A Multi-Constraint Planning Approach for Offshore Test Tasks for an Intelligent Technology Test Ship

: A hierarchical population task planning method is presented to enhance the test efficiency and reliability of intelligent technology test ships under various tasks and complex limitations. Firstly, a mathematical model of the vehicle path problem for multi-voyage vessel testing is developed, which aims to minimize the ship’s fixed and fuel costs, taking into account the energy and space constraints of an intelligent technology test vessel, as well as practical factors such as the dependencies and temporal relationships between test tasks. Second, to fairly minimize constraint complexity in the planning process, an offshore test task planning architecture based on the concept of hierarchical population is explored and built. This architecture separates task planning into four levels and allocates the tasks to distinct populations. Using this information, a grouping genetic algorithm is suggested based on the characteristics of the population. This algorithm uses a unique coding method to represent task clusters and narrows the range of possible solutions. The issue of the conventional grouping genetic algorithm’s vast search space is resolved. Lastly, simulation verification is carried out, and the results show that the method can effectively solve the problem of offshore test task planning for intelligent technology test ships under multi-constraint conditions. It reduces test cost and improves test efficiency.


Introduction
The journey of marine equipment, from its inception as a conceptual design to its finalization as a finished product, necessitates a myriad of sea trials and tests.The establishment of a sea test environment serves to ameliorate the expenses incurred in testing marine technologies and equipment while fostering breakthroughs in technology and facilitating industrial advancements [1].In 2012, the European Union launched the "MUNIN" project for unmanned maritime vehicles, which has led countries around the world to recognize that testing under real sea conditions is crucial to ensure the functionality, safety, and reliability of future intelligent ships and to set up their offshore test sites [2,3].
In the early days of maritime history, ship testing at sea mainly referred to simple performance and safety tests, which were simple processes and did not require the support of professional test teams and tools.With the rapid development of marine science and technology, the content of modern ship inspection tasks at sea has become more complex and comprehensive [4].Ship testing requires professional equipment, technology, and knowledge.At the same time, the difficulty of inspection tasks is also increasing, covering a range of factors such as safety, stability, fuel consumption, and so on.In addition, with an increasing emphasis on environmental protection and energy conservation, offshore inspection tasks now also have to take into account multiple aspects of a ship's environmental friendliness, energy conservation, productivity, and cost-effectiveness [5,6].
To meet the increasingly demanding testing requirements, a new type of offshore test platform has emerged-the intelligent technology test ship.Intelligent technology test ships are specialized ships designed for conducting experimental research on intelligent systems and new equipment.They are equipped with innovative energy-saving devices and are capable of autonomous navigation.These ships can perform various functions, including remote control driving, autonomous navigation in open water, and autonomous docking and undocking.The primary purpose of intelligent technology test ships is to test and verify innovative energy-saving systems and intelligent systems for ships.They can also be used for basic hydrodynamic performance research.By establishing testing and verification standards for onboard green intelligent technology tests, these vessels aim to effectively promote the development of innovative energy-saving technology and intelligent technology for ships, thereby contributing to autonomous and controllable vessel development.
To ensure the reliability and test efficiency of intelligent technology test ships, researchers (Luo et al. [7] and Bai et al. [8]) have already conducted in-depth studies on the reliability of intelligent ships.In comparison, there are relatively few studies on test efficiency.With the development of intelligent ships, test task requirements are more and more detailed and complex, and their applications are more extensive.Factors such as obtaining more useful data in the test task, reducing the cost of the test task, and achieving an efficient completion of the sea test task, as well as reasonably allocating the test ship's tasks and reasonably planning the test ship's activities are important to using the experimental ship efficiently and improving the benefit of the task issue as a whole.
The above-mentioned real-life questions motivated this study.When carrying out an offshore test task, the intelligent technology vessel must carry specialized equipment to the designated testing site to fulfill its mission.However, due to limited space and fuel, the test ship is unable to carry all test equipment and reach all test sites simultaneously.Therefore, the test ship performs the test tasks on different voyages.The effective completion of a maritime test mission often involves multiple sets of equipment, multiple test sites, multiple ranges, and multiple routes.Therefore, it is necessary to prepare the test plan for the sea test task in advance and specify the tasks and routes to be performed by the test ship in each voyage.In summary, the test task planning method for intelligent technology test ships is also a key method for achieving the improvement of the overall autonomy of the test vessels and test system.The study of this problem has important theoretical significance and engineering application value.
This paper proposes a multi-constraint planning approach for offshore test tasks for intelligent technology test ships to solve the ship routing and task assignment problem under multi-constraint conditions.In this paper, firstly, a mathematical model of the vessel routing problem for multi-voyage maritime testing is established, which aims to minimize the fixed and fuel costs of the ship and takes into account the energy and space constraints of the smart technology test vessel, as well as practical factors such as the dependency and temporal relationship between the test tasks.Secondly, a hierarchical population and genetic algorithm for offshore test tasks is designed, which divides the offshore test tasks into four levels using hierarchical planning and distributes the constraints to different levels for processing; the large-scale task planning problem is divided into small-scale task planning problems by dividing the tasks into different communities.Finally, for the characteristics of the task communities, a proposed grouping genetic algorithm based on population characteristics is designed, which uses a two-part coding and constructs two cross-domain genetic evolution operators to ensure that the task communities are not destroyed during genetic evolution, thus greatly reducing the range of feasible solutions and improving the convergence speed of the algorithm.The simulation results show that this method can effectively solve the multiconstrained offshore test task scheduling problem.
The remainder of this paper is organized as follows.Section 2 reviews the relevant literature, Section 3 describes the components of the problem, Section 4 describes the proposed methodology, Section 5 performs the case simulation, and Section 6 gives the conclusion.

Literature Review
The connotation of the maritime test task planning problem for smart technology test ships studied in this paper is to assign a set of test tasks to different ship voyages and plan the paths for each voyage to achieve the goal of minimizing the total cost while satisfying constraints such as fuel constraints, space capacity constraints, time constraints, dependency constraints, and other constraints.According to the classification of this type of problem in the literature [9], the above problem is a vehicle routing problem (VRP).Therefore, this section first gives a brief introduction to vehicle routing problems and analyses the current state of the research on these problems in the field of ships.Subsequently, the current state of the research on the solution methods and algorithms related to this work is reviewed.

Vehicle Routing Problem
The VRP is also known as the vehicle path problem.In a VRP, a set of vehicle routes must be planned to satisfy a given number of customers while optimizing a set of objectives, such as minimizing the total distance traveled, minimizing the total travel time, minimizing the total cost, and so on.
Since the actual demand is constantly changing, vehicle routing problems can be divided into the following categories by introducing different constraints: (1) Vehicle routing problems with loading constraints: vehicle routing problems with load constraints can be classified into capacitated vehicle routing problem (CVRPs) and heterogeneous fleet vehicle routing problem (HFVRPs), depending on the type and load of the vehicle.In CVRPs, vehicles are of the same type and volume [10].In contrast to CVRPs, HFVRPs involve vehicles with different volumes and operating costs [11]; (2) Vehicle path problems for simultaneous goods collection and delivery: traditional VRPs require vehicles to return to the origin after completing the delivery, but in real operations, vehicles are also required to complete the task of picking up goods while delivering them.This gives rise to vehicle routing problems with simultaneous pickups and deliveries (VRPSPDs) [12]; (3) Vehicle route problem with time windows (VRP with time windows or VRPTW): in this problem, the customer wants to receive goods within a certain period and the transport vehicle has to arrive at the distribution point within this period.VRPTWs can be further classified into two categories: hard time windows and soft time windows.Hard time windows require that the vehicle cannot arrive later than the end of the time window, and if it arrives too early, the vehicle must wait [13].Soft time windows, on the other hand, are flexible concerning the time window requirement, allowing vehicles to arrive slightly later, but at an additional cost to the vehicle [14]; (4) The clustered vehicle routing problem (CluVRP) was first proposed by Sevaux and Sorensen [15] in 2008.It requires grouping customers or locations into different clusters based on their similarity to each other and then planning vehicle routes for each cluster.Each cluster can be viewed as a small subproblem containing a set of customers or locations that need to be served on the same path.For each cluster, vehicle paths must then be planned in such a way that the customers within each cluster are served while meeting certain constraints (e.g., time windows, capacity limits, etc.).
In addition to the above four common types of VRP variants, there are also problems such as dynamic VRP, wo-echelon VRP, multi-depot VRP, etc. [16].The ship test task planning problem discussed in this paper belongs to the category of clustered vehicle routing problems (CluVRPs) considering capacity constraints, where the clusters in the problem are grouped according to the dependencies between tasks, where dependencies refer to the fact that between certain test tasks, due to the sharing of part of the equipment and testers, it is necessary to arrange this part of the test tasks in the same voyage of the intelligent technology test ship, which is in line with the characteristics of CluCRP.

Current State of Research on the Problems of Shipping Lanes
Gao et al. [17] built a feeder ship route planning model (FSRPTTW, feeder ship routing problem) with nonlinear tidal time window constraints based on the VRPTW model, decomposed the problem into a main problem and subproblems by using the Dantzig-Wolfe method, and designed a column generation algorithm to solve the problem.Shen [18] applied a genetic algorithm to the study of route optimization of inland waterway vessels, established a route optimization model for inland waterway vessels without time window constraints to minimize the delivery cost, and designed a genetic algorithm for the model.Korsvik and Fagerholt [19] proposed a forbidden search heuristic to solve the vessel routing and scheduling problem in bulk cargo transportation.Shane et al. [20] proposed a branching price reduction algorithm to solve the maritime pickup and delivery problem with time windows and split loads.Tao [21] combined a particle swarm algorithm and ant colony algorithm to solve the car two-path problem with capacity constraints and save on the cost of ship logistics.Christiansen [22] solved the ship routing and scheduling problem with time windows using the Dantzig-Wolfe decomposition approach and branchbounding method.Lee and Kim [23] proposed a mixed integer planning model and a heuristic method based on adaptive large neighborhood search to solve the ship routing problem with delivery windows.Christiansen and Nygreen [24] used the Dantzig-Wolfe decomposition method combined with a variant of the multi-vehicle pickup and delivery problem (m-PDPTW) and a multi-inventory model to solve the inventory pickup and delivery problem with time windows (IPDPTW).Al-Hamad [25] developed an efficient genetic algorithm (GA) variant to solve the ship routing and scheduling problem for heterogeneous ships with time windows.Pang et al. [26] proposed a heuristic algorithm to solve the ship routing problem with berthing time conflict avoidance constraints.
In order to describe the literature content more clearly, the above literature was categorized and the results are shown in Table 1.

Current Status of Research on Solution Methods
From a solution perspective, solution methods can be classified into two categories, centralized and distributed, depending on the information interaction strategy [27,28].For vehicle path problems, centralized approaches are commonly used, but the scalability of centralized task scheduling is limited due to the computational power and performance constraints of the central scheduler [29].To improve the efficiency of centralized task planning, hierarchical ideas can be used.Lin et al. [30] adopted hierarchical processing to solve the task planning problem, which can effectively solve small-scale task planning problems.However, when faced with larger-scale tasks, this planning method is not efficient.Zhang et al. [31] proposed methods based on the idea of hierarchical planning to simplify the multi-traveler problem into multiple single-traveler problems, thus simplifying the complex task planning problem.However, these methods do not consider the problem of task types.Zhang et al. [32] explored a hierarchical swarm task planning method for multiple unmanned aerial vehicles (UAVs) to perform reconnaissance-attack-relay tasks.This method solves time constraints in the task planning process by layering and solving tasks step by step and using a strategy of excessive agents.Zhang et al. [33] investigated the AUV task planning and replanning architecture based on hierarchical planning, which divides each task into multiple subtasks and completes the tasks step by step, thus improving the efficiency and reliability of task execution.Fu et al. [34] proposed a hierarchy-based approach to transforming a complex large-scale mission planning problem into multiple small-scale problems by considering the number, types, and interrelationships of UAV resources and dividing a large UAV swarm into multiple subswarms.By using the idea of hierarchical structure, problem-solving efficiency can be improved, which has important application value for large-scale mission planning.
The solution methods commonly used in centralized task planning models can be divided into exact methods and heuristic methods.Exact methods include the Hungarian algorithm, integer linear programming, etc. [35,36].They can find the optimal solution to a problem, but as the problem's size increases, the solution cost increases exponentially.In contrast, heuristic algorithms compromise between time complexity and optimal solution, so they are widely studied in the field of task planning.Optimization algorithms such as genetic algorithms [37], artificial bee colony algorithms [38], ant colony algorithms [39], and bionic algorithms are widely used in task allocation.
Genetic algorithms are commonly used to solve centralized task planning models with strong robustness and global search capability.Current research on genetic algorithms focuses on how to transform VRPs into a form suitable for chromosome encoding and how to improve the efficiency of genetic algorithms to obtain the optimal solution.Common chromosome coding methods include single chromosome, double chromosome, and combined chromosome [40][41][42][43][44].Baghel [45] proposed a GGA-SS algorithm to achieve better results by grouping, crossover, and mutation operations on travelers.Chen et al. [46] selected a two-part chromosome coding method and studied and designed the crossover operator and mutation operator for this coding method which were able to achieve better computational results.However, these algorithms still have some problems, such as search space redundancy and inefficiency of the genetic evolution operator.These problems limit the performance and effectiveness of the algorithms and require further improvement and optimization.
Based on the findings from the literature review, it can be concluded that researchers have addressed the ship routing problem using various approaches.However, it is evident that the research on ship test task planning is relatively limited, with little literature focusing on clustered vehicle routing problems similar to the scope of this paper.Furthermore, when dealing with complex task planning problems, most researchers employ hierarchical processing to simplify the problem.Some researchers have also categorized the population based on different vehicle types.Nevertheless, there are no subdivisions based on task types.Lastly, in terms of grouping the grouping genetic algorithm, most researchers concentrate on studying the coding method of the grouping genetic algorithm and developing different crossover and mutation operators.In this paper, on the basis of the original coding method, we combine the population in the planning method with the group of the grouping genetic algorithm and introduce new constraints on the genetic evolution of the grouping genetic algorithm.This narrows down the feasible solution range for the clustered vehicle routing problem and improves convergence efficiency.

Problem Statement
Task testing is a complex type of systems engineering, and the actual planning process involves several factors such as ship speed, ship quality, and mission type.In this paper, we study the basic planning method of the intelligent technology test ship task planning problem.To model complex problems, specific assumptions are made about the actual task-planning problems, as follows: (1) Ships offer the same resource conditions for each voyage; (2) Each test task is scheduled only once and cannot be stopped midway until the mission is completed; (3) Ignoring the specifics of the test ship's task execution, arrival at the task position indicates completion of the task; (4) Each voyage of the intelligent technology test ship can be defined as traversing the task location during the task execution process, starting from any task location and returning to the starting task location after completing all tasks, forming a circular path; (5) The distance between any two tasks satisfies the space requirements for intelligent technology test ship navigation and mission execution operations; (6) The distance between any two tasks on the map is expressed as Euclidean distance between locations.
Assuming that there are n tasks to be processed, the task planning-oriented problem abstracts the path network of the set ship as an undirected graph G = (V, E).Where V = {0} ∪ {1, 2, 3, . . . ,n} ∪ {n + 1} denotes the set of all vertices, and 0 and n + 1 denote the vertices when the dock is used as the starting point and the end point, respectively, and V 1 = {1, 2, 3, . . . ,n} is the set of task location points.E = E 1 ∪E 2 ∪E 3 denotes the set of all paths, E 1 = {(i, j)|i = 0, j ∈ V 1 } denotes the set of paths from the dock to the task location point, denotes the set of paths from two different task location points, and E 3 = {(i, j)|i ∈ V 1 , j = n + 1} denotes the set of paths from the task location point to the dock.S = {1, 2, . . . ,M} denotes a set of ship journeys out of a set of M journeys, and the number of actual journeys involved is denoted by m. t i denotes the time required to perform task i; t denotes the time required to maintain the ship at the dock; d ij denotes the distance from task location point i to task pivot point j; and v denotes the speed of the ship.At present, most fuel consumption models agree to assume that the fuel consumption rate of the main engine (ton/n miles) is a power function of sailing speed v, namely c1vc2, where v is the sailing speed (knots) and c1 and c2 are positive coefficients [47].Therefore, we assume that δ 1 represents the fuel consumption rate of the ship, δ 1 = c1vc2.In addition, after consulting with experts, it was determined that when performing the test task, the fuel consumption rate was 70% of the usual working condition, that is, 0.7 δ 1 .U represents the unit price of fuel (USD/ton).P 1 i denotes the size of the bow space required for task i, i ∈ V 1 ; P 2 i denotes the size of the bow space required for task i, i ∈ V 1 ; P 3 i denotes the size, i ∈ V 1 ; P 1 denotes the size of the bow space; P 2 denotes the size of the stern space; P 3 denotes the size of the superstructure space; P 4 denotes mailbox capacity; and f denotes the fixed cost of the vessel per voyage.
d i denotes the voyage on which task i is performed, i ∈ V 1 ; x ijk = 1 denotes that the ship travels from vertex i to vertex j on voyage k, ∀(i, j) ∈ E, k = 1, 2, . . ., m; otherwise x ijk = 0. u i indicates that node i is the nth visited node from the starting point, u i ≥ 0. N indicates the number of nodes, N = n+ 1.
In Equation ( 1), F denotes the objective function of the optimization and the first term represents the fixed costs, including ship rent, labor costs, insurance costs, and equipment costs.The second term is the fuel cost of navigation, which is mainly the cost of the fuel generated by the ship navigating between vertices.The third item is the fuel cost of the test, which is mainly the fuel cost generated by the ship during the execution of the test mission.Equation (2) indicates that the ship leaving a terminal has to return to the terminal; Equation (3) indicates that the number of times the ship enters a location point is equal to the number of times it leaves the location point; Equation (4) indicates that the ship only goes to each task location point once; Equation (5) indicates that the ship only departs from each task location point once; Equation (6) represents constraints on the order of access to task location points; Equation (7) indicates that the queue time for nodes, other than the starting and ending nodes, which are not allowed to be accessed, should be zero; Equation (8) represents the binary constraint of the decision variable x ijk ; Equation (9) indicates that the fuel consumed by the ship while sailing and performing tasks cannot exceed the capacity of the fuel tank; Equations ( 10)- (12) indicate that the space occupied by the ship's equipment on board for each voyage cannot exceed the space of the ship's bow space, stern space, and superstructure space; Equation (13) indicates the dependency relationship between tasks, which means that task i and task j must be arranged in the same voyage; and Equation (14) indicates the sequential relationship between tasks, which means that the voyage for performing task i must be preceded by the voyage for performing task j.

Methods
In this paper, the proposed planning method and the general planning process are described in Section 4.1; the hierarchical structure and ethnic group assignment in the planning method are introduced in Sections 4.2 and 4.3; and the improved grouping genetic algorithm is proposed and described in detail in Section 4.4.

Planning Methods and Processes
The task planning problem of the offshore testing setup studied in this article has multiple types of testing tasks and complex constraints.To better approach the actual situation, a task planning method for offshore testing based on a hierarchical population and grouped genetic algorithm is proposed.This method disperses the complex and numerous constraints of task planning problems into different levels of abstraction through hierarchical processing, reducing the difficulty of the problem.By assigning ethnic groups, the originally large-scale task planning problem is divided into smaller-scale task planning problems on a family basis, reducing the size of the problem.By improving the grouping genetic algorithm, the range of feasible solutions during solving is significantly reduced, improving the efficiency of solving.Through the above three measures, the problem of planning offshore testing tasks for intelligent technology test ships can be effectively solved.
The task planning process consists of three parts: population distribution, atomic task planning, and task execution.Taking two types of tasks as examples, a task planning flowchart was constructed, as shown in Figure 1.

Hierarchical Structure
Hierarchical processing is the process of decomposing the task planning problem into different levels of subproblems, and focusing on and dealing with different details of the problem at different levels so that the complex and numerous constraints of the task planning problem are distributed to different levels of subproblems according to their level of abstraction.The difficulty of the problem is reduced.The schematic diagram of hierarchical task planning processing is shown in Figure 2.
According to the requirements of test task planning for intelligent technology test ships, the planning method is divided into an input layer, a population distribution layer, an atomic task planning layer, and a task execution layer.
The input layer mainly manages and preprocesses task and resource information into a standard form that can be read by the next layer.Task information includes the types, (1) Task preprocessing: The tester enters the task information and resource information in a specified format, which is processed into a standard form that can be accepted by the group allocation layer.The information is then sent to the population task planning layer to await group allocation; (2) Population distribution: The population distribution layer obtains the input information of the input layer in real time, processes the task information, performs the first population allocation, divides the test tasks into two populations according to the task type, and then sorts the populations according to their temporal relationship.Secondly, the population distribution layer performs the second population allocation on the two populations in groups of populations.According to the dependencies between tasks within a population, the test tasks are divided into multiple atomic populations.Finally, the population distribution layer combines them to form a complete population distribution plan and sends it to the atomic task planning layer.During population allocation, it is necessary to update the status of all tasks in real time and allocate tasks to corresponding task populations in real time; (3) Atomic task planning: The atomic task allocation module obtains the population distribution plan of the population distribution layer in real time and uses a grouping genetic algorithm to solve it to obtain the task execution scheme and send it to the task execution layer; (4) Task execution: The intelligent technology test ship receives the task execution plan starts to perform the test task and feeds back the task progress and ship resource status to the input layer in real time.When all tasks have been completed, the test task planning ends.

Hierarchical Structure
Hierarchical processing is the process of decomposing the task planning problem into different levels of subproblems, and focusing on and dealing with different details of the problem at different levels so that the complex and numerous constraints of the task planning problem are distributed to different levels of subproblems according to their level of abstraction.The difficulty of the problem is reduced.The schematic diagram of hierarchical task planning processing is shown in Figure 2.  In the case of layered processing, the processing of layer 1 and layer 2 focus on the operator's decision, and the unified allocation and planning are carried out ing to our resources and the actual situation, while the processing of layer 3 reli on the results of the automatic calculation.The whole task planning process is i until a satisfactory result is finally achieved.The layered approach can focus on d problems at different levels, extract and simplify the factors involved in the probl combine the respective advantages of manual processing and automatic calculatio

Population Distribution
Populations are based on multiple task types, where tasks of the same type signed to the same population, and each population is further subdivided to assig with dependencies to the same atomic populations.Multiple "tasks" form "small According to the requirements of test task planning for intelligent technology test ships, the planning method is divided into an input layer, a population distribution layer, an atomic task planning layer, and a task execution layer.
The input layer mainly manages and preprocesses task and resource information into a standard form that can be read by the next layer.Task information includes the types, numbers, and logical relationships of tasks.Resource information includes the energy and space resources of the intelligent technology test ship.The population distribution layer forms a population distribution plan based on task types, timing relationships, and dependency relationships.The atomic task planning layer generates the final task planning scheme by accepting the population distribution plan and planning using the group genetic algorithm.The task execution layer executes the test task according to the planning scheme and feeds back the execution process and the remaining resources of the ship to the input layer for re-planning the task in case of special circumstances.
In the case of layered processing, the processing of layer 1 and layer 2 focuses more on the operator's decision, and the unified allocation and planning are carried out according to our resources and the actual situation, while the processing of layer 3 relies more on the results of the automatic calculation.The whole task planning process is iterative until a satisfactory result is finally achieved.The layered approach can focus on different problems at different levels, extract and simplify the factors involved in the problem, and combine the respective advantages of manual processing and automatic calculation.

Population Distribution
Populations are based on multiple task types, where tasks of the same type are assigned to the same population, and each population is further subdivided to assign tasks with dependencies to the same atomic populations.Multiple "tasks" form "small populations" and multiple "small populations" form "large communities".A schematic diagram of the population distribution process is shown in Figure 3.In the case of layered processing, the processing of layer 1 and layer 2 focuse on the operator's decision, and the unified allocation and planning are carried out a ing to our resources and the actual situation, while the processing of layer 3 relie on the results of the automatic calculation.The whole task planning process is it until a satisfactory result is finally achieved.The layered approach can focus on d problems at different levels, extract and simplify the factors involved in the proble combine the respective advantages of manual processing and automatic calculatio

Population Distribution
Populations are based on multiple task types, where tasks of the same type signed to the same population, and each population is further subdivided to assig with dependencies to the same atomic populations.Multiple "tasks" form "small p tions" and multiple "small populations" form "large communities".A schematic d of the population distribution process is shown in Figure 3.  Population distribution requires the real-time acquisition of task information, after which the test tasks are assigned to different populations, and the task populations are sorted, after which the populations are divided into smaller atomic populations according to the relationships between the tasks.There are three main aspects to the population distribution process.The first aspect is the type of task.Tasks of the same type are placed in the same population according to task type.The second aspect is the time relationship according to the actual needs of the task populations, and the types of tasks with strict time requirements are placed at the top.The third aspect is the dependency relationship: there is a dependency relationship between some tasks of the same population, which must be arranged in the same voyage of the ship.This requires them to be arranged in the same atomic population.In the atomic task planning layer, the atomic population corresponds to the group of the grouping genetic algorithm, which reduces the range of feasible solutions and overcomes the problem of a traditional grouping genetic algorithm with a large search space.

Gene Coding and Population Initialization
As the problem size increases, the solution cost increases exponentially, and it is not possible to use the exhaustive method to obtain the optimal task assignment.At present, heuristic algorithms are commonly used to obtain a feasible solution [45][46][47][48][49].According to the literature [45], combined with the actual situation during the test tasks of the intelligent technology test ship, a two-stage coding process was designed to encode the task allocation using integer coding.Nonrepeating natural integers are used to chromosomally encode all tasks.It is assumed that the order of the three groups of execution task combinations is as follows: {[T 1 , T 2 , T 3 , T 4 , T 5 ], [T 6 , T 7 , T 8 ], [T 9 , T 10 , T 11 , T 12 ]}.The chromosome coding is shown in Figure 4.
according to the actual needs of the task populations, and the types of tasks with strict time requirements are placed at the top.The third aspect is the dependency relationship: there is a dependency relationship between some tasks of the same population, which must be arranged in the same voyage of the ship.This requires them to be arranged in the same atomic population.In the atomic task planning layer, the atomic population corresponds to the group of the grouping genetic algorithm, which reduces the range of feasible solutions and overcomes the problem of a traditional grouping genetic algorithm with a large search space.

Gene Coding and Population Initialization
As the problem size increases, the solution cost increases exponentially, and it is not possible to use the exhaustive method to obtain the optimal task assignment.At present, heuristic algorithms are commonly used to obtain a feasible solution [45][46][47][48][49].According to the literature [45], combined with the actual situation during the test tasks of the intelligent technology test ship, a two-stage coding process was designed to encode the task allocation using integer coding.Nonrepeating natural integers are used to chromosomally encode all tasks.It is assumed that the order of the three groups of execution task combinations is as follows: {  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  }.The chromosome coding is shown in Figure 4.The shaded area in Figure 3 is the second part of the chromosome, indicating that the first five test points, 1, 2, 3, 4, and 5, are the first group; the middle three test points, 6, 7, and 8, are the second group; and the last four test points, 9, 10, 11 and 12, are the third group.

Adaptation Function
In genetic algorithms, the individual superiority or inferiority of a population is evaluated using a fitness function, which is the basis for genetic selection.In the process of task planning, it is necessary to obtain the optimal mission combination through the ship's fixed cost and voyage cost and calculate a quote.In the process of task planning, the ship's fixed cost and voyage cost are mainly considered, and by calculating the mission benefit and mission cost of different mission combination schemes, the combination scheme with the highest benefit and lowest cost is taken as the planning result.The objective function can be expressed as follows: The shaded area in Figure 3 is the second part of the chromosome, indicating that the first five test points, 1, 2, 3, 4, and 5, are the first group; the middle three test points, 6, 7, and 8, are the second group; and the last four test points, 9, 10, 11 and 12, are the third group.

Adaptation Function
In genetic algorithms, the individual superiority or inferiority of a population is evaluated using a fitness function, which is the basis for genetic selection.In the process of task planning, it is necessary to obtain the optimal mission combination through the ship's fixed cost and voyage cost and calculate a quote.In the process of task planning, the ship's fixed cost and voyage cost are mainly considered, and by calculating the mission benefit and mission cost of different mission combination schemes, the combination scheme with the highest benefit and lowest cost is taken as the planning result.The objective function can be expressed as follows:

Selection of Operator Design
The selection operator is mainly responsible for the preservation of excellent individuals and good genes in the parent population, and this paper adopts the following selection mechanism: For a newly generated population, individuals are sorted according to their adaptive value from high to low.The highest-ranking individuals directly enter the mating pool, and the remaining individuals are selected according to a roulette mechanism to increase the probability of individuals with high adaptive value entering the mating pool and to eliminate individuals with low adaptive value as much as possible.The number of individuals entering the mating pool is equal to the number of individuals eliminated from the mating pool.

Local Path Crossing Operator
The exchange process of the local path crossing operator is shown in Figure 5.It is implemented as follows: Step 7: Generate offspring 2 by selecting the crossover portion for parent samp and sequentially copying the unselected genes in parent sample 1 into offspring 2.
The local path crossover operator only exchanges two gene segments from the sa group within a chromosome, not between different groups, so it cannot change the or of the groups.Step 1: Select two parents using the roulette method; Step 2: Randomly select the same group as the exchange region in parent sample 1 and parent sample 2; Step 3: Select the crossover part in the crossover region of parent sample 1; Step 4: Generate offspring 1 according to the crossover part; offspring 1 inherits the subexchange region of parent sample 1; Step 5: Sequentially copy the genes not selected in parent sample 2 into offspring 1; Step 6: Sequentially copy the unselected genes in parent sample 2 into offspring 1; Step 7: Generate offspring 2 by selecting the crossover portion for parent sample 2 and sequentially copying the unselected genes in parent sample 1 into offspring 2.
The local path crossover operator only exchanges two gene segments from the same group within a chromosome, not between different groups, so it cannot change the order of the groups.

Cross-Domain Path Crossing Operator
To improve chromosome diversity and speed up the convergence of the algorithm, a cross-domain path crossover operator was designed.The exchange process of the crossdomain path crossover operator is shown in Figure 6.It is implemented as follows: Step 1: Select two parents using the roulette method; Step 2: Randomly select a group as the crossover part in parent sample 1; Step 3: Generate offspring 1 according to the crossover part; Step 4: Copy the unselected genes in parent sample 2 into offspring 1 in order; Step 5: Then, select the crossover part to generate offspring 2 to parent sample 2, and sequentially copy into offspring 2.
As for the cross-domain path crossover operator in the crossover process, the two segments of chromosome genes in the crossover are synchronous.This can keep the same group of constraints of the test task from being destroyed, reduce repeated calculations due to constraints not being satisfied, and speed up the convergence of the algorithm.
sequentially copy into offspring 2.
As for the cross-domain path crossover operator in the crossover process, the segments of chromosome genes in the crossover are synchronous.This can keep the s group of constraints of the test task from being destroyed, reduce repeated calcula due to constraints not being satisfied, and speed up the convergence of the algorithm

Two-Opt Local Path Variation Operator
The two-opt neighborhood search algorithm is a local search algorithm for opti ing path problems.It attempts to optimize all possible solutions by swapping path tween two nodes in the current path until a better path is obtained.It is suitable for mizations where the initial solution is known.
The implementation of the two-opt local path variation operator is shown in Fi 7. It is implemented as follows: Step 1: Randomly select a group in a chromosome as a mutable region; Step 2: Randomly select two points m and n in the mutable region, where m < n Step 3: Add the unchanged path before m to the new path and flip the coding o path between m and n to add it to the new path; Step 4: Add the unchanged path after n to the new path.The mutation process of the cross-domain path mutation operator is shown in Figure 8.It is implemented as follows: Step 1: Two groups in a given chromosome are randomly selected to be the exchange group; Step 1: Randomly select a group in a chromosome as a mutable region; Step 2: Randomly select two points m and n in the mutable region, where m < n; Step 3: Add the unchanged path before m to the new path and flip the coding of the path between m and n to add it to the new path; Step 4: Add the unchanged path after n to the new path.

Cross-Domain Path Variation Operator
The mutation process of the cross-domain path mutation operator is shown in Figure 8.It is implemented as follows: The mutation process of the cross-domain path mutation operator is shown in Figure 8.It is implemented as follows: Step 1: Two groups in a given chromosome are randomly selected to be the exchange group; Step 2 The positions of two segments of genes in the exchange group are swapped; Step 3: New genes are generated according to the swapped position.The cross-domain path variation operator, which also ensures that the two segments of the chromosome are synchronized when exchanging genes during the exchange process, avoids breaking the constraints by adhering to the cohort constraints of the test task, which in turn reduces repetitive computations and speeds up the convergence of the algorithm.Step 1: Two groups in a given chromosome are randomly selected to be the exchange group; Step 2 The positions of two segments of genes in the exchange group are swapped; Step 3: New genes are generated according to the swapped position.The cross-domain path variation operator, which also ensures that the two segments of the chromosome are synchronized when exchanging genes during the exchange process, avoids breaking the constraints by adhering to the cohort constraints of the test task, which in turn reduces repetitive computations and speeds up the convergence of the algorithm.

Improvement of the Grouping Genetic Algorithm Process
The flow chart of the improved grouping genetic algorithm is shown in Figure 9, and the specific steps are as follows: Step 1: Initialize and determine population size N, two crossover probabilities C1 and C2, two mutation probabilities V1 and V2, and the number of iterations G; Step 2: In this paper, for chromosome coding, we use real-number coding: the dock is 0, and the other test points are 1, 2, 3. ..; Step 3: Adopt the greedy strategy and assign the test tasks to different voyages; Step 4: Calculate the fitness of each chromosome in the initial population according to the fitness function given in Section 4.4.2; Step 5: Select the chromosome populations according to the replication roulette approach described in Section 4.4.3; Step 6: Perform the crossover operation as described in Sections 4.4.4 and 4.4.5; Step 7: Perform the mutation operation as described in Sections 4.4.6 and 4.4.7; Step 8: Judge the stopping condition of the algorithm.If it meets the stopping condition, go to step 9; otherwise, select individuals according to the selection mechanism, complete the mutation and crossover operation to produce new individuals, and perform the population update to obtain the next generation of the population.Go back to step 3; Step 9: The algorithm ends, the best individual is obtained, and the corresponding test task program is output.
Step 5: Select the chromosome populations according to the replication roule proach described in 4.4.3; Step 6: Perform the crossover operation as described in 4.4.4 and 4.4.5; Step 7: Perform the mutation operation as described in 4.4.6 and 4.4.7; Step 8: Judge the stopping condition of the algorithm.If it meets the stopping tion, go to step 9; otherwise, select individuals according to the selection mechanism plete the mutation and crossover operation to produce new individuals, and perfo population update to obtain the next generation of the population.Go back to step Step 9: The algorithm ends, the best individual is obtained, and the correspo test task program is output.

Simulation Verification
The simulation experiment platform was AMD Ryzen 7 6800H/16 GB/64-bit Win11 operating system Thinkpad notebook.The programming tool was Python 3.11 (64-bit).The parameters of the grouping genetic algorithm were set as follows: the group size was set to 100, the local path crossing operator probability wass 0.8, the cross-domain path crossing operator probability was 0.6, the two-opt local path variation operator had a probability of 0.1, and the cross-domain path variation operator had a probability of 0.05.The number of iterations was 1000.

Example Introduction and Solution
To verify the effectiveness of the research method in this paper, two sets of task scenarios with different scales were set up to analyze actual cases of sea test tasks.
Case 1: Suppose the intelligent technology test ship is required to perform 28 test tasks in different sea areas, of which 8 tasks are basic performance tests of the vessel and need to be prioritized, while the remaining 20 are test trials of intelligent devices.One pier location will be tasked with supplying materials for the intelligent technology test vessel.The distribution of the dock location and each sea test area is shown in Figure 10, where point 0 is the dock location and points 1-28 are the 28 test sea areas to be reached.
set to 100, the local path crossing operator probability wass 0.8, the cross-domain path crossing operator probability was 0.6, the two-opt local path variation operator had a probability of 0.1, and the cross-domain path variation operator had a probability of 0.05.The number of iterations was 1000.

Example Introduction and Solution
To verify the effectiveness of the research method in this paper, two sets of task scenarios with different scales were set up to analyze actual cases of sea test tasks.
Case 1: Suppose the intelligent technology test ship is required to perform 28 test tasks in different sea areas, of which 8 tasks are basic performance tests of the vessel and need to be prioritized, while the remaining 20 are test trials of intelligent devices.One pier location will be tasked with supplying materials for the intelligent technology test vessel.The distribution of the dock location and each sea test area is shown in Figure 10, where point 0 is the dock location and points 1-28 are the 28 test sea areas to be reached.The task dependencies are shown in Table 2. Based on the above, in the population distribution phase of task planning, the tasks given at the same time were analyzed and categorized, and the tasks were divided into The task dependencies are shown in Table 2. Based on the above, in the population distribution phase of task planning, the tasks given at the same time were analyzed and categorized, and the tasks were divided into two populations and seven atomic populations.Among them, the first population A contains two atomic populations with a total of eight test tasks, and the second population B contains five atomic populations with a total of eighteen test tasks.The distribution of the two populations is shown in Figures 11 and 12.
Then, atomic task planning is performed, and the best genes are obtained based on constraint Equations ( 2)-( 10), the objective function Equation (13), and the improved grouped genetic algorithm to obtain the best task planning results.
two populations and seven atomic populations.Among them, the first population A contains two atomic populations with a total of eight test tasks, and the second population B contains five atomic populations with a total of eighteen test tasks.The distribution of the two populations is shown in Figures 11 and 12.Then, atomic task planning is performed, and the best genes are obtained based on constraint Equations ( 2)-( 10), the objective function Equation (13), and the improved grouped genetic algorithm to obtain the best task planning results.
This paper was used to generate chromosome genes.Each chromosome contains two segments of genes, and the two segments correspond to each other.The first segment of genes is the order of task execution and the second segment of genes is the grouping of genes.The initial genes are generated using the roulette method and are cross-mutated; the final optimal gene sequence is generated as shown in Figure 13; and the task path is shown in Figure 14.two populations and seven atomic populations.Among them, the first population A contains two atomic populations with a total of eight test tasks, and the second population B contains five atomic populations with a total of eighteen test tasks.The distribution of the two populations is shown in Figures 11 and 12.Then, atomic task planning is performed, and the best genes are obtained based on constraint Equations ( 2)-( 10), the objective function Equation (13), and the improved grouped genetic algorithm to obtain the best task planning results.
This paper was used to generate chromosome genes.Each chromosome contains two segments of genes, and the two segments correspond to each other.The first segment of genes is the order of task execution and the second segment of genes is the grouping of genes.The initial genes are generated using the roulette method and are cross-mutated the final optimal gene sequence is generated as shown in Figure 13; and the task path is shown in Figure 14.This paper was used to generate chromosome genes.Each chromosome contains two segments of genes, and the two segments correspond to each other.The first segment of genes is the order of task execution and the second segment of genes is the grouping of genes.The initial genes are generated using the roulette method and are cross-mutated; the final optimal gene sequence is generated as shown in Figure 13; and the task path is shown in Figure 14.First, as can be seen from the schematic diagram of the best genes in Figure 12, each gene is divided into two segments: the first segment is the serial number of the task, reflecting the order of execution of the task.From segments in the genes, it can be seen that the first eight positions of each gene contain all the tasks in population A, indicating that the tasks in population A will be given priority to be executed by the intelligent technology test ship.The tasks in population B are assigned to the back of the gene in this segment, indicating that the tasks in population B will begin to be executed after all the tasks in population A have been executed, which satisfies the sequential time constraints of the test tasks.
Secondly, as can be seen from the task path schematic in Figure 13, after the test tasks have been assigned voyages, the tasks with dependency constraints are all assigned to the same voyage, which satisfies the cohorting constraint of the test tasks.
Finally, in the atomic mission planning process, the optimization of the objective function must be pursued under the premise of satisfying the multiple resource constraints of the intelligent technology test vessel.As shown in the task path schematic in Figure 13, taking the trip with path [0, 28,16,14,15,0] as an example, since the planning objective includes minimizing the ship's fixed cost as well as minimizing the trip cost, the intelligent technology test ship will assign tasks of different atomic groups to the same trip as far as the ship's resources allow, thus reducing the execution cost of the tasks.
Our simulation experiments show that the task planning method for offshore testing based on a hierarchical population and grouped genetic algorithm can effectively accomplish task allocation for complex task types under multiple constraints.
Case 2: A task scale was added to Case 1 to test the stability of the task planning method.The new task scenario adds 10 new tasks and randomly changes the location of the test sea area as well as the task limitations; the new task assumes that the intelligent technology test vessel is required to test 38 test tasks in different sea areas, of which 10 tasks are urgent and need to be prioritized for testing, another 18 tasks are allocated more time and can be tested normally, and the last 10 tasks are the least urgent and are placed at the end of the test.The location distribution of the dock and each sea test area is shown in Figure 15, First, as can be seen from the schematic diagram of the best genes in Figure 12, each gene is divided into two segments: the first segment is the serial number of the task, reflecting the order of execution of the task.From segments in the genes, it can be seen that the first eight positions of each gene contain all the tasks in population A, indicating that the tasks in population A will be given priority to be executed by the intelligent technology test ship.The tasks in population B are assigned to the back of the gene in this segment, indicating that the tasks in population B will begin to be executed after all the tasks in population A have been executed, which satisfies the sequential time constraints of the test tasks.
Secondly, as can be seen from the task path schematic in Figure 13, after the test tasks have been assigned voyages, the tasks with dependency constraints are all assigned to the same voyage, which satisfies the cohorting constraint of the test tasks.
Finally, in the atomic mission planning process, the optimization of the objective function must be pursued under the premise of satisfying the multiple resource constraints of the intelligent technology test vessel.As shown in the task path schematic in Figure 13, taking the trip with path [0, 28,16,14,15,0] as an example, since the planning objective includes minimizing the ship's fixed cost as well as minimizing the trip cost, the intelligent technology test ship will assign tasks of different atomic groups to the same trip as far as the ship's resources allow, thus reducing the execution cost of the tasks.
Our simulation experiments show that the task planning method for offshore testing based on a hierarchical population and grouped genetic algorithm can effectively accomplish task allocation for complex task types under multiple constraints.
Case 2: A task scale was added to Case 1 to test the stability of the task planning method.The new task scenario adds 10 new tasks and randomly changes the location of the test sea area as well as the task limitations; the new task assumes that the intelligent technology test vessel is required to test 38 test tasks in different sea areas, of which 10 tasks are urgent and need to be prioritized for testing, another 18 tasks are allocated more time and can be tested normally, and the last 10 tasks are the least urgent and are placed at the end of the test.The location distribution of the dock and each sea test area is shown in Figure 15, with point 0 indicating the location of the dock, and points 1-38 indicating the 38 sea test areas that need to be reached.
The task dependencies are shown in Table 3.
with point 0 indicating the location of the dock, and points 1-38 indicating the 38 sea test areas that need to be reached.The task dependencies are shown in Table 3.Based on the above, the first stage of population distribution is first carried out, and the test task set is divided into three populations (A, B, and C) based on the task types and sequential time constraints.Based on the three populations, the second stage of population distribution is carried out, and the results of the assignments for each of the three populations are shown in Figures 16-18.
Subsequently, atomic task planning is performed.The resulting optimal gene schematic is shown in Figure 19 and the task path is shown in Figure 20.
Firstly, as can be seen from the optimal gene schematic in Figure 14, the intelligent technology test ship executes the tasks of population A first, then those of population B, and finally those of population C.This enables the test tasks to be assigned and executed by the correct population and satisfies the sequential time constraints of the test tasks.Secondly, the test tasks with dependencies are all assigned to the same voyage of the ship, which satisfies the same group constraint of the test tasks.Finally, during the atomic task planning process, it can be seen in the task path schematic in Figure 15   Based on the above, the first stage of population distribution is first carried out, and the test task set is divided into three populations (A, B, and C) based on the task types and sequential time constraints.Based on the three populations, the second stage of population distribution is carried out, and the results of the assignments for each of the three populations are shown in Figures 16-18.function.Our simulation experiments show that the task planning method for offshore testing based on hierarchical populations and a grouping genetic algorithm is still able to effectively complete the task allocation of complex task types under multiple constraints in the case of multiple communities.Subsequently, atomic task planning is performed.The resulting optimal gene schematic is shown in Figure 19 and the task path is shown in Figure 20.Firstly, as can be seen from the optimal gene schematic in Figure 14, the intelligent technology test ship executes the tasks of population A first, then those of population B, and finally those of population C.This enables the test tasks to be assigned and executed by the correct population and satisfies the sequential time constraints of the test tasks.Secondly, the test tasks with dependencies are all assigned to the same voyage of the ship, which satisfies the same group constraint of the test tasks.Finally, during the atomic task planning process, it can be seen in the task path schematic in Figure 15     Regarding solving the multi-traveler problem with cohort constraints, it can be seen in Table 3 that the algorithms in this paper have the longest running time and optimal results with the same number of iterations, and the minimum cost is reduced by 3% to 14%.In comparison with the traditional grouping genetic algorithm, the use of evolutionary operators that do not destroy the grouping situation can improve the relevance of the problem solution while reducing the number of actual voyages of the ship.The optimal value of the three algorithms' convergence curves is shown in Figures 20 and 21.It can be seen that the improved grouping genetic algorithm converges faster and has fewer iterations.As the problem size increases, the solution time does not change much.

Conclusions
With the advancement of intelligent ships, the complexity and specificity of test tasks have increased.As testing personnel formulate the sequence of test tasks for these ships, they face numerous intricate constraints.Given this scenario, the development of task planning technology for intelligent technology test ships becomes crucial.However, the existing literature lacks sufficient research on multi-constraint sea test mission planning.
To address this research gap, this paper introduces a mathematical model for the ship test The obtained results of the simulation comparing the algorithm developed in this paper with the multi-chromosome genetic algorithm and the traditional grouping genetic algorithm for 1000 iterations can be seen in Table 4. Regarding solving the multi-traveler problem with cohort constraints, it can be seen in Table 3 that the algorithms in this paper have the longest running time and optimal results with the same number of iterations, and the minimum cost is reduced by 3% to 14%.In comparison with the traditional grouping genetic algorithm, the use of evolutionary operators that do not destroy the grouping situation can improve the relevance of the problem solution while reducing the number of actual voyages of the ship.The optimal value of the three algorithms' convergence curves is shown in Figures 20 and 21.It can be seen that the improved grouping genetic algorithm converges faster and has fewer iterations.As the problem size increases, the solution time does not change much.

Conclusions
With the advancement of intelligent ships, the complexity and specificity of test tasks have increased.As testing personnel formulate the sequence of test tasks for these ships, they face numerous intricate constraints.Given this scenario, the development of task planning technology for intelligent technology test ships becomes crucial.However, the existing literature lacks sufficient research on multi-constraint sea test mission planning.
To address this research gap, this paper introduces a mathematical model for the ship test multi-constraint vehicle routing problem.The objective of the model is to minimize the ship's fixed and fuel costs while considering practical factors such as limited ship space, fuel constraints, dependencies, and time relationships.To tackle this problem, a hierarchical population group planning method is proposed, alongside a grouping genetic algorithm that leverages ethnic group characteristics.The primary contributions of this paper primarily lie in the following three aspects: (1) This paper analyzes the characteristics of multi-constraint planning for offshore test tasks using intelligent technology test ships.Based on the characteristics of these tasks, a suitable mathematical model is proposed; (2) According to the needs of sea test tasks, a hierarchical population planning method is designed, which is divided into groups according to the types of tasks.Large-scale problems are decomposed into small-scale problems, which reduces the complexity of the existing planning methods; (3) Based on the original grouping genetic algorithm, we designed a new algorithm that incorporates ethnic group characteristics.In this algorithm, a new constraint is introduced during the genetic evolution process to preserve ethnic group characteristics and narrow down the range of feasible solutions.The simulation results demonstrate that our improved grouping genetic algorithm outperforms both the two-part chromosome genetic algorithm and the traditional grouping genetic algorithm.Moreover, it is better capable of adapting to practical problems.
However, currently, there are some deficiencies in our research on multi-constraint planning for sea test missions performed by intelligent technology test ships.We are conducting additional research in the following areas: (1) dynamic planning: our current planning method focuses on static planning and does not consider dynamic scenarios.To address this, we are developing a dynamic task planning method that is suitable for this problem; (2) expanded data sets: in order to enhance the robustness and accuracy of our research, we plan to enrich and improve our data set and perform calculations using larger data sets in future studies; (3) planning software development: we are in the process of developing planning software specifically designed for smart technology test ships.This software will provide advanced planning capabilities and facilitate efficient decisionmaking during test missions; (4) integration with ship planning software: to further enhance the effectiveness of our mission planning, we will integrate our planning software with other ship planning software.This integration will improve resource allocation and coordination between different ship systems.By addressing these areas for improvement, we aim to enhance the capabilities and efficiency of multi-constraint planning for sea test missions of intelligent technology test ships.

Figure 1 .
Figure 1.Schematic diagram of task planning process.

Figure 1 .
Figure 1.Schematic diagram of task planning process.

Figure 5 .
Figure 5. Schematic diagram of local path crossing operator.

Figure 5 .
Figure 5. Schematic diagram of local path crossing operator.

Figure 6 .
Figure 6.Schematic diagram of cross-domain path crossing operator.

Figure 6 . 25 Figure 7 .
Figure 6.Schematic diagram of cross-domain path crossing operator.4.4.6.Two-Opt Local Path Variation OperatorThe two-opt neighborhood search algorithm is a local search algorithm for optimizing path problems.It attempts to optimize all possible solutions by swapping paths between two nodes in the current path until a better path is obtained.It is suitable for optimizations where the initial solution is known.The implementation of the two-opt local path variation operator is shown in Figure7.It is implemented as follows:Processes 2024, 12, 392 14 of 25

Figure 7 .
Figure 7. Schematic diagram of the two-opt local path variation operator.

Figure 8 .
Figure 8. Schematic diagram of the local path variation operator.Figure 8. Schematic diagram of the local path variation operator.

Figure 8 .
Figure 8. Schematic diagram of the local path variation operator.Figure 8. Schematic diagram of the local path variation operator.

Figure 9 .
Figure 9. Flowchart of the improved grouping genetic algorithm.

Figure 9 .
Figure 9. Flowchart of the improved grouping genetic algorithm.

Figure 11 .
Figure 11.Distribution of test locations for population A-Case1.

Figure 12 .
Figure 12.Distribution of test locations for population B-Case1.

Figure 11 .
Figure 11.Distribution of test locations for population A-Case1.

Figure 11 .
Figure 11.Distribution of test locations for population A-Case1.

Figure 12 .
Figure 12.Distribution of test locations for population B-Case1.

Figure 12 .
Figure 12.Distribution of test locations for population B-Case1.
Figure Distribution of test locations-Case2.

Figure 16 .
Figure 16.Distribution of test locations for population A-Case2.

Figure 16 .
Figure 16.Distribution of test locations for population A-Case2.

Figure 16 .
Figure 16.Distribution of test locations for population A-Case2.

Figure 17 .
Figure 17.Distribution of test locations for population B-Case2.

Figure 18 .
Figure 18.Distribution of test locations for population C-Case2.

Figure 17 .
Figure 17.Distribution of test locations for population B-Case2.

Figure 16 .
Figure 16.Distribution of test locations for population A-Case2.

Figure 17 .
Figure 17.Distribution of test locations for population B-Case2.

Figure 18 .
Figure 18.Distribution of test locations for population C-Case2.

Figure 18 .
Figure 18.Distribution of test locations for population C-Case2.

Figure 21 .
Figure 21.Comparison results of the 3 algorithms in Case 1.

Figure 21 .
Figure 21.Comparison results of the 3 algorithms in Case 1.

Figure 21 .
Figure 21.Comparison results of the 3 algorithms in Case 1.

Figure 22 .
Figure 22.Comparison results of the 3 algorithms in Case 2.

Figure 22 .
Figure 22.Comparison results of the 3 algorithms in Case 2.

Table 1 .
Literature summary table.

Table 4 .
Comparison of the computational variability of the three algorithms.