An Improved Iterated Greedy Algorithm for Solving Collaborative Helicopter Rescue Routing Problem with Time Window and Limited Survival Time

Abstract

Research on helicopter dispatching has received considerable attention, particularly in relation to post-disaster rescue operations. The survival chances of individuals trapped in emergency situations decrease as time passes, making timely helicopter dispatch crucial for successful rescue missions. Therefore, this study investigates a collaborative helicopter rescue routing problem with time window and limited survival time constraints, solving it using an improved iterative greedy (IIG) algorithm. In the proposed algorithm, a heuristic initialization strategy is designed to generate an efficient and feasible initial solution. Then, a feasible-first destruction-construction strategy is applied to enhance the algorithm’s exploration ability. Next, a problem-specific local search strategy is developed to improve the algorithm’s local search effectiveness. In addition, the simulated annealing (SA) method is integrated as an acceptance criterion to avoid the algorithm from getting trapped in local optima. Finally, to evaluate the efficacy of the proposed IIG, 56 instances were generated based on Solomon instances and used for simulation tests. A comparative analysis was conducted against six efficient algorithms from the existing studies. The experimental results demonstrate that the proposed algorithm performs well in solving the post-disaster rescue helicopter routing problem.

1. Introduction

In the modern world, natural disasters occur frequently, posing significant challenges for post-disaster rescue efforts that have garnered substantial societal attention [1,2]. Recently, numerous scholars have investigated the post-disaster rescue problem across various contexts and scenarios [3,4,5]. Disasters often cause severe road damage, impeding ground transportation, and delaying rescue operations in affected areas. In contrast, helicopters can rapidly transport rescuers and materials to disaster zones [6,7,8], making helicopter scheduling a critical yet complex problem in post-disaster rescues.

Rescue tasks in post-disaster situations typically involve transporting rescue materials and evacuating survivors. A major challenge is that the survival chances of victims decrease over time. Thus, efficient helicopter scheduling for rescue missions has become an urgent issue [9]. The helicopter scheduling problem is a variant of the vehicle routing problem with time windows (VRPTW). However, existing research on rescue routing rarely incorporates the dual constraints of time windows and survivors’ diminishing life strength. Therefore, we extended the VRPTW problem to consider the limited survival time of victims and established a new problem model. Currently, the iterative greedy (IG) algorithm has gained popularity due to its simplicity, strong exploitation capability, and fast convergence, attracting significant attention from researchers in various fields [10,11,12,13]. In addition, it has shown excellent performance in solving many other complex problems, which has laid a solid foundation for addressing helicopter rescue routing challenges. Thus, this paper proposes an IIG algorithm to solve the collaborative helicopter rescue routing problem. The main contributions of the paper are as follows:

(1) A heuristic initialization strategy was designed to generate an efficient and feasible initial solution;

(2) A feasible-first destruction-construction strategy was applied to enhance the exploration ability of IIG;

(3) Three problem-specific local search operators were developed to improve the algorithm’s local search performance;

(4) An improved acceptance criterion for the evolutionary solution was investigated to prevent the algorithm from getting trapped in local optima.

The remainder of this paper is structured as follows: Section 2 presents a literature review of VRPTW; Section 3 provides a concise overview of the problem description and formulation; Section 4 outlines the various components of the proposed algorithm; Section 5 discusses the simulation environment and the parameter settings, along with a comparative analysis with several effective algorithms. Finally, we summarize the conclusions of this study and present the future work in Section 6.

2. Literature Review

The VRPTW is a classical optimization problem. Solomon [14] first proposed a push-forward insertion heuristic algorithm to solve the VRPTW in 1987. Since then, numerous innovative algorithms have been developed to address the VRPTW. For example, Potvin et al. [15] proposed a parallel insertion heuristic method to construct parallel paths in VRP, while Ioannou et al. [16] introduced a greedy insertion heuristic based on a look-ahead heuristic to enhance traditional vehicle routing methods. However, these algorithms do not guarantee population diversity. Li et al. [17] addressed this issue by combining the three aforementioned heuristic methods and developing an improved push-forward insertion heuristic (PFIH) strategy to maintain population diversity in solving the VRPTW in prefabricated systems. In addition, Bray et al. [18] conducted a survey on metaheuristics for the VRPTW and analyzed the effectiveness of various approaches. Cai et al. [19] developed a combined evolutionary multitask algorithm for solving multi-objective VRPTW, and Zhang et al. [20] proposed a constraint-aware policy optimization for VRPTW. Liu et al. [21] employed an adaptive large neighborhood search (ALNS) algorithm for the VRPTW, while Saksuriya et al. [22] introduced a hybrid novice local search for VRPTW. Xu et al. [23] presented a three-phase heuristic approach for multi-trip VRPTW and resource synchronization on heterogeneous facilities.

As research on VRPTW has deepened, its applications have broadened significantly. Ahmed et al. [24] developed a hybrid algorithm based on a modified rank-based ant system for heterogeneous fixed fleet open VRPTW. Vida et al. [25] presented an efficient hybrid genetic search with advanced diversity control for a large range of time-constrained vehicle routing problems, introducing several features to manage the temporal dimension. Bezerra et al. [26] improved a variable neighborhood search-based algorithm with adaptive local search for VRPTW and multi-depot problems, with an aim to reduce vehicle fleet size. Liu et al. [27] proposed a hybrid brainstorm optimization algorithm for the VRPTW to minimize the number of unserved customers. Wen et al. [28] presented an improved ALNS algorithm to effectively solve large-scale cases of green VRPTW with multiple depots. Matijević [29] investigated a general variable neighborhood search metaheuristic to minimize the total distance traveled. Gao et al. [30] designed a hybrid genetic algorithm with a large neighborhood search to minimize transport distance and energy consumption. Zhang et al. [31] developed an improved neighborhood search algorithm to reduce total costs, including vehicle dispatch and transportation expenses. Although there are numerous examples of the VRPTW being applied in various fields, the practical application of the helicopter dispatching problem after a disaster presents unique challenges. As research on VRPTW progresses, neural networks have emerged as a new direction for solving this problem. Ma et al. [32] proposed a pointer neural network to address VRPTW, utilizing a global attention mechanism to improve convergence speed and results. Lee et al. [33] trained an artificial neural network to predict costs, i.e., distance and time, in VRPTW.

In summary, research on VRPTW remains as a key topic of discussion, with significant implications for various fields. However, few studies have explored the application of VRPTW to post-disaster rescue helicopter routing. In addition, current post-disaster rescue strategies predominantly focus on vehicular methods, often overlooking the critical role of helicopters in rescue operations. Therefore, it is necessary to design unique algorithm strategies tailored to the characteristics of helicopter rescue missions. Although substantial progress has been made in VRP research over the past few decades, it remains an open problem. In particular, the IG algorithm, holds significant potential for further development in solving VRP. To better approximate real-world disaster rescue scenarios, this study investigates a model for helicopter scheduling in disaster rescue. The model considers multiple factors, including the life strength of the survivor, time window constraints, helicopter loading capacity, transportation of wounded individuals and materials, and travel distance. Additionally, an IIG algorithm is proposed to address the helicopter scheduling problem in post-disaster rescue tasks.

3. Problem Formulation

3.1. Problem Description

In this study, the collaborative helicopter rescue routing problem is formulated as an expansion of the VRPTW, incorporating limited survival time (referred to as the R-VRPTWLST). All rescue locations have time window constraints and material requirements, with each survivor having limited life strength. In R-VRPTWLST, two types of helicopters, i.e., transport and medical helicopters, are used to complete rescue tasks. Transport helicopters primarily deliver rescue materials to each rescue location. Medical helicopters are specifically designed to transport survivors and are critical in rescue operations. To make a great effort to rescue survivors, we have stipulated that when the transport helicopter’s load is less than half of its maximum capacity, it can simultaneously perform medical helicopter rescue tasks.

The R-VRPTWLST problem is based on the following assumptions:

(1) All helicopters depart from and return to a single rescue center;

(2) Each type of helicopter visits each rescue site only once;

(3) The total demand for supplies of the rescue sites along the helicopter’s route should not exceed the maximum carrying capacity of the helicopter;

(4) The number of survivors to be rescued from the rescue sites along the helicopter’s route should not exceed the helicopter’s maximum carrying capacity;

(5) Helicopters need to meet the time window requirements at each rescue site;

(6) The return time of each helicopter to the rescue center should not exceed its maximum travel time;

(7) The life strength of each survivor transported by a helicopter needs to exceed the corresponding threshold. Once a survivor is picked up by the medical helicopter, he receives initial treatment and his life strength remains stable so he can be safely transported to the rescue center.

3.2. Problem Formulation

The R-VRPTWL is modeled as a directed complete graph G = (N, A), where N = {0} ∪ R represents the set of nodes and A denotes the set of arcs. Here, node 0 represents the rescue center, and R is the set of rescue sites. The set A = {(i, j) | i, j ∈ N, i ≠ j} contains the arcs connecting the nodes in N, with d_ij indicating the distance between nodes i and j. Each rescue site i ∈ N has a time window [a_i, b_i], within which the start service time u_ik of helicopter k at the rescue site i must be fulfilled. The t_i denotes the service time at the rescue site i, dm_i represents the demanding material to be delivered at the site i, dp_i is the number of survivors waiting to be rescued at the site i, and lt_i denotes the minimum life strength for survivors at the site i. The set H = TH ∪ MH represents all helicopters, where TH = {1, …, m₁} denotes the set of transport helicopters, MH= {m₁ + 1, …, m₁ + m₂} denotes the set of medical helicopters. The transport helicopter has a maximum materiel transport capacity cm_k and a minimum life strength threshold requirement tt_k. The medical helicopter has a maximum carrying capacity cp_k and a minimum life strength threshold requirement mt_k.

Table 1. Notation used in the R-VRPTWLST model.

Notations	Description
0	Index of the rescue center
N	Set of all nodes, including the rescue sites and the rescue center
R	Set of all rescue locations
H	Set of all helicopters
TH	Set of transport helicopters
MH	Set of medical helicopters
ai	Earliest possible rescue time of location i, i ∈ N
bi	Latest possible rescue time of location i, i ∈ N
dij	Distance between locations i and j, i,j ∈ N, i ≠ j
dmi	Demand for the supplies of location i, i ∈ N
dpi	The number of survivors waiting to be rescued at the rescue site i, i ∈ N
lti	Life strength for survivors at the rescue site i, i ∈ N
cmk	Maximum capacity for the material of transport helicopter k, k ∈ TH
ttk	Minimum life strength threshold of transport helicopter k, k ∈ TH
cpk	Maximum capacity for casualty care of medical helicopter k, k ∈ MH
mtk	Minimum life strength threshold of medical helicopter k, k ∈ MH
ti	Service duration for rescue site i, i ∈ N
uik	Start time of service for rescue site i, i ∈ N, k ∈ H

summarizes the notation used in the R-VRPTWLST model.

Decision variable:

x_ijk: if a helicopter k visits and serves the node i and travels directly to the node j, x_ijk = 1; otherwise, x_ijk = 0;

Objective:

$$Min{\displaystyle \sum _{i\in N}{\displaystyle \sum _{j\in N}{\displaystyle \sum _{k\in H}{d}_{ij}{x}_{ijk}}}}$$

(1)

Constraints:

$$\begin{array}{cc}{\displaystyle \sum _{k\in TH}{\displaystyle \sum _{j\in R}{x}_{ijk}=1}}& \forall i\in N\end{array}$$

(2)

$$\begin{array}{cc}{\displaystyle \sum _{k\in MH}{\displaystyle \sum _{j\in R}{x}_{ijk}\le 1}}& \forall i\in N\end{array}$$

(3)

$$\begin{array}{cc}{\displaystyle \sum _{j\in R}{x}_{0jk}=1}& \forall k\in H\end{array}$$

(4)

$$\begin{array}{cc}{\displaystyle \sum _{i\in R}{x}_{i0k}=1}& \forall k\in H\end{array}$$

(5)

$$\begin{array}{cc}{\displaystyle \sum _{i\in N}{\displaystyle \sum _{j\in N}d{m}_i{x}_{ijk}\le c{m}_k}}& \forall k\in TH\end{array}$$

(6)

$$\begin{array}{cc}{\displaystyle \sum _{i\in N}{\displaystyle \sum _{j\in N}d{p}_i{x}_{ijk}\le c{p}_k}}& \forall k\in TH\end{array}$$

(7)

$$\begin{array}{cc}{a}_i\le u_{ik}\le b_i& \forall i\in N,\forall k\in H\end{array}$$

(8)

$$l{t}_i{e}^{-0.037u_{ik}}\ge t{t}_k,\forall i\in N,\forall k\in TH$$

(9)

$$l{t}_i{e}^{-0.037u_{ik}}\ge m{t}_k,\forall i\in N,\forall k\in MH$$

(10)

$$\begin{array}{cc}{x}_{ijk}\in \left\{0,1\right\}& \forall i,j\in N,\forall k\in \end{array}H$$

(11)

Equations (1)–(11) outlines the constraints and objectives of the model as follows: Equation (1) aims to minimize the total driving distance of all helicopters. Equation (2) guarantees that each rescue site is serviced exactly once by transport helicopters. Equation (3) specifies that each rescue site is serviced exactly once by medical helicopters. Equations (4) and (5) ensure that all helicopters depart from and ultimately return to the rescue center. Equations (6) and (7) enforce the constraint that helicopters do not exceed their maximum carrying capacity when transporting supplies and rescuing injured individuals along their route. Equation (8) requires that the service start time of the transport helicopter at each rescue site falls within the specified time window. Equations (9) and (10) guarantee that the life strength of survivors remains above the threshold. Finally, Equation (11) restricts the decision variables.

4. Improved Iterative Greedy Algorithm

4.1. Framework of the IIG

In this section, the IIG algorithm is designed to solve the R-VRPTWLST. The framework of the proposed IIG is outlined in Algorithm 1. The main steps of the IIG are as follows. First, a feasible and efficient initial solution is generated using the heuristic initialization strategy, as described in line 1 of Algorithm 1. In each iteration, destruction and construction operators are employed to enhance the exploration ability of the IIG, corresponding to lines 3 and 4 of Algorithm 1. Next, the problem-specific local search strategy is applied to improve local search performance, as shown in line 5 of Algorithm 1. Finally, the SA method is embedded as the acceptance criterion to prevent the IIG from getting trapped in local optima, as indicated in lines 6 to 12 of Algorithm 1.

Algorithm 1: The framework of the proposed IIG algorithm
Input: the information on rescue locations
Output: the best solution
1	Generate an initial solution X by using the initialization strategy
2	while the termination condition is not satisfied do
3	Xr = Destruction (X);
4	X’ = Construction (Xr, Xd);
5	X* = Local search (X’);
6	for the optimal solution found up to now do
7	Y = Acceptance criterion (X, X*);
8	Temperature = T0*sum (uij)/n*m*10;
9	if  X’< X*  then
10	X* = X’;
11	end
12	end
13	end
14	return the best solution X*

4.2. Solution Representation

In this study, each solution is encoded as a two-dimensional array. The first dimension indicates the helicopter number, while the second dimension lists the rescue locations assigned to each helicopter. Figure 1 presents this solution representation, where the sequence {0, 2, 7, 9, 5, 0} shows a helicopter starting from the rescue center, visiting rescue locations 2, 7, 9, 5, and returning to the rescue center. The sequences {0, 6, 4, 1, 3, 0} and {0, 10, 8, 0} represent the routes for helicopters H2 and H3, respectively. It is crucial to note that the selection of each rescue site in these routes adheres to the constraints specified by the R-VRPTWLST model.

4.3. Initialization Strategy

The quality of the initial solution significantly influences the performance of the IIG. Considering the problem characteristics of R-VRPTWLST and the different life strengths of survivors at different rescue sites, we proposed a heuristic initialization strategy. First, rescue locations are ordered in ascending sequence based on survivors’ life strength and stored in the set S_t. Then, each rescue site from set S_t is iteratively inserted into the route at the position that minimizes the overall distance. Algorithm 2 outlines the detailed steps of this heuristic initialization strategy.

Algorithm 2: Heuristic initialization strategy
Input: instance information
Output: an initial solution
1	Arrange all rescue sites in ascending order based on survivors’ life strength and place them into set St;
2	for each site i within the set St do
3	for each helicopter k do
4	for each position u do
5	if rescue site i can be added to the current position u then
6	Store the position u into set Su;
7	end
8	end
9	end
10	Choose the optimal position from the set Su and add site i to the position u;
11	if the insert operation of the site i is failed then
12	Add a new helicopter and allocate the site i to this helicopter;
13	end
14	end
15	Store the generated solution

4.4. Feasible-First Destruction and Construction Strategy

The feasible-first destruction-construction strategy aims to broaden the search scope of the algorithm and enhance its exploitation ability. Algorithm 3 outlines the steps of this strategy. First, all rescue sites are allocated to set S_n. Next, a rescue site is randomly selected from S_n and moved to set S_r. Then, the distance between each rescue site in S_n and the selected site is calculated. The set S_n is sorted based on these distances, and (n×d−1) rescue sites from S_n are transferred to S_r, where d represents the destruction length and n is the total number of rescue sites. Subsequently, the set S_r is removed from the current solution. Finally, each site in S_r is inserted into the position in the current solution that minimizes the distance.

Algorithm 3: Feasible-first destruction-construction strategy
Input: a solution
Output: a new solution
1	Add each rescue site to set Sn;
2	Select one rescue site i randomly from Sn and add this rescue site into set Sr;
3	Delete the chosen rescue site i from Sn;
4	for each rescue site j in Sn do
5	Calculate the distance for each arc a(i, j);
6	end
7	Sort the set Sn in ascending order based on all arcs a(i, j);
8	Move (n×d−1) rescue sites ahead of Sn to Sr;
9	Delete set Sr from the current solution;
10	for each rescue site i in the set Sr do
11	for each helicopter k do
12	for each position u do
13	if site i can be added to the current position u then
14	Store the position u into set Su;
15	end
16	end
17	end
18	Select the optimal position from the set Su and add site i to the position u;
19	if insert the site i failed then
20	Add a new helicopter and allocate the site i to this helicopter;
21	end
22	end

4.5. Problem-Specific Local Search Strategy

To improve the exploitation ability of the IIG algorithm, a problem-specific local search strategy has been developed. This strategy employs three types of search operators, i.e., the swap operator, insertion operator, and route reconstruction operator. Algorithm 4 provides the pseudocode for this strategy. To balance problem characteristics with neighborhood search capabilities, one of the three operators is randomly selected for helicopter scheduling. Figure 2 provides examples of these local search operators based on problem characteristics. The specific steps for each operator are as follows:

(1) Swap operator: First, randomly select a helicopter k from the set H of all helicopters, with each helicopter having an equal probability of being chosen. Next, within helicopter k, randomly select a rescue site i and swap it with the remainder rescue sites orderly. The solution corresponding to the best fitness value during the swap process is saved. This process is repeated m times, where m is the number of helicopters used in the current solution.

(2) Insertion operator: First, randomly select a helicopter k. Next, identify the rescue site i with the largest anteroposterior distance within the helicopter k. Insert rescue site i into the current route at the position that minimizes the distance. This process is repeated m times.

(3) Route reconstruction operator: First, randomly select a helicopter k and transfer the associated rescue sites to the set S_r. Then, remove S_r from the current solution. Finally, reinsert each site in S_r into the current route position that minimizes the distance. This process is repeated m times.

Algorithm 4: The local search strategy
Input: a solution
Output: an improved solution
1	r = rand ()%2, cnt = 0;
2	switch r do
3	case 0:
4	while  cnt  <  m  do
5	Randomly select a helicopter k;
6	Randomly select a rescue site i in the helicopter k;
7	for each of the other rescue site j in the helicopter k do
8	Swap rescue site i and rescue site j;
9	end
10	Select the best solution;
11	end
12	case 1:
13	while  cnt < m  do
14	Selected the rescue site i with the largest anteroposterior distance in the helicopter k;
15	Remove the rescue site i from the current solution;
16	for each helicopter k in the current solution do
17	for each position u do
18	if site i can be added to the current position u then
19	Store the position u into set Su;
20	end
21	end
22	end
23	Choose the optimal position from the set Su and add site i to the position u;
24	end
25	case 2:
26	while  cnt  < m  do
27	Randomly select helicopter k and the rescue sites along its route are stored in the set Sr;
28	Delete the set Sr from the current solution;
29	for each rescue site i in the set Sr do
30	for each helicopter k in the current solution do
31	for each position u do
32	if site i can be added to the current position u then
33	Store the position u into set Su;
34	end
35	end
36	end
37	Choose the optimal position from the set Su and add site i to the position u;
38	if insert the site i failed then
39	Add one helicopter and allocate the site i to this helicopter;
40	end
41	end
42	end
43	end

4.6. SA-Based Acceptance Criterion

The SA-based acceptance criterion is applied to prevent the IIG from falling into local optima. According to this criterion, the new solution X_new is accepted with a probability of p, as given by Equation (12) as follows:

$$p=\min \left\{\exp \left({\displaystyle \frac{C_{\max }(X)-C_{\max }(X_{new})}{temperature}}\right)\right.,\left.1\right\}$$

(12)

where X represents the current route, C_max(X) and C_max(X_new) denote the total distances of the current and new routes, respectively. The temperature is calculated as follows:

$$temperature=T\times {\displaystyle \frac{{\displaystyle \sum _{i=R}{\displaystyle \sum _{k=H}{u}_{ik}}}}{10\times m\times n}}$$

(13)

where m represents the number of helicopters, and n denotes the number of rescue sites. T is a constant temperature value. Specifically, if C_max(X_new) ≤ C_max(X), the solution X_new replaces X. Otherwise, if C_max(X_new) > C_max(X) and a random number θ is generated such that θ ≤ p (θ ∈ [0, 1]), the solution X_new replaces X. This approach allows slightly inferior solutions to be accepted with a high probability. Consequently, the final solution achieved by the algorithm may not always be the optimal solution. Therefore, it is crucial to maintain a record of the best solution throughout the iterative process.

5. Experiment Results

5.1. Experimental Instances

To evaluate the effectiveness of the proposed IIG, 56 instances were generated based on the well-known Solomon benchmark instances for the VRPTW. These instances included three types, i.e., random instances, cluster instances, and semi-cluster instances and each instance contained 100 rescue sites. The expanded instances maintain the same number of customers, locations, service times, and time windows as the Solomon instances. Additionally, for each rescue site, a random number of survivors and their corresponding initial life strength were generated. All algorithms were implemented in MATLAB R2022a and run on an Intel Core i5-12490 PC with 16 GB of RAM. To ensure fair comparisons, a uniform maximum CPU time of 100 s was used as a stopping criterion for all of the algorithms. The final solutions, obtained after 30 independent runs for each comparison, were collected for performance analysis.

5.2. Parameters Setting

This study primarily considers two key parameters, i.e., the destruction length ‘d’ and the temperature parameter for acceptance criterion ‘T’.

Table 2. The levels of the two parameters.

Parameters	Levels
	1	2	3	4
d	0.1	0.2	0.3	0.4
T	0.2	0.3	0.4	0.5

presents the various values applied to these two parameters. The design of experiment (DOE) Taguchi method [34,35] was used to construct an orthogonal array L₁₆ and the proposed IIG was executed independently 30 times for each parameter combination. The response variable was collected as the average fitness value.

Table 3. The response variable of the four parameters.

d	T	Average Values
1	1	1862.75
1	2	1886.63
1	3	1896.69
1	4	1892.37
2	1	1704.23
2	2	1754.04
2	3	1712.81
2	4	1716.56
3	1	1799.72
3	2	1803.39
3	3	1776.82
3	4	1798.90
4	1	1867.13
4	2	1841.73
4	3	1819.77
4	4	1866.40

presents the experimental results, while Figure 3 illustrates the main effects plots for these parameters. The analysis indicated that the proposed IIG achieves optimal performance with parameter settings of d = 0.2 and T = 0.4.

5.3. Effectiveness of the Local Search Strategy

To evaluate the performance of the local search strategy, an IIG algorithm without the local search strategy (IIG_NL) was developed and compared to the IIG algorithm. The two compared algorithms were run independently 30 times on 56 instances, and the average fitness values for each instance are collected. The relative percentage increase (RPI) is used to measure the algorithm’s performance, which is calculated by Equation (14) as follows:

$$RPI={\displaystyle \frac{(f_c-f_b)}{f_b}}\times 100$$

(14)

where f_b is the best value obtained by all compared algorithms, and f_c denotes the best value achieved by a specified algorithm.

Table 4. Comparison results of the IIG and IIG_NL.

Instances	Better Values	Algorithms		RPIs
		IIG	IIG_NL	IIG	IIG_NL
rc101	1646.27	1646.27	1762.03	0.00	7.03
rc102	1723.15	1723.15	1802.15	0.00	4.58
rc103	1861.14	1861.14	1975.20	0.00	6.13
rc104	2014.80	2014.80	2159.82	0.00	7.20
rc105	1802.07	1802.07	1946.52	0.00	8.02
rc106	1741.29	1741.29	1872.15	0.00	7.52
rc107	1893.48	1893.48	2033.92	0.00	7.42
rc108	2005.98	2005.98	2215.52	0.00	10.45
rc109	1924.37	1924.37	1999.77	0.00	3.92
rc201	1971.68	1971.68	2058.68	0.00	4.41
rc202	1926.09	1926.09	2000.24	0.00	3.85
rc203	1929.75	1993.15	1929.75	3.29	0.00
rc204	2037.70	2037.70	2175.06	0.00	6.74
rc205	1996.43	1996.43	2102.14	0.00	5.29
rc206	1947.04	1947.04	2011.36	0.00	3.30
rc207	2016.15	2016.15	2112.56	0.00	4.78
rc208	2046.07	2046.07	2147.54	0.00	4.96
rr101	2866.62	3022.86	2866.62	5.17	0.00
rr102	2019.82	2033.62	2019.82	0.68	0.00
rr103	1754.20	1794.85	1754.20	2.32	0.00
rr104	1735.86	1735.86	1806.37	0.00	4.06
rr105	2065.52	2065.52	2111.00	0.00	2.20
rr106	1981.65	1981.65	2024.05	0.00	2.14
rr107	1855.79	1855.79	1898.20	0.00	2.29
rr108	1699.01	1699.01	1815.62	0.00	6.86
rr109	1938.55	1938.55	2035.11	0.00	4.98
rr110	1807.02	1807.02	1921.21	0.00	6.32
rr111	1764.78	1764.78	1883.87	0.00	6.75
rr112	1763.50	1763.50	1901.52	0.00	7.83
rr201	2106.75	2106.75	2286.69	0.00	8.54
rr202	1924.95	1924.95	2055.48	0.00	6.78
rr203	1811.40	1811.40	2002.64	0.00	10.56
rr204	1612.58	1612.58	1766.06	0.00	9.52
rr205	1859.54	1859.54	2008.47	0.00	8.01
rr206	1857.82	1857.82	2016.67	0.00	8.55
rr207	1701.91	1701.91	1825.76	0.00	7.28
rr208	1598.95	1598.95	1735.23	0.00	8.52
rr209	2026.87	2336.35	2026.87	15.27	0.00
rr210	1782.63	1782.63	1905.31	0.00	6.88
rr211	1735.94	1735.94	1859.30	0.00	7.11
rrc101	2472.05	2472.05	2642.48	0.00	6.89
rrc102	2309.12	2309.12	2387.87	0.00	3.41
rrc103	2165.93	2165.93	2312.67	0.00	6.77
rrc104	2045.94	2045.94	2186.47	0.00	6.87
rrc105	2454.70	2454.70	2590.14	0.00	5.52
rrc106	2333.60	2333.60	2498.16	0.00	7.05
rrc107	2159.70	2159.70	2329.19	0.00	7.85
rrc108	2172.61	2172.61	2379.28	0.00	9.51
rrc201	2741.56	2741.56	2931.13	0.00	6.91
rrc202	2560.35	2560.35	2803.59	0.00	9.50
rrc203	2396.92	2396.92	2646.57	0.00	10.42
rrc204	2245.18	2245.18	2502.46	0.00	11.46
rrc205	2836.28	3165.74	2836.28	11.62	0.00
rrc206	2614.92	2614.92	2820.25	0.00	7.85
rrc207	2380.57	2380.57	2502.09	0.00	5.10
rrc208	2248.49	2248.49	2426.44	0.00	7.91
Mean	2016.77	2018.88	2144.75	0.11	6.32

presents the comparison results, with the first two columns indicating the instances and the better values achieved by two compared algorithms. The subsequent four columns display the average fitness values and RPI values obtained by the two algorithms. It can be seen from Table 4 that: (1) The IIG achieved 50 optimal average fitness values out of the 56 instances, while the IIG_NL obtained only 3 optimal values and (2) the average RPI achieved by IIG is 0.11, which is lower than that of IIG_NL, demonstrating that the proposed IIG outperforms IIG_NL and the local search strategy enhances the algorithm’s exploitation capability. A multifactor analysis of variance (ANOVA) was conducted to evaluate the significance of the differences between the two algorithms. Figure 4 presents the ANOVA results, which indicate that the stability of the IIG is significantly better than that of IIG_NL, with a p-value of 1.4181 × 10⁻¹⁷, far less than 0.05, demonstrating a significant difference between the IIG and IIG_NL.

5.4. Effectiveness of the SA-Based Acceptance Criterion

The IIG algorithm without the SA-based acceptance criterion (IIG_NS) was developed to compare with the IIG and to verify the effectiveness of the SA-based acceptance criterion.

Table 5. Comparison results of the IIG and IIG_NS.

Instances	Better Values	Algorithms		RPIs
		IIG	IIG_NS	IIG	IIG_NS
rc101	1535.46	1646.27	1535.46	7.22	0.00
rc102	1688.98	1723.15	1688.98	2.02	0.00
rc103	1821.79	1861.14	1821.79	2.16	0.00
rc104	2014.80	2014.80	2048.45	0.00	1.67
rc105	1687.56	1802.07	1687.56	6.79	0.00
rc106	1741.29	1741.29	1880.81	0.00	8.01
rc107	1893.48	1893.48	1924.29	0.00	1.63
rc108	2002.70	2005.98	2002.70	0.16	0.00
rc109	1924.37	1924.37	1964.96	0.00	2.11
rc201	1930.64	1971.68	1930.64	2.13	0.00
rc202	1926.09	1926.09	1931.06	0.00	0.26
rc203	1960.17	1993.15	1960.17	1.68	0.00
rc204	2027.05	2037.70	2027.05	0.53	0.00
rc205	1993.13	1996.43	1993.13	0.17	0.00
rc206	1947.04	1947.04	1950.87	0.00	0.20
rc207	2016.15	2016.15	2031.26	0.00	0.75
rc208	2020.31	2046.07	2020.31	1.28	0.00
rr101	3005.39	3022.86	3005.39	5.81	0.00
rr102	2033.62	2033.62	2043.89	0.00	0.51
rr103	1794.85	1794.85	1798.33	0.00	0.19
rr104	1735.86	1735.86	1737.67	0.00	0.10
rr105	2056.11	2065.52	2056.11	0.46	0.00
rr106	1978.56	1981.65	1978.56	0.16	0.00
rr107	1855.79	1855.79	1865.20	0.00	0.51
rr108	1699.01	1699.01	1753.45	0.00	3.20
rr109	1938.55	1938.55	1985.92	0.00	2.44
rr110	1807.02	1807.02	1837.60	0.00	1.69
rr111	1764.78	1764.78	1799.71	0.00	1.98
rr112	1763.50	1763.50	1801.79	0.00	2.17
rr201	2106.75	2106.75	2208.56	0.00	4.83
rr202	1924.95	1924.95	2011.90	0.00	4.52
rr203	1811.40	1811.40	1884.65	0.00	4.04
rr204	1612.58	1612.58	1675.13	0.00	3.88
rr205	1859.54	1859.54	1935.95	0.00	4.11
rr206	1857.82	1857.82	1943.45	0.00	4.61
rr207	1701.91	1701.91	1797.41	0.00	5.61
rr208	1598.95	1598.95	1673.06	0.00	4.63
rr209	2025.39	2336.35	2225.39	4.99	0.00
rr210	1782.63	1782.63	1859.44	0.00	4.31
rr211	1735.94	1735.94	1788.57	0.00	3.03
rrc101	2472.05	2472.05	2544.58	0.00	2.93
rrc102	2309.12	2309.12	2352.05	0.00	1.86
rrc103	2165.93	2165.93	2239.96	0.00	3.42
rrc104	2045.94	2045.94	2093.10	0.00	2.31
rrc105	2454.70	2454.70	2540.50	0.00	3.50
rrc106	2333.60	2333.60	2413.30	0.00	3.42
rrc107	2159.70	2159.70	2235.99	0.00	3.53
rrc108	2172.61	2172.61	2252.53	0.00	3.68
rrc201	2741.56	2741.56	2838.36	0.00	3.53
rrc202	2560.35	2560.35	2665.13	0.00	4.09
rrc203	2396.92	2396.92	2515.60	0.00	4.95
rrc204	2245.18	2245.18	2365.22	0.00	5.35
rrc205	2886.66	3165.74	2886.66	8.82	0.00
rrc206	2614.92	2614.92	2746.17	0.00	5.02
rrc207	2380.57	2380.57	2450.92	0.00	2.96
rrc208	2248.49	2248.49	2347.66	0.00	4.41
Mean	2011.12	2018.88	2059.98	0.45	2.37

presents the average fitness and RPI values achieved by the two compared algorithms. From Table 5 we can see that (1) the IIG achieved 41 optimal average fitness values out of 56 instances, significantly outperforming the IIG_NS. (2) In the last row, the average RPI of the IIG is 0.45, less than that achieved by the IIG_NS. These results validate the efficacy of the SA-based acceptance criterion. Figure 5 presents the ANOVA results for the two algorithms, showing that the IIG exhibits better stability compared to the IIG_NS. The p-value is 1.5457 × 10⁻⁴, which is significantly less than 0.05, indicating a significant difference between the IIG and IIG_NS algorithms.

5.5. Effectiveness Evaluation against the Known Optimal Solutions

The main difference between our R-VRPTWLST and the VRPTW lies in the following two aspects: first, survivors at the site are given limited survival time constraints, and second, the transport helicopters can also be used to transfer survivors under given conditions, i.e., when the transport helicopter’s load is less than half of its maximum capacity and the survivors’ life strength could endure the subsequent journey until they are transported to the rescue center. In these cases, transport helicopters deliver rescue supplies along the way to pick up survivors. Although the travel route of transport helicopters may not be optimal, if several survivors are transported along the way, it reduces the number of sites that medical helicopters need to arrive to, indirectly reducing the travel route of medical helicopters. As we take the sum of the route lengths flown by two types of helicopters as the optimization objective, we present a joint rescue problem involving these two types of helicopters, i.e., R-VRPTWLST.

In order to further decouple the R-VRPTWLST problem and evaluate the effectiveness of the IIG, we employed the Solomon instances’ best solution to create instances of the R-VRPTWLST, where problem parameters are chosen so that the given route is still feasible. Since the optimal solutions of Solomon instances in the literature are values, rather than the specific route sequence, to simplify this creation process, we removed the constraint of limited survival time for survivors. Specifically, we set the survivor’s life strength value to infinity to ensure that limited survival time does not become a condition that affects helicopter path optimization. This rescue path planning problem is named R-VRPTWLST-ILS and the corresponding instance is marked as ‘r***_ILS’. Additional experiments were conducted to solve these instances using the proposed IIG algorithm.

Table 6. Comparison results of IIG on two types of R-VRPTWLST instances.

VRPTW		R-VRPTWLST-ILS				R-VRPTWLST
Solomon Instances	Optimal Values	Created Instances	THD Values	MHD Values	OD Values	Instances	THD Values	MHD Values	OD Values
c101	828.94	rc101_ILS	828.94	297.32	1126.26	rc101	1077.21	501.45	1578.66
c102	828.94	rc102_ILS	828.94	354.14	1183.07	rc102	1152.61	483.86	1636.47
c103	828.06	rc103_ILS	872.47	396.32	1268.78	rc103	1322.09	454.68	1776.77
c104	824.78	rc104_ILS	889.39	328.32	1217.71	rc104	1363.43	493.28	1856.71
c105	828.94	rc105_ILS	865.09	337.55	1202.64	rc105	1208.96	540.41	1719.88
c106	828.94	rc106_ILS	828.94	342.59	1171.53	rc106	1042.91	503.33	1546.24
c107	828.94	rc107_ILS	863.70	387.76	1251.46	rc107	1184.06	610.14	1794.20
c108	828.94	rc108_ILS	828.94	328.90	1157.84	rc108	1308.34	503.64	1811.98
c109	828.94	rc109_ILS	921.55	315.65	1237.20	rc109	1395.12	520.08	1915.19
c201	591.56	rc201_ILS	591.56	269.09	860.65	rc201	1139.93	650.83	1790.76
c202	591.56	rc202_ILS	602.86	251.05	853.91	rc202	1166.14	582.65	1748.79
c203	591.17	rc203_ILS	621.21	259.64	880.84	rc203	1266.77	563.16	1829.92
c204	590.60	rc204_ILS	636.55	281.21	917.75	rc204	1276.85	587.60	1864.45
c205	588.88	rc205_ILS	588.88	261.87	850.75	rc205	1330.27	545.47	1875.74
c206	588.49	rc206_ILS	589.34	319.94	909.28	rc206	1252.41	604.03	1856.45
c207	588.29	rc207_ILS	588.32	263.74	852.07	rc207	1247.46	643.71	1891.17
c208	588.32	rc208_ILS	589.48	265.53	855.01	rc208	1271.96	605.51	1877.47
r101	1645.79	rr101_ILS	1670.84	35.61	1706.45	rr101	1698.84	495.00	2193.85
r102	1486.12	rr102_ILS	1496.39	65.44	1561.83	rr102	1616.63	414.36	2030.98
r103	1292.68	rr103_ILS	1298.04	179.67	1477.71	rr103	1357.38	378.46	1735.84
r104	1007.24	rr104_ILS	1095.22	239.44	1334.65	rr104	1254.61	474.88	1729.50
r105	1377.11	rr105_ILS	1434.26	78.09	1512.36	rr105	1607.16	419.56	2026.72
r106	1251.98	rr106_ILS	1310.58	208.38	1518.96	rr106	1517.11	374.07	1891.18
r107	1104.66	rr107_ILS	1155.45	205.36	1360.81	rr107	1228.81	523.98	1752.79
r108	960.88	rr108_ILS	1027.31	215.01	1242.31	rr108	1082.84	563.21	1646.06
r109	1194.73	rr109_ILS	1217.34	164.01	1381.35	rr109	1422.37	475.53	1897.91
r110	1118.59	rr110_ILS	1164.19	119.73	1283.93	rr110	1329.92	420.04	1749.96
r111	1096.72	rr111_ILS	1151.99	196.36	1348.35	rr111	1277.32	432.00	1709.33
r112	982.14	rr112_ILS	1029.39	180.29	1209.67	rr112	1272.21	425.08	1697.29
r201	1252.37	rr201_ILS	1298.18	403.21	1701.39	rr201	1509.41	537.99	2047.40
r202	1191.70	rr202_ILS	1195.17	391.44	1586.61	rr202	1296.83	534.97	1831.80
r203	939.54	rr203_ILS	997.57	315.94	1313.51	rr203	1187.71	509.94	1697.65
r204	825.52	rr204_ILS	867.36	435.27	1302.64	rr204	1035.20	535.18	1570.38
r205	994.42	rr205_ILS	1107.30	354.30	1461.61	rr205	1369.13	452.50	1821.63
r206	906.14	rr206_ILS	986.62	354.26	1340.87	rr206	1272.80	547.01	1819.81
r207	893.33	rr207_ILS	918.39	421.88	1340.27	rr207	1171.63	520.17	1691.80
r208	726.75	rr208_ILS	774.68	136.96	911.65	rr208	1034.95	509.60	1544.55
r209	909.16	rr209_ILS	978.53	519.05	1497.59	rr209	1242.30	564.37	1806.67
r210	939.34	rr210_ILS	1022.58	348.64	1371.23	rr210	1294.41	472.94	1767.35
r211	892.71	rr211_ILS	915.17	321.97	1237.14	rr211	1149.17	545.19	1694.36
rc101	1696.94	rrc101_ILS	1720.54	200.93	1921.47	rrc101	1879.91	537.97	2417.88
rc102	1554.75	rrc102_ILS	1562.46	237.48	1799.94	rrc102	1716.54	550.34	2266.88
rc103	1261.67	rrc103_ILS	1332.53	320.72	1653.24	rrc103	1518.68	266.25	2084.93
rc104	1135.48	rrc104_ILS	1214.76	349.70	1564.46	rrc104	1460.12	564.49	2024.61
rc105	1629.44	rrc105_ILS	1665.63	226.02	1891.65	rrc105	1758.08	630.80	2388.88
rc106	1424.73	rrc106_ILS	1481.60	314.20	1795.80	rrc106	1591.53	574.08	2165.61
rc107	1230.48	rrc107_ILS	1260.39	333.83	1594.21	rrc107	1495.99	593.71	2089.70
rc108	1139.82	rrc108_ILS	1208.48	374.53	1583.01	rrc108	1516.41	617.01	2133.42
rc201	1406.91	rrc201_ILS	1451.38	451.13	1902.52	rrc201	1798.77	711.08	2509.85
rc202	1367.09	rrc202_ILS	1404.60	480.92	1885.53	rrc202	1682.92	724.20	2407.12
rc203	1049.62	rrc203_ILS	1131.16	452.35	1583.51	rrc203	1538.99	723.38	2262.37
rc204	798.41	rrc204_ILS	877.53	133.71	1011.25	rrc204	1576.20	648.57	2224.77
rc205	1297.19	rrc205_ILS	1390.44	430.08	1820.52	rrc205	1887.67	627.22	2515.19
rc206	1146.32	rrc206_ILS	1197.44	219.54	1416.97	rrc206	1811.11	693.55	2504.66
rc207	1061.14	rrc207_ILS	1082.12	181.30	1263.42	rrc207	1487.43	675.30	2162.73
rc208	828.14	rrc208_ILS	894.30	329.86	1224.16	rrc208	1556.21	604.91	2161.12

presents the comparison results of the IIG used to deal with two types of R-VRPTWLST instances. In Table 6, the first two columns provide traditional Solomon instances and their corresponding known optimal values. The third to sixth columns provide the corresponding results of using the IIG to solve the R-VRPTWLST-ILS instances, where THD value represents the traveled distance of the transport helicopters, THD value describes the traveled distance of the medical helicopters, and OD value gives the total distance of the two types of helicopters. The seventh to tenth columns present the results of the IIG solving the R-VRPTWLST problem. Since these R-VRPTWLST-ILS’s instances are created based on Solomon instances’ best solution, it is simply a VRPTW problem for the rescue route planning of transporting rescue supplies by transport helicopters. However, considering the coupling relationship between the two types of helicopters during rescue missions, we took the total travel distance of the two types of helicopters as the optimization objective. Comparing the second and fourth columns in Table 6, the IIG algorithm still achieved six optimal values on transport helicopters, while the differences between the values achieved on the other fifty instances and the optimal values were all less than 12%. These results demonstrate that, although the IIG was originally designed for the R-VRPTWLST, it could also perform effectively on R-VRPTWLST-ILS, and even the traditional VRPTW problems. Comparing columns 4–6 and 8–10 in Table 6, respectively, after adding the constraint of life strength, the travel distances of the R-VRPTWLST are greater than those of the R-VRPTWLST-ILS. These results further verify that the traditional VRPTW scheme is not suitable for rescue tasks that integrate life strength, that is, rescue route planning problems that put human life first. In the face of R-VRPTWLST, we need to improve algorithms based on problem characteristics and solve them.

5.6. Comparison with Two Efficient Heuristic Algorithms

Currently, the optimal solution for Solomon instances is constantly being refreshed by many researchers. Among them, Rochat et al. [36] proposed the diversification and intensification technique (denoted as DI), while Bent et al. [37] proposed the two-stage hybrid algorithm (denoted as TSH). These two heuristic algorithms obtained the highest number of optimal solutions across all 56 Solomon instances. As discussed above, our test instances were expanded from Solomon instances. Therefore, to evaluate the performance of our algorithm, we selected the DI and TSH algorithms to solve R-VRPTWLST and compared the results with those obtained by the IIG. For each expanded instance, the compared algorithm was run independently 30 times, and the optimal fitness value is recorded in

Table 7. Comparison results of the optimal fitness values obtained by the IIG, DI, and TSH.

Instances	Best-Known	Algorithms			RPIs
		IIG	DI	TSH	IIG	DI	TSH
rc101	1578.66	1578.66	1676.79	1732.33	0.00	0.06	0.10
rc102	1636.47	1636.47	1765.23	1961.10	0.00	0.08	0.20
rc103	1776.77	1776.77	1856.43	1813.18	0.00	0.04	0.02
rc104	1856.71	1856.71	1901.38	2030.77	0.00	0.02	0.09
rc105	1719.88	1719.88	1817.41	1832.61	0.00	0.06	0.07
rc106	1546.24	1546.24	1765.18	1977.82	0.00	0.14	0.28
rc107	1794.20	1794.20	1894.03	2047.19	0.00	0.06	0.14
rc108	1811.98	1811.98	2035.64	2064.53	0.00	0.12	0.14
rc109	1901.80	1915.19	1901.80	2184.02	0.01	0.00	0.15
rc201	1790.76	1790.76	1993.16	2037.44	0.00	0.11	0.14
rc202	1748.79	1748.79	1961.77	2093.95	0.00	0.12	0.20
rc203	1829.92	1829.92	1977.69	2038.81	0.00	0.08	0.11
rc204	1864.45	1864.45	2057.87	2039.72	0.00	0.10	0.09
rc205	1875.74	1875.74	2081.99	2098.25	0.00	0.11	0.12
rc206	1815.31	1856.45	1815.31	2026.22	0.02	0.00	0.12
rc207	1891.17	1891.17	2097.25	2084.41	0.00	0.11	0.10
rc208	1807.60	1877.47	2115.49	1807.60	0.04	0.17	0.00
rr101	2066.81	2193.85	2398.23	2066.81	0.06	0.16	0.00
rr102	1973.20	2030.98	1973.20	2070.14	0.03	0.00	0.05
rr103	1701.48	1735.84	1701.48	1889.48	0.02	0.00	0.11
rr104	1709.25	1729.50	1709.25	1791.51	0.01	0.00	0.05
rr105	1899.16	2026.72	2084.20	1899.16	0.07	0.10	0.00
rr106	1758.03	1891.18	1946.39	1758.03	0.08	0.11	0.00
rr107	1752.79	1752.79	1807.77	1826.68	0.00	0.03	0.04
rr108	1646.06	1646.06	1807.17	1763.29	0.00	0.10	0.07
rr109	1813.14	1897.91	1981.40	1813.14	0.05	0.09	0.00
rr110	1749.96	1749.96	1905.14	1956.02	0.00	0.09	0.12
rr111	1709.33	1709.33	1774.36	1916.03	0.00	0.04	0.12
rr112	1697.29	1697.29	1842.53	1882.52	0.00	0.09	0.11
rr201	2047.40	2047.40	2125.49	2177.37	0.00	0.04	0.06
rr202	1831.80	1831.80	2007.04	1996.64	0.00	0.10	0.09
rr203	1697.65	1697.65	1890.90	1927.94	0.00	0.11	0.14
rr204	1570.38	1570.38	1701.69	1929.08	0.00	0.08	0.23
rr205	1821.63	1821.63	1876.99	2178.59	0.00	0.03	0.20
rr206	1819.81	1819.81	1962.74	1959.20	0.00	0.08	0.08
rr207	1691.80	1691.80	1804.68	1961.28	0.00	0.07	0.16
rr208	1544.55	1544.55	1708.31	1923.56	0.00	0.11	0.25
rr209	1727.37	1806.67	1727.37	1915.69	0.05	0.00	0.11
rr210	1767.35	1767.35	1834.68	2073.05	0.00	0.04	0.17
rr211	1694.36	1694.36	1809.62	1760.74	0.00	0.07	0.04
rrc101	2417.88	2417.88	2513.64	2452.82	0.00	0.04	0.01
rrc102	2263.87	2266.88	2391.28	2263.87	0.00	0.06	0.00
rrc103	1906.91	2084.93	2268.32	1906.91	0.09	0.19	0.00
rrc104	2024.61	2024.61	2132.79	2176.89	0.00	0.05	0.08
rrc105	2253.53	2388.88	2531.48	2253.53	0.06	0.12	0.00
rrc106	2165.61	2165.61	2424.50	2399.76	0.00	0.12	0.11
rrc107	2089.70	2089.70	2165.82	2148.35	0.00	0.04	0.03
rrc108	1979.60	2133.42	1979.60	2328.98	0.08	0.00	0.18
rrc201	2509.85	2509.85	2607.89	2611.76	0.00	0.04	0.04
rrc202	2407.12	2407.12	2503.57	2863.43	0.00	0.04	0.19
rrc203	2262.37	2262.37	2440.52	2471.48	0.00	0.08	0.09
rrc204	2224.77	2224.77	2370.84	2394.29	0.00	0.07	0.08
rrc205	2380.76	2515.19	2380.76	2770.01	0.06	0.00	0.16
rrc206	2504.66	2504.66	2627.29	2556.88	0.00	0.05	0.02
rrc207	2162.73	2162.73	2461.72	2656.62	0.00	0.14	0.23
rrc208	2161.12	2161.12	2325.33	2466.77	0.00	0.08	0.14
Mean	1904.50	1929.38	2039.65	2089.79	0.01	0.07	0.10

. In Table 7, the first column represents the expanded instances and the second column provides the best fitness value achieved among the compared algorithms for each instance. The next six columns provide the optimal fitness values and corresponding RPI values of the compared algorithms. It can be seen that (1) the proposed IIG obtained 39 optimal fitness values out of 56 instances, significantly outperforming the other two algorithms. (2) The average RPI value of the IIG algorithm is 0.01, which is 0.14 and 0.1 times those of the DI and TSH algorithms, respectively. This demonstrates that the proposed IIG algorithm is effective in solving the R-VRPTWLST. Figure 6 presents an ANOVA comparison of the three algorithms, showing a p-value of 6.529 × 10⁻¹⁶, which is much lower than 0.05, indicating significant differences among the three algorithms.

5.7. Comparisons with Several Efficient Algorithms

To further evaluate the performance of the IIG in solving the collaborative helicopter rescue routing problem, four efficient algorithms were selected for comparison, i.e., the improved artificial bee colony (IABC) algorithm [17], the ALNS algorithm [38], the IG algorithm based on a learning-based variable neighborhood search (VNIG) [39], and the self-adaptive iterated greedy (SAIG) algorithm [40]. All four algorithms were implemented and run on the same PC under the same stopping condition. For each instance, the algorithms were run independently 30 times for comparison and the average fitness values were collected.

Table 8. Comparison results of the average fitness values obtained by the compared algorithms.

Instances	Best-Known	Algorithms					RPIs
		IIG	IABC	ALNS	VNIG	SAIG	IIG	IABC	ALNS	VNIG	SAIG
rc101	1646.27	1646.27	1797.77	1725.45	1752.38	1906.85	0.00	9.20	4.81	6.45	15.83
rc102	1723.15	1723.15	1774.46	1790.25	1876.03	1937.90	0.00	2.98	3.89	8.87	12.46
rc103	1861.14	1861.14	1884.37	1890.11	1988.72	1965.67	0.00	1.25	1.56	6.85	5.62
rc104	1917.06	2014.80	1921.70	1985.40	1917.06	2264.61	5.10	0.24	3.56	0.00	18.13
rc105	1802.07	1802.07	1888.92	1908.45	2113.25	2039.27	0.00	4.82	5.90	17.27	13.16
rc106	1741.29	1741.29	1796.88	1780.32	2023.76	1949.68	0.00	3.19	2.24	16.22	11.97
rc107	1893.48	1893.48	1984.87	1934.17	2060.32	2298.90	0.00	4.83	2.15	8.81	21.41
rc108	1932.38	2005.98	1977.62	1932.38	2052.50	2409.94	3.81	2.34	0.00	6.22	24.71
rc109	1875.56	1924.37	1941.96	1878.39	1875.56	2273.27	2.60	3.54	0.15	0.00	21.20
rc201	1971.68	1971.68	1990.10	2148.42	2283.46	2102.45	0.00	0.93	8.96	15.81	6.63
rc202	1926.09	1926.09	1980.53	1971.66	1983.05	2112.97	0.00	2.83	2.37	2.96	9.70
rc203	1893.91	1993.15	1893.91	1978.87	2117.33	2296.98	5.24	0.00	4.49	11.80	21.28
rc204	2037.70	2037.70	2097.60	2060.78	2212.94	2324.72	0.00	2.94	1.13	8.60	14.09
rc205	1868.28	1996.43	1868.28	2020.63	1980.78	2634.99	6.85	0.00	8.15	6.02	41.04
rc206	1947.04	1947.04	1977.86	1984.33	2062.08	2649.70	0.00	1.58	1.92	5.91	36.09
rc207	1938.67	2016.15	1938.67	2055.05	2059.97	2600.96	4.00	0.00	6.00	6.26	34.16
rc208	1997.49	2046.07	2000.75	1997.49	2039.01	2696.20	2.43	0.16	0.00	2.08	34.98
rr101	2684.64	3022.86	2684.64	3096.27	3162.39	3109.69	0.13	0.00	15.00	18.00	16.00
rr102	2033.62	2033.62	2077.13	2169.49	2213.74	2167.79	0.00	2.14	6.68	8.86	6.60
rr103	1794.85	1794.85	1870.71	1896.33	1972.20	1945.93	0.00	4.23	5.65	9.88	8.42
rr104	1735.86	1735.86	1738.09	1903.84	1743.39	1968.11	0.00	0.13	9.68	0.43	13.38
rr105	2065.52	2065.52	2127.28	2195.04	2271.79	2226.11	0.00	2.99	6.27	9.99	7.77
rr106	1920.16	1981.65	2027.87	2057.64	2145.01	1920.16	3.20	5.61	7.16	11.71	0.00
rr107	1792.15	1855.79	1792.15	1938.33	1836.75	2015.37	3.55	0.00	8.16	2.49	12.46
rr108	1699.01	1699.01	1792.71	1925.74	1737.92	1848.04	0.00	5.51	13.34	2.29	8.77
rr109	1938.55	1938.55	2019.06	2036.17	2166.92	2223.12	0.00	4.15	5.04	11.78	14.68
rr110	1807.02	1807.02	1824.12	1971.65	1832.97	1906.87	0.00	0.95	9.11	1.44	5.53
rr111	1764.78	1764.78	1803.27	1932.27	1918.23	1925.53	0.00	2.18	9.49	8.70	9.11
rr112	1763.50	1763.50	1792.94	1937.91	1838.86	1965.29	0.00	1.67	9.89	4.27	11.44
rr201	2059.73	2106.75	2072.58	2404.25	2059.73	2571.16	2.28	0.62	16.73	0.00	24.83
rr202	1924.95	1924.95	1971.59	2130.51	2002.56	2256.09	0.00	2.42	10.68	4.03	17.20
rr203	1811.40	1811.40	1860.81	2040.99	2064.73	2104.38	0.00	2.73	12.67	13.98	16.17
rr204	1612.58	1612.58	1680.88	1809.39	1711.52	1806.95	0.00	4.24	12.20	6.14	12.05
rr205	1859.54	1859.54	1871.79	2050.15	2029.72	2195.36	0.00	0.66	10.25	9.15	18.06
rr206	1857.82	1857.82	1897.48	2010.60	2072.26	2203.35	0.00	2.13	8.22	11.54	18.60
rr207	1701.91	1701.91	1748.88	1890.49	1713.59	1906.55	0.00	2.76	11.08	0.69	12.02
rr208	1598.95	1598.95	1628.37	1738.48	1728.63	1755.39	0.00	1.84	8.73	8.11	9.78
rr209	2129.60	2336.35	2541.20	2129.60	2505.66	2886.51	10.00	19.00	0.00	17.66	35.57
rr210	1782.63	1782.63	1871.75	2021.39	1966.07	2129.54	0.00	5.00	13.39	10.29	19.46
rr211	1735.94	1735.94	1767.11	1908.10	1847.12	2052.21	0.00	1.80	9.92	6.40	18.22
rrc101	2472.05	2472.05	2714.31	2583.59	2581.84	2551.06	0.00	9.80	4.51	4.44	3.20
rrc102	2309.12	2309.12	2413.71	2393.93	2552.78	2414.67	0.00	4.53	3.67	10.55	4.57
rrc103	2165.93	2165.93	2298.33	2342.60	2318.29	2195.55	0.00	6.11	8.16	7.03	1.37
rrc104	2045.94	2045.94	2193.79	2221.56	2258.76	2468.93	0.00	7.23	8.58	10.40	20.67
rrc105	2454.70	2454.70	2624.85	2681.30	2489.15	2498.68	0.00	6.93	9.23	1.40	1.79
rrc106	2333.60	2333.60	2487.12	2489.82	2384.91	2546.56	0.00	6.58	6.69	2.20	9.12
rrc107	2159.70	2159.70	2272.74	2356.52	2218.53	2608.24	0.00	5.23	9.11	2.72	20.77
rrc108	2172.61	2172.61	2202.60	2336.79	2286.66	2319.16	0.00	1.38	7.56	5.25	6.75
rrc201	2610.57	2741.56	2612.54	2770.38	2610.57	3249.81	5.02	0.08	6.12	0.00	24.49
rrc202	2422.05	2560.35	2422.05	2731.28	2591.50	2905.86	5.71	0.00	12.77	7.00	19.98
rrc203	2396.92	2396.92	2426.70	2586.54	2622.43	2658.06	0.00	1.24	7.91	9.41	10.89
rrc204	2245.18	2245.18	2327.69	2466.37	2338.17	2375.21	0.00	3.67	9.85	4.14	5.79
rrc205	3165.74	3165.74	3342.36	3398.85	3361.03	4134.57	0.00	6.00	7.00	6.00	31.00
rrc206	2614.92	2614.92	2755.82	2784.37	2724.47	3053.46	0.00	5.39	6.48	4.19	16.77
rrc207	2380.57	2380.57	2466.01	2423.22	2409.92	2534.57	0.00	3.59	1.79	1.23	6.47
rrc208	2248.49	2248.49	2362.23	2401.33	2587.38	2428.97	0.00	5.06	6.80	15.07	8.03
Mean	1996.71	2018.88	2056.52	2141.19	2135.33	2279.45	1.06	2.95	7.26	7.04	14.33

reports the comparison results of the average fitness values and RPI values obtained by five algorithms. From this table, that the following is evident: (1) the proposed IIG achieved 42 best average fitness values out of 56 instances, whereas the four compared algorithms obtained only 14 optimal values. (2) As shown in the last row of Table 8, the average RPI value of the IIG is 1.06, showing a significant improvement compared to the other algorithms. These findings demonstrate that the proposed IIG exhibits competitive performance in solving the considered problems compared to the four algorithms. Figure 7 presents the ANOVA for the obtained results in Table 8. The p-value is 1.5273 × 10⁻³⁵, far less than 0.05, indicating that the proposed IIG demonstrates a significant improvement in performance over the compared algorithms. Furthermore, the IIG shows greater stability in achieving an optimal solution.

To provide a clear and intuitive representation of the evolution process for all compared algorithms, Figure 8 presents the convergence curves for four selected instances, i.e., rc105, rr103, rrc107, and rrc208. All the compared algorithms were executed for 100 s on the aforementioned PC. From Figure 8, it can be observed that: (1) in most cases, the IIG algorithm provided the best initial values, demonstrating the effectiveness of the proposed heuristic initialization strategy in generating high-quality initial solutions. (2) As the runtime increases, the fitness values obtained by IIG are consistently the lowest, indicating that IIG effectively guides solutions toward better outcomes. (3) The final fitness values of the convergence curves confirm the effectiveness of the destruction-construction strategy, problem-specific local search strategy, and acceptance criterion. Additionally, Figure 9 illustrates the helicopter routing charts generated by the IIG for four selected instances. The red square in Figure 9 represents the rescue center, while the circle denotes the rescue site. For each instance, the blue solid line represents the route of the medical helicopter, and the red dashed line indicates the route of the transport helicopter. These results further validate that the IIG produces superior quality solutions when solving the collaborative helicopter rescue routing problem.

6. Conclusions and Future Works

This paper addressed the helicopter rescue routing problem by considering the survivors’ limited survival time constraint and modeling it as R-VRPTWLST. An IIG algorithm was proposed to solve this problem. First, a two-dimensional vector encoding method was employed considering the specific features of this problem. Then, a heuristic initialization strategy was designed to generate a high-quality solution. Additionally, three problem-specific search operators were developed to enhance the local search capability of the IIG. Moreover, the SA strategy was used as an acceptance criterion to deal with potential optimal solutions. Finally, the proposed IIG was compared with two efficient heuristic algorithms and four advanced algorithms to evaluate its effectiveness. Compared with two heuristic algorithms, the proposed IIG achieved 39 optimal solutions out of 56 instances. Meanwhile, the average RPI of the IIG was 0.14 and 0.11 times those of the DI and TSH, respectively. Compared with four advanced algorithms, the proposed IIG achieved 42 best average fitness values out of 56 instances. Additionally, the average RPI of the IIG is 0.359, 0.146, 0.151, and 0.074 times those of the IABC, ALNG, VNIG, and SAIG, respectively. These results demonstrate that the proposed IIG outperforms the other algorithms in solving the R-VRPTWLST.

The limitations of this study are as follows: (1) the objective function focuses solely on minimizing the total flight distance of helicopters, neglecting other important objectives, such as energy consumption during rescue operations. (2) This study failed to focus on a specific rescue scenario, and some constraints and assumptions are somewhat idealized, such as overlooking a hierarchical and phased manner of rescuing survivors and the multi-trip rescue problem. (3) The proposed IG did not consider the 3D rescue routing problem, neglecting environmental conditions, such as weather, during the execution of the rescue task.

Future research could explore the following aspects: (1) the constraints of the rescue problem need to be continuously improved to better align with real-world scenarios. For example, the multi-trip rescue and the limited flight time under constrained resources should be investigated. (2) The assumptions underlying the rescue problem require further refinement to enhance the practical applicability of the R-VRPTWLST. This could include enriching the rescue instance configuration, such as classifying survivors into multiple categories based on the severity of their injuries, prioritizing the transfer of critically injured survivors, and allowing a site to be visited multiple times. Additionally, the differential changes in the survivors’ life strength before and after being picked up by two types of helicopters at a rescue site should also be integrated into problem modeling. (3) The multi-objective helicopter rescue problem, which involves balancing various conflicting objectives, such as minimizing rescue time and energy consumption while maximizing flight safety and the quality of rescue tasks, will be investigated in the coming years. At that time, 3D rescue routing problems affected by environmental conditions will also be considered. Future multi-objective optimization algorithms will provide the decision-maker with more diverse schemes to address urgent rescue problems.