pm CWL\right|\times \mathit{sin}a)$$

(7)

Here, $a_{-c}$ denotes the slope coefficient for the left-offset crosswind leeway component ($-CWL$), analogous to $a_{+c}$ for the right-offset case. Since $a_{-c}$ is less than 0 and $W_{10\mathrm{m}}$ is greater than 0, and under normal circumstances, $a_{-c}\times W_{10\mathrm{m}}$ determines the principal value of $-CWL$. Therefore, $-CWL$ takes a negative value. Thus, the above four formulas can be uniformly combined into the following formula, which is the general calculation formula for the wind-induced drift velocities $u_{wind}$ and $v_{wind}$ obtained by floating objects under the AP98 model in the actual wind field are shown in Equations (8) and (9).

$$u_{wind}=DWL\times \mathit{sin}\alpha +CWL\times \mathit{cos}\alpha$$

(8)

$$v_{wind}=DWL\times \mathit{cos}\alpha -CWL\times \mathit{sin}\alpha$$

(9)

3.3. Drift Integration Method Under the Action of Wind and Current Based on AP98 Model

There are many dynamic factors affecting marine floating objects. When taking PIW as the distress target for simulation, the main influences considered are current and wind. The integral formula for current-induced drift can be calculated using Equations (10)–(12).

$$x(t)=x(t_0)+{\int }_{t_0}^{t}u(x,t)dt+{\delta }_x$$

(10)

$$y(t)=y(t_0)+{\int }_{t_0}^{t}v(x,t)dt+{\delta }_y$$

(11)

$${\delta }_x=\xi \sqrt{2{\kappa }_x\mathsf{\Delta }t}, {\delta }_y=\zeta \sqrt{2{\kappa }_y\mathsf{\Delta }t}$$

(12)

In the formula, $x\left(t\right)$ and $y\left(t\right)$ respectively represent the position coordinates in the east–west and north–south directions. $\mathsf{\Delta }t$ is the time step of the simulation. $\delta$ is the physical error correction. $\xi$ and $\zeta$ are two random numbers selected from the standard normal distribution. ${\kappa }_x$ and ${\kappa }_y$ are the eddy diffusion coefficients on the sea surface. As for the vector decomposition of the wind pressure model of PIW, it has been analyzed in the section above. Considering the wind and current velocities in the AP98 model comprehensively, the total drift velocity formula of the target is obtained, that is, Equations (13) and (14).

$$u_{total}=u_{current}+u_{wind}=u_{current}+DWL\times \mathit{sin}\alpha +CWL\times \mathit{cos}\alpha$$

(13)

$$v_{total}=v_{current}+v_{wind}=v_{current}+DWL\times \mathit{cos}\alpha +CWL\times \mathit{sin}\alpha$$

(14)

Given the initial drift time $t_0$ and the simulated drift time $t$ of the target, by calculating the magnitude of the drift amount during this period, the simulated coordinate position can be obtained, that is, Equations (15) and (16).

$$x\left(t\right)=x\left(t_0\right)+{\int }_{t_0}^t{u}_{total}\left(x,t\right)dt+{\delta }_x=$$

$$x(t_0)+{\int }_{t_0}^t(u_{current}+DWL\times \mathit{sin}\alpha +CWL\times \mathit{cos}\alpha )(x,t)dt+{\delta }_x$$

(15)

$$y\left(t\right)=y\left(t_0\right)+{\int }_{t_0}^t{v}_{total}\left(y,t\right)dt+{\delta }_y=$$

$$y(t_0)+{\int }_{t_0}^t(v_{current}+DWL\times \mathit{cos}\alpha -CWL\times \mathit{sin}\alpha )(y,t)dt+{\delta }_y$$

(16)

3.4. Probability Model of Monte Carlo Simulation Method

According to the motion theory, after completing the model construction of the distress target and collecting the data of sea conditions and distress-related information, the coordinate position of the target at the corresponding time can be calculated. However, in the real process, due to the variable physical structure of the drifting target itself or the uncertainty of the marine environment, it is difficult to accurately simulate the actual drift trajectory without error. Therefore, when planning the search area, a rough scene estimation should be made considering the influence of various errors. We adopt the Monte Carlo method to fit the simulation solution of the problem through experiments on the results.

The Monte Carlo method is superior to the traditional analytical method in calculating the search area and even determining the target inclusion probability model. Not only because the calculation process of the analytical method is too complicated, but also because when the Monte Carlo method simulates random particles, the particles are independent of each other and can flexibly allocate the time, position, quantity and even drift characteristics of the particles to simulate the diversity of the state of the distress target in the actual search process, making it more representative of the complex search and rescue scenarios in reality, which is difficult for the traditional analytical method to achieve. The basic steps for using the Monte Carlo method to generate a simulation solution for the search area are described below.

Assuming that the drift simulation function of the distress target is $g$, and the integral of the search area is as follows:

$$I={\int }_{D}g\left(x\right)dx$$

(17)

$${\int }_{D}g\left(x\right)dx={\int }_D{\displaystyle \frac{g\left(x\right)}{f\left(x\right)}}f(x)dx=E(h(X))$$

(18)

where $h\left(x\right)=g\left(x\right)/f\left(x\right)$, the probability density function of the random variable $X$ is $f\left(x\right)$, and $f\left(x\right)$ is a properly selected function and is greater than 0 on the support of $g\left(x\right)$.

In computer simulation, (pseudo) random numbers $X_1,X_2,X_3,\dots ,X_N$ are generated. They are independently and identically distributed and come from the distribution corresponding to $f\left(x\right)$.

Use ${\widehat{I}}_N$ to estimate $I$. In Equation (19), $I$ denotes the expected integral of the leeway drift function $g\left(x\right)$ over the search domain $D$, where $g\left(x\right)$ is the probability density of the drifting target’s location at a given time.

$${\widehat{I}}_N={\displaystyle \frac{1}{N}}{\sum }_{i=1}^{N}h(X_i)$$

(19)

This study assumes that individuals who have fallen overboard have no active swimming capacity or have become physically exhausted—a condition that aligns with most sudden maritime incident scenarios. Should the individuals retain some swimming ability, the actual drift trajectories could deviate from the predictions of the AP98 model. Future work could address this potential source of error by incorporating a probabilistic behavioral model that accounts for swimming direction preferences based on the individual’s remaining physical stamina.

4. Maritime SAR Coverage Path Planning Based on Mixed Integer Linear Programming

To solve the problem in the full-coverage path planning of maritime search and rescue (SAR), a team of UAVs of size $K$ needs to continuously cover a set of target grid maps. Each UAV has an endurance capacity of $L$. In an environment where recharging is possible, there is a group of supply stations scattered, with unlimited endurance capacity. If there is an edge, its cost is less than or equal to $L$. A necessary condition to ensure a feasible solution is to provide UAV coverage with endurance capacity for each target. In addition to the time required to travel to the target site, UAVs also spend a certain amount of time charging at the warehouse. Assume that this charging time is proportional to the flight time, that is, the endurance consumption. Let the proportionality constant be $Q$ (assuming that the proportionality constants of all warehouses are the same). In addition, we require that the rescue UAV starts from the supply station and automatically returns to the supply station at the end. The overall design of the specific Mixed Integer Linear Programming (MILP) is described below.

Equality constraints: $A_{eq}{x}_{val}=b_{eq}$, where $A_{eq}$ is the coefficient matrix of equality constraints, $x_{val}$ is the variable vector, and $b_{eq}$ is the constant vector on the right side of equality constraints. In the maritime SAR scenario, these equality constraints include constraint conditions such as ensuring that only one UAV arrives at and leaves each target, the UAV starts and ends at the starting position, and each UAV visits a target and then leaves.

Inequality constraints: $A_{ineq}{x}_{val}\le b_{ineq}$, where $A_{ineq}$ is the coefficient matrix of inequality constraints, $x_{val}$ is the variable vector, and $b_{ineq}$ is the constant vector on the right side of inequality constraints. These inequality constraints include constraint conditions such as endurance restrictions and capacity limitations.

Variable bounds constraints: $lb\le x_{val}\le ub$, where $lb$ is the lower bound vector of variables and $ub$ is the upper bound vector of variables. The upper and lower bounds of variables are set according to different types of variables, and integer variables have different ranges of upper and lower bounds.

4.1. Objective Function of MILP Model

In maritime rescue missions, the design of the objective function is crucial for rationally planning the paths of rescue UAVs. The objective function of this model is dedicated to minimizing a comprehensive index that combines the probability of the existence of target nodes (possibly the locations to be rescued), the search range limitations of UAVs, and penalty terms to ensure the effectiveness and feasibility of the rescue plan. The probability of the existence of target nodes occupies an important position in the objective function. By assigning different weights, it reflects the possibility of distress signals appearing at various locations to be rescued. A higher probability of target existence means that this location is more likely to require the timely arrival of rescue resources. Considering the limited endurance and search efficiency of UAVs, a specific search radius is defined for each UAV. In the objective function, the weights of corresponding variables are adjusted according to whether node pairs are within the search range of UAVs. If the node pairs are within the search range and involve target nodes, then the weights will be dynamically adjusted according to specific circumstances to encourage UAVs to conduct more targeted searches within their effective search range. In order to encourage UAVs to cover all areas in as few steps as possible, a penalty term is introduced. The existence of penalty terms encourages the path selection of rescue UAVs to be more definite and to avoid repetitive and useless decisions.

$$\mathit{min}Z=1+{\sum }_{k=0}^{K-1}{\sum }_{i=0}^{N-1}{\sum }_{j=0}^{N-1}{f}_{kij}{x}_{ij}^k+{\sum }_{k=0}^{K-1}{\sum }_{i=0}^{N-1}{\sum }_{j=0}^{N-1}{C}_p{x}_{ij}^k$$

(20)

$$f_{kij}=\left\{\begin{array}{c}-a\times f_{decay}\times po{s}_{ij},i<T,j<T,\left|x_i^k-x_{i-1}^k\right|\le 1,\left|y_j^k-y_{j-1}^k\right|\le 1,\sqrt{{\left(x_i^k-x_s^k\right)}^2+{\left(y_j^k-y_s^k\right)}^2}\le R_k\hfill \\ -f_{decay}\times po{s}_{ij},i<T,j<T,\sqrt{{\left(x_i^k-x_s^k\right)}^2+{\left(y_j^k-y_s^k\right)}^2}\le R_k\hfill \\ -\beta \times f_{decay}\times po{s}_{ij},i<T,j<T,\sqrt{{\left(x_i^k-x_s^k\right)}^2+{\left(y_j^k-y_s^k\right)}^2}>R_k\hfill \\ 0,others\hfill \end{array}\right.$$

(21)

$$f_{decay}=\left\{\begin{array}{l}{d}_0,t=0\\ d_t=d_0\times e^{\lambda t}\end{array}\right.$$

(22)

The optimization objective $Z$ represents the minimization of the comprehensive cost or maximization of the benefit when the UAVs execute the maritime search and rescue mission under the premise of satisfying all constraints. $f_{kij}$ is a variable related to UAV path planning and is used to describe the flow or path selection situation of the UAV from node $i$ to node $j$. Let $K$ represents the number of UAVs, $N$ represents the total number of nodes including target nodes $T$ and supply stations $D$. $x_{ij}^k$ indicates whether UAV $k$ chooses to go to node $j$ from $i$. and $po{s}_{ij}$ represents the $POS$ probability corresponding to the grid at $(i,j)$. $C_p$ represents the penalty coefficient. $f_{decay}$ is used to dynamically adjust the weight of node probability in the objective function. At the early stage of optimization, more emphasis is laid on fewer steps and time; as the number of iterations rises, the weight increases, enabling the probability of node existence to play a more significant role in decision-making. We hope that UAVs can cover all target areas in the fewest steps possible. Let $R_k$ be the scalar flight-radius limit of UAV $k$, where $(x_s^k,y_s^k)$ is its starting depot. In real situations, considering that UAVs have a maximum communication radius, this is used to define constraints. This radius is therefore employed to bound the UAV’s instantaneous coordinates $(x_i^k,y_j^k)$ within a circular disk of radius $R_k$.

Consequently, early in the mission, the optimizer is strongly attracted to high-$POC$ nodes within the reachable radius, whereas later iterations witness a progressive loss of attraction as $e^{\lambda t}$ to 0. This mechanism prevents excessive revisitation of already-covered high-probability cells and encourages systematic coverage of the remaining search area. In summary, Equation (21) enforces spatial feasibility via the radius $R_k$, while Equation (22) introduces time-based diminishing rewards, together yielding a spatio-temporal adaptive cost structure that balances exploitation of high-likelihood regions with exhaustive coverage of the entire reachable domain.

Moreover, we also need to take into account the flight time of UAVs. In the actual maritime SAR scenario, the rescue time for people in distress is limited and valuable. UAVs need to achieve full-coverage search in the shortest time to enhance the rescue success rate.

Let $T$ be the set of target nodes and $D$ be the set of supply depots. Then, the total number of nodes is $N=\left|T\right|+\left|D\right|$, where $\left|T\right|$ and $\left|D\right|$ denote the cardinality of the sets. Here, $x_{ij}$ represents the edge between two nodes connected by the UAV, $c_{ij}$ represents the time required for the UAV to cross these two nodes, and $d_i$ represents the waiting time at the supply station $i$. Then, the minimum upper bound required for all UAVs to complete a full-coverage search can be represented by the following formula:

$$T^{*}=\mathit{min}\underset{k\in K}{max}{\sum }_{i\in N}{\sum }_{j\in N}(c_{ij}+d_i)x_{ij}^k$$

(23)

In order to minimize the required time in a way that conforms to MILP constraints while minimizing the objective function, the paper uses a single decision variable $T_{max}$ to minimize it and writes the conditions of maximizing constraints for it.

Based on the above derivation, the formula now becomes the following:

$${\sum }_{i\in N}{\sum }_{j\in N}(1+q_k)c_{ij}{x}_{ij}^k\le T_{max}$$

(24)

Among them, $q_k$ is the ratio of the time required for UAV $k$ to recharge, given its travel time.

4.2. Integer Restriction Constraint

When it comes to targets, these constraints force $x_{ij}^k$ to be binary and an integer between 0 and 1. When connecting two supply stations with a path. This means that for each pair of nodes, when rescue resources (such as rescue UAVs) decide whether to move from one node to another, there are only two choices: either choose this path, in which case the variable value is 1; or do not choose this path, in which case the variable value is 0.

$$x_{ij}^k\in \left\{\mathrm{0,1}\right\},\forall i\in T,\forall j\in T,k\in K$$

(25)

$${\sum }_{i,j}{x}_{ij}^k\in \left\{\mathrm{0,1},2,\dots ,\left|T\right|\right\},\forall i,j\in D,k\in K$$

(26)

The above restriction means that each UAV can only choose whether to move from point $i$ to point $j$ at a time. In addition, for rescue UAVs, they can go to the supply station at most $T$ times. $T$ represents the target node and $D$ represents the supply point. There are a total of $K$ UAVs here.

4.3. Node Degree Restriction Constraint

In the maritime search and rescue (SAR) scenario, degree constraints play a crucial role in mixed integer linear programming (MILP) path planning. Degree constraints are usually applied in the formulation of the traveling salesman problem (TSP) or vehicle routing problem (VRP), and their purpose is to control the flow through nodes. Specifically, let $S_k\in D$ denote the starting depot assigned to UAV $k$; the degree constraints then stipulate that every target node is entered and exited exactly once by a single UAV, while each UAV must both originate from and return to its designated depot $S_k$. By enforcing these rules, the model ensures balanced inflow and outflow at every node, and guarantees that UAV routes form closed loops beginning and ending at $S_k$, and thereby achieves rational allocation and efficient use of rescue resources, reduces waste and conflicts, and provides a strong guarantee for the success of maritime SAR tasks.

$${\sum }_{k\in K}{\sum }_{j\in N}{x}_{ij}^k=1,\forall i\in T\backslash S_k$$

(27)

$${\sum }_{k\in K}{\sum }_{j\in N}{x}_{ji}^k=1,\forall i\in T\backslash S_k$$

(28)

$${\sum }_{j\in N}{x}_{S_{k}j}^k\le 1,\forall k\in K$$

(29)

$${\sum }_{j\in N}{x}_{j{B}_k}^k\le 1,\forall k\in K$$

(30)

$${\sum }_{i\in N}(x_{ij}^k-x_{ji}^k)=0,\forall j\in N,\forall k\in K$$

(31)

$S_k$ in constraints (27) and (28) is defined as the starting point of UAV $k$, same as the end point $B_k$. Constraints (27) and (28) ensure that each target is reached and left by only one UAV to guarantee the rational allocation of rescue resources and avoid multiple UAVs flocking to the same target while ignoring other targets. For target node $i$, constraint (27) requires that the sum of path selection variables from node $j$ to target node $i$ is 1. Constraint (28) is similar. Constraints (29) and (30) ensure that the UAV on a mission starts and ends at the starting position. Usually, the starting position is a base or supply station, which is convenient for management and monitoring.

Constraint (31) ensures that each UAV will leave the target after visiting it. In maritime SAR tasks, the mission of UAVs is to respond quickly and rescue targets. This constraint guarantees that after completing the rescue mission for a target, the UAV will not stay at the target location but continue to the next target or return to the base. In conclusion, in the MILP path planning of the maritime SAR scenario, degree constraints ensure through constraints (27)–(31) that each target is reached and left by only one UAV, that the UAV starts and ends at the starting position, and that each UAV will leave the target after visiting it. These constraints help to rationally allocate rescue resources, improve rescue efficiency, and ensure the successful completion of maritime SAR tasks.

4.4. Maximum Capacity and Flow Restriction Constraints

In maritime search and rescue (SAR) tasks, UAVs typically have limited resources such as endurance, battery capacity, or the quantity of rescue equipment. The purpose of capacity and flow constraints is to ensure that when UAVs are performing tasks, they can manage and allocate these resources reasonably while avoiding ineffective paths or sub-loops.

Eliminating sub-loops is crucial for maritime SAR tasks. If a UAV becomes stuck in a sub-loop during a mission, that is, repeatedly visiting certain nodes without being able to cover the entire target set, it will waste precious time and resources and reduce rescue efficiency. Through capacity and flow constraints, sub-loops can be effectively eliminated to ensure that the path of the UAV forms a single closed loop, thereby covering all target nodes and returning to the starting position or an appropriate supply station.

$${\sum }_{i\in N\backslash B_k}(p_{B_{k}i}^k-p_{i{B}_k}^k)={\sum }_{i\in T\backslash B_k,j\in N}{x}_{ij}^k,\forall k\in K$$

(32)

$${\sum }_{j\in N\backslash \left\{i\right\}}(p_{ji}^k-p_{ij}^k)={\sum }_{j\in N}{x}_{ij}^k,\forall i\in T\backslash S_k,k\in K$$

(33)

$${\sum }_{j\in N\backslash \left\{i\right\}}(p_{ji}^k-p_{ij}^k)=0,\forall i\in D,k\in K$$

(34)

$$0\le p_{ij}^k\le \left|T\right|x_{ij}^k,\forall i,j\in N,k\in K$$

(35)

In the maritime SAR scenario, there are constraints on both the flow quantity $p_{ij}^k$—representing the UAV’s resource state when traversing from node $i$ to node $j$ and the node flow resource allocation at each node.

Constraint (32): In the maritime SAR scenario, this constraint represents the flow through the starting node. For each UAV $k$, the amount of resources obtained when starting from the starting node $S_k$ is equal to the number of targets allocated to this UAV (i.e., ${\sum }_{i\in T\backslash B_k,j\in N}{x}_{ij}^k$, where $x_{ij}^k$ is a binary variable indicating whether the UAV moves from node $i$ to node $j$). Constraint (33): In the maritime SAR scenario, this constraint ensures that if the corresponding target is included in the allocation set of the UAV, the capacity is reduced by according to this constraint. For each target node $i$ ($i\in T\backslash B_k$, representing the target nodes in the target node set except for the starting position of UAV $k$) and UAV $k$, when the UAV arrives at the target node $i$ from another node $j$, the amount of flow resources will decrease.

Constraint (34): where $j\in N\backslash \left\{i\right\}$. In the maritime SAR scenario, this constraint means that when the UAV passes through an endurance supply station (a node in set $D$), the amount of resources does not change. This is to prevent detours during recharging and damage the continuity of the UAV’s actions. Constraint (35): This constraint ensures that the target capacity of each UAV does not exceed $\left|T\right|$ (the total number of target nodes). In the maritime SAR scenario, this means that the amount of resources carried by the rescue UAV at any time cannot exceed the total number of target nodes multiplied by a unit resource quantity. If each target is restricted to be visited only once, then the only way for this set of constraints to be effective is for the target set to consist of a single connected sequence.

4.5. Maximum Endurance Constraints

In the current full-coverage maritime SAR scenario of UAVs, in order to ensure that the UAV has enough endurance to complete the comprehensive monitoring task of the vast sea area and to ensure that the UAV has enough endurance to continuously traverse the itinerary, constraints must be added to force the UAV to detour for recharging at an appropriate time.

$$r_j-r_i+f_{ij}\le M(1-x_{ij}^k),\forall i,j\in T,k\in K$$

(36)

$$r_j-r_i+f_{ij}\ge -M(1-x_{ij}^k),\forall i,j\in T,k\in K$$

(37)

$$r_j-L+f_{ij}\ge -M(1-x_{ij}^k),\forall i\in D,j\in T,k\in K$$

(38)

$$r_j-L+f_{ij}\le M(1-x_{ij}^k),\forall i\in D,j\in T,k\in K$$

(39)

$$r_j-f_{ij}\ge -M(1-x_{ij}^k),\forall i\in T,j\in D,k\in K$$

(40)

$$0\le r_i\le L,\forall i\in T$$

(41)

Among them, $r_i$ is a key decision variable representing the remaining endurance amount of the UAV when leaving the target. In the actual maritime rescue scenario, accurately grasping this remaining endurance amount is crucial for planning a reasonable flight path. $f_{ij}$ represents the endurance cost between nodes $i$ and $j$. This cost is affected by multiple factors such as flight distance, wind direction, sea conditions, etc. When planning the path, these endurance costs must be fully considered to ensure that the UAV will not have problems due to endurance exhaustion during flight. $M$ is a relatively large constant value and can be selected as $L+{\mathit{max}}_{\mathit{ij}\in N} f_{ij}$. It can ensure that in the calculation of constraint conditions, various situations can be effectively handled to ensure the validity of the constraints.

Constraint conditions (36) and (37) can be represented by the constraint $r_i-r_j=f_{ij}$ when $x_{ij}^k=1$. In the full-coverage path planning for maritime rescue SAR, this pair of constraints ensures that the endurance loss between two nodes is equal to the endurance cost of traveling between them.

Constraints (38) and (39) can be described as $L-r_j=f_{ij}$ and establish the condition that the endurance level at the visited target after leaving the supply station is equal to the endurance capacity minus the traversal endurance cost. Similarly, constraint (40) can also be represented as $r_i\ge f_{ij}$ and limits the endurance lost when approaching a supply station to at most the cost of departing from the previous target. In the maritime rescue scenario, when the UAV approaches the supply station, it must be ensured that its remaining endurance is sufficient to support it to safely reach the depot. Finally, in Equation (41), we forcibly limit each endurance level parameter to be between 0 and $L$.

4.6. Establishment of the Mixed Integer Linear Programming (MILP) Model

The mixed integer linear programming (MILP) model plays an important role in solving complex optimization problems. The following

Table 3. Primary parameter settings of the MILP model.

Parameter Name	Value	Parameter Explanation
Outputflag	1 (output)	Turn on log output to view information during the solution process.
Method	2 (Barrier)	Try different solution methods, which may have an impact on variable relaxation.
BranchDir	−1 (branch direction)	Automatically select the branch direction, which may help with relaxation during the solution process.
Presolve	2 (Aggressive)	Enable Aggressive preprocessing to improve solution efficiency.
TimeLimit	100	Set the time limit to 100 s.
NodeLimit	500	Limit the number of nodes.
MIPGap	0.01 (difference)	Set the MIP gap to 1%.
MIPFocus	1 (feasible)	Mainly focus on finding a feasible solution.

is an explanation of the parameter settings of this MILP model to better understand the solution process and optimization objective of the model.

In the initial model, this paper adopts the default model-solving approach to conveniently obtain a specific and feasible solution. Subsequently, the parameter will be optimized and iterated.

5. Experimental Verification and Discussion

To verify the effectiveness of the model solution in this paper, the AP98 model is first used to simulate the drift trajectory probability map of the drowning person. In this paper, it is assumed that after a period of drifting, the probability map is static (at this time, the rescue UAVs arrive at the search and rescue area). After the UAVs arrive at the site, path planning and search are carried out according to the search probability map at the current time point until full coverage search of the area is achieved.

Table 4. Parameter settings of the case study.

Type of Parameter	Parameter Name	Value	Parameter Explanation
Parameters for drift trajectory prediction	$DWL$	0.125	Downwind regression coefficient
	$CWL$	0.175	Crosswind regression coefficient
	$T$	5	Drift time (5 h)
	$Dt$	1/60	Time step (60 s)
	$N\_simulations$	10,000	Number of Monte Carlo simulations
	$Bodymass$	70	Body mass (kg)
UAV parameters	$E$	10 (km)	Endurance of the UAVs
	$V$	14 m/s	Speed of the UAVs
	Rk	5 (km)	Maximum flight radius of UAV $k$
	qk	0.1	The ratio of charging time to flight time
	$K$	5	The number of UAVS
	$POD$	1	The possibility that a search unit can detect a SAR target
Parameters of the model	$C_p$	5	Penalty coefficient corresponding to the cumulative number of steps
	$d_0$	1	Initial weight of the attenuation coefficient
	$\lambda$	−0.1	Step attenuation factor

is the parameter table of the AP98 drift simulation model. The diffusion coefficient is calculated dynamically according to the variance of wind speed and current speed, as well as the simulation time, as shown below. At this time, the specific parameters of $POD$ are listed in Table 1.

5.1. Prediction of Maritime Drift Trajectory Probability

This paper uses the Monte Carlo simulation method to simulate the drift trajectory of a person who has fallen into the water and the probability of each area. Considering the actual situation, parameters such as the range and change probability of wind speed, wind direction, and ocean current, as well as physical parameters of the human body, are set. The downwind and crosswind drifts are calculated, and then the wind drift velocity and ocean current velocity components are obtained. The position is updated by considering the effects of various forces, and the search grid probability is updated. Finally, a probability distribution map of the drift area of the person who has fallen into the water is obtained.

In this paper, taking the water entry point of the human body model as the starting point, the 4 h drift trajectory of PIW is calculated by using the simulated ocean drift parameters, as shown in Figure 3. The 4 h drift trajectory and particle distribution diagram of PIWs are generated through simulation. In this study, it is assumed that when the rescue UAVs arrive in 4 h, the possible position distribution data afterwards is constant, and the search time of the rescue UAVs is ignored.

Therefore, this paper extracts a partial area within the overall area as the experimental probability distribution map of this paper. The specific distribution is shown in Figure 3 below. Each red dot indicates that there is a presumed person in the water at that point during the Monte Carlo simulation.

5.2. The Results of UAV Path Planning Solved by MILP Model

This paper generates supply points and the starting points of UAVs in an actual probability graph by inputting parameters such as the number of target nodes, the number of supply points, and the number of UAVs. The initial grid size is set to 8 × 8. Gurobi (version 11.0.3) is used to solve the mixed-integer linear programming problem. The goal is to minimize the total time and maximize the coverage probability. In path planning, the MILP model in this report can automatically determine the optimal number of UAVs, automatically allocate search areas according to the actual optimal number of UAVs, and implement the optimal path planning algorithm.

All the solution processes in this paper are completed on a 12th generation Intel Core i5-12500H processor (manufactured by Lenovo, Shenzhen, China). This processor has 12 physical cores and 16 logical processors. In the solution process, up to four threads are used. A model with 90,447 rows, 26,236 columns, and 272,017 non-zero elements is optimized. In the process of using Gurobi to solve MILP, the iterative process refers to a series of procedures for solving mixed-integer linear programming problems using the interior point method (barrier method). Initially, the algorithm generates an initial solution. In each iteration, the algorithm improves the quality of the solution so that the function value is closer to the optimal value. When the number of iterations reaches the upper limit or the change in the objective function value is less than a certain threshold, the algorithm stops iterating and outputs the current approximate optimal solution. The following Figure 4 is the solution iteration process of this paper. It can be clearly seen that as the number of iterations increases, the values of the Primal Objective, Residual, and the dual form all gradually decrease and stabilize, indicating that an approximate optimal solution is reached at this time.

In the barrier log output plotted in Figure 4, Gurobi reports five diagnostic quantities whose rigorous definitions are given below. $\mathit{Pr}i malOb{j}^{\left(t\right)}=c^T{x}^{\left(t\right)}$, where $x^{\left(t\right)}$ is the primal iterate at the iteration $t$. $DualOb{j}^{\left(t\right)}=b^T{y}^{\left(t\right)}$, where $y^{\left(t\right)}$ is the dual iterate. Primal Residual normalized quantity measures the maximum violation of primal feasibility. Dual Residual is the dual slack vector.

To facilitate unambiguous cross-referencing, all subsequent results in this section were generated under three clearly delineated benchmark configurations, as specified below:

Original-MILP: The proposed MILP model was solved using Gurobi’s default parameter set. The resulting UAV routes are rendered in red in Figure 5; and its quantitative metrics correspond to the data presented in Table 3.

PS (parallel-scanning): A classical geometric method that partitions the search domain into rectangular strips designed to be non-overlapping. However, its actual flight trajectories exhibit pronounced overlaps (shown in black in Figure 6)—a phenomenon indicating that the prescribed strip assignment fails to achieve optimality (shown in Figure 7).

Optimized-MILP: The same MILP formulation as Original-MILP, but solved with tuned solver settings—specifically, MIPFocus = 2 and Method = 3. These parameter values were selected based on the ablation study reported in

Table 5. Comparison of solution results of different emphases and methods.

MIPFocus	Method	Gap (%)	Time (s)	Objective Values
0	0	1.0486	22.27	−1.4136 × 104
0	1	0.9571	8.36	−1.4142 × 104
0	2	0.9986	10.27	−1.4145 × 104
0	3	0.9492	21.17	−1.4154 × 104
1	0	1.0168	16.30	−1.4151 × 104
1	1	0.9513	9.70	−1.4154 × 104
1	2	1.0701	10.92	−1.4151 × 104
1	3	0.9374	10.45	−1.4155 × 104
2	0	0.9992	12.88	−1.4152 × 104
2	1	0.8528	9.55	−1.4159 × 104
2	2	0.9998	13.56	−1.4152 × 104
2	3	0.7891	9.46	−1.4162 × 104
3	0	0.9760	16.23	−1.4152 × 104
3	1	0.9200	12.95	−1.4155 × 104
3	2	0.9970	16.52	−1.4152 × 104
3	3	0.8582	21.93	−1.4159 × 104

(resulting in a gap of 0.7891%).

In this case, five Depots and six UAVs are initialized. The model in this paper can automatically dispatch the number of UAVs to meet the requirement of full coverage. The obtained optimized path results are shown in Figure 5. It can be seen that after the positions of the initial Depots are selected, the UAVs start searching from this point. The only four UAVs do not encounter desperate situations such as collisions or repeated searches. In the algorithm designed in this paper, all UAVs will return to the Depot after completing the search task, which is in line with common sense. In addition, it can be seen that if the UAV encounters insufficient battery power during the search process, it will automatically go to the Depot for replenishment, so as to have sufficient battery power to continue to complete the corresponding task.

At present, the commonly used search path planning method is parallel search, that is, searching in one direction until reaching the boundary of the search area. This method will have overlapping paths in this case, as shown by the black path in Figure 6. The parallel search path shown in Figure 6 is a baseline method for comparison. While it is possible to improve the parallel method by early turning, the proposed MILP model focuses on global optimization under realistic constraints (e.g., endurance, depot access, non-overlapping paths), which may not always align with greedy local improvements. Overlapping paths are a great waste of resources and time in real search and rescue. Therefore, the advancement of the mixed-integer linear programming algorithm in this paper can be compared.

By comparing the cumulative search and rescue success probability $CPOS$ of the two methods, it can be known that in the early stage, when the number of steps corresponding to the four UAVs is less than 12, there is no obvious difference in cumulative probability. After that, the cumulative search and rescue success probability corresponding to the MILP method in this paper is significantly higher than that of the traditional parallel search method. This is because UAVs will fall into an overlapping situation in the parallel search method, resulting in reduced efficiency and being unrealistic in practice. In addition, Figure 7 also shows the average search and rescue success probability Average-$CPOS$. Although the parallel search method is too aggressive in the early stage, resulting in a relatively high Average-$CPOS$, it is not as good as the MILP method in the later stage. This is because it cannot consider whether it will fall into overlapping in the future. The specific average $CPOS$ is calculated as follows:

$$\overline{CPOS}={\displaystyle \frac{CPO{S}_N}{N}}={\displaystyle \frac{{\sum }_{n=1}^{N}PO{S}_n}{N}}$$

(42)

Multi-UAVs collaborative search and rescue is adopted for all the above cases. For the MILP model in this paper, a single UAV is also used to verify the applicability of our scheme. Still under the influence of five supply points $D$, as shown in Figure 8, that after starting from the initial point, the UAV gradually passes by four supply points to replenish endurance and completes the full-coverage search and rescue task.

5.3. Comparison of Different Solution Methods and the Best Solution Set

When solving the MILP model, Gurobi has various process solution methods, such as MIPFocus and Method. They, respectively, represent the strategy of the mixed integer programming solver and the method for optimizing convex optimization problems. In this paper, we take this case as an experiment to compare the advantages and disadvantages and solution time of different methods. Here, the gap refers to the difference between the lower bound of the current optimal solution and the objective value of the best known solution divided by the lower bound of the current optimal solution. A smaller value represents a better solution set effect.

In the solution of the MILP model, MIPFocus from 0 to 3, respectively, represents that the solver strikes a balance between feasible solutions and optimal solutions, gives priority to finding feasible solutions, gives priority to finding optimal solutions, and gives priority to optimizing boundaries as the goal. The corresponding Method from 0 to 3, respectively, indicates that the solution uses the primal simplex method, the dual simplex method, the Barrier algorithm, and the parallel method.

Three benchmark configurations were implemented for rigorous comparison: (i) the classical parallel-scanning (PS) model, which minimizes total flight time under implicit endurance limits by assigning non-overlapping rectangular strips; (ii) the default MILP that optimizes the composite objective “total flight time + penalty for uncovered high-probability cells” subject to node-degree, endurance, flow-capacity, and reachable-set constraints detailed in Section 4, solved with Gurobi default parameters; and (iii) the optimized MILP that retains the identical objective and constraints of but adopts tuned Gurobi settings—Method = 3 (parallel barrier), MIPFocus = 2, Presolve = 2, NodeLimit = 500, TimeLimit = 100 s, and MIPGap = 0.01. Optimal values for PS were obtained analytically from closed-form strip geometry; those for default MILP and optimized MILP were extracted from the best incumbent returned by Gurobi 10.0 on an Intel i5-12500H (12 cores) once the relative MIP gap reached ≤1%.

Analyzing the results in Table 5, it can be known that when the Method is defined, focusing on the balance between feasible solutions and optimal solutions generally leads to a weaker gap than other goals. Among them, the form focusing on the optimal solution brings a lower gap, indicating that this solution goal is optimal. When MIPFocus is defined, it can be clearly found that the parallel solution method is significantly better than the other three methods. And when the optimal solution is the goal and the parallel solution method is used together, the current best gap is achieved. At the same time, the solution time is relatively better.

In order to verify the effectiveness of the optimal solution method, in addition to comparing the difference from the optimal solution, it is also necessary to test the reality of the solution in practice. As can be seen from the comparison chart of cumulative search and rescue success probability average search and rescue efficiency in Figure 9, the average search and rescue efficiency corresponding to the optimal parameter solution is significantly higher than the original solution in the first few steps, indicating that the reduction in its gap value is truly effective in reality.

To recap, the empirical evidence presented in Section 5 is anchored to three mutually exclusive configurations: the overlapping trajectories of the classical parallel-scanning method (Figure 6), the red routes delivered by the default-parameter MILP (Figure 5), whose performance curves appear as “Original” in Figure 9 and whose metrics populate Table 3, and the overlap-free, parameter-tuned Optimized-MILP whose gains are documented in the last rows of Table 5 and visualized in Figure 9 as “Optimized-MILP”.

6. Conclusions

At the outset, we assumed (i) that persons-in-water do not swim voluntarily, (ii) that UAV endurance and speed remain constant, and (iii) that wind and current conditions are static during the operation. The results support these assumptions for the investigated 4 h scenarios: observed trajectories matched the no-swimming drift model, endurance variations stayed within the model tolerance, and environmental data showed negligible change.

This paper comprehensively considers the realistic situation of personnel falling into the sea during the search and rescue process, including the simulation of personnel drifting trajectories. Based on the AP98 model, the Monte Carlo simulation method is used to simulate the regional probability map. The method of mixed-integer linear programming is adopted to completely solve the entire path planning problem. Realistic constraints such as cumulative search and rescue success probability, search and rescue time, search and rescue radius tendency, endurance limitation, and non-overlapping paths are considered. The solution of this method remains stable and unique under the same conditions in all aspects and has high safety and practicability. Compared with the traditional parallel search method, this method performs excellently in terms of average search efficiency and whether it falls into path overlap, and can automatically allocate different search areas for different UAVs. Different solution tendencies and methods are selected for comparative experiments. The maximum gap difference in the obtained optimal solution is up to 0.2595%, and the solution time is reduced by a maximum of 12.81 s. The average search efficiency is significantly better than the default solution in the first five steps.

However, the current MILP model omits two potentially significant factors. First, it treats the UAVs’ starting positions as fixed, whereas in real operations the departure point—such as a mothership—may itself be in motion. Incorporating dynamic starting-point constraints (e.g., a moving platform whose position evolves while the UAV is airborne) would enhance the model’s generality. Second, the model neglects voluntary swimming by persons in the water. Future work should integrate physiological data, for instance, statistical distributions of individual swimming speeds, to reduce trajectory-prediction error and further improve rescue effectiveness.