These time windows are the type of time window with uncertainties which is manifested by the uncertainty of the number and length of time windows.
Third, in the orienteering problem with variable profits OPVP , the profit from visiting each node and the visiting time are related by a concave or convex function [ 8 ].
However, for the pesticide spraying process, the profit from each farmland after spraying pesticides also changes with the temperature [ 9 ]. The relation between the efficacy of the sprayed farmland and the time is not fixed on one function, and it could be another functional relationship.
In terms of the model solution, solving the TOP has been proven to be a typical NP-hard nondeterministic polynomial-time hard problem [ 10 ]. Although the exact algorithm can be used to obtain the optimal TOP solution, it is difficult to obtain the optimal solution within the polynomial solvable time when the scale of the problem increases. Therefore, we can only use the heuristic algorithm to obtain the solution [ 11 ]. At present, there are many heuristic algorithms used to solve the TOP.
Among them, the GA has been proven to be an effective heuristic algorithm for solving the TOP [ 12 ]. It is very effective in solving standard benchmark instances and can obtain better results by adjusting the corresponding parameter configuration. In solving practical problems, the GA is also used as an efficient algorithm for solving the problem of task assignment and trajectory optimization [ 5 ].
In most cases, it exhibits better results [ 13 ] and shorter solution times [ 14 ] than other algorithms. The rest of this paper is organized as follows. The researches related to this topic are reviewed and analyzed in Section 2. The GA based on the model solution algorithm is detailed in Section 4. The numerical experiments and comparative experiments conducted are described in Section 5 , and conclusions are presented in Section 6.
TOP is an extension of the orienteering problem OP. In the TOP, several members are given, and each member starts from the same starting point within the specified time and score to the same ending point. In this process, after the target is visited by a member for the first time, the member can obtain the appropriate score. Each member needs to visit as many targets as possible, so that the total score of all members can be maximized [ 17 ].
The TOP has two characteristics [ 4 , 18 ]: the objective function is the maximum total profit and all targets are visited, at most, once. Clearly, for such problems, it is difficult to build the vehicle routing problem VRP model because the goal of the VRP is to use the minimum number of vehicles to serve all the vertices or to use the minimum total travel distance with a fixed number of vehicles [ 19 , 20 ].
Currently, the TOP has been widely used in solving tourist trip design problems [ 6 , 10 , 21 ], mobile crowdsourcing problems [ 22 — 24 ], UAV task allocation problems [ 25 , 26 ], pharmaceutical sales representative planning problems [ 27 ], and resource management allocation problem during wildfires [ 28 ]. In the above-mentioned application scenarios, the path length between targets is generally considered to be fixed, such as the distance between different POIs, the distance between hospitals, and the distance between the locations on wildfire.
To solve these types of problems, one only needs to select and combine the existing routes to maximize the total profits [ 4 , 18 ].
The Dubins path is a feasible trajectory of the minimum length over a bounded curvature trajectory at a constant rate [ 29 ], and it has been widely used in the field of UAV trajectory planning [ 5 , 30 — 32 ].
In addition, for multi-UAV pesticide spraying assignment problems, due to the many possible points for UAVs to enter and exit the farmland, there are multiple Dubins paths between farmlands. When all of the farmlands must be sprayed with pesticides, the problem can be regarded as a DTSP [ 33 — 35 ] and can be solved by using decoupling methods and transformation methods. However, in the case where UAVs cannot spray pesticides for all farmlands due to the constraints of flight distances and profits of targets, the problem is described as a Dubins traveling salesman problem DTOP.
To solve this model, we need to determine the visiting order of the targets under the condition that the trajectory length is changeable. At the same time, when the visiting of targets must be completed within a time window, the TOP is extended to the TOP with time windows [ 36 ]. In the TOP-TW, each vertex has a fixed time window [ 11 , 38 , 39 ], and the time window constraints require that the visit to the vertex must start within the specified time [ 40 ].
According to the different standards of classification, the time windows are divided into the following two categories. The first is to determine different time windows according to the available visiting time of each target. For example, the opening hours of different points are different, and the working hours of the same point are intermittent. Therefore, multiple fixed time windows are generated [ 6 ].
As for problems with the above two cases occurring at the same time, literature [ 41 ] includes a study of tourist trip problems under the constraints of opening hours of points and tourist time. There are related studies in other models on multiple time windows, such as the VRP about multiple time windows [ 42 , 43 ], and the TSP of multiple time windows [ 44 ].
However, for UAV task allocation problems, due to the fact that the time when the farmland can be sprayed with pesticides changes with the temperature, the resulting time windows have the characteristic of uncertainty. In terms of the profit of the targets, the goal of the OP and TOP is to maximize the profits of all targets after the selection of the routes.
Under normal circumstances, the profit of each target is fixed [ 45 , 46 ]. However, the spraying efficacy for each farmland changes with time in UAV task allocation problems.
The model of this type of problem is similar to the OPVP. In the OPVP, the relationship between the profit of target and time can be a concave or convex function, such as the relationship between the profit of being able to catch the fish and time in fishing operations or the relationship between the profit of the time length of viewing a program and time [ 8 ].
These relationships can randomly change with the normal distribution function [ 47 ]. However, in the process of UAV pesticide spraying task allocation, the profit of each farmland after spraying the pesticides does not necessarily change with time and completely exhibits the above-described concave or convex functional relationship.
There may be a diminishing profit relationship [ 48 ] or any other type of functional relationship. Currently, the GA [ 49 ], branch-and-cut algorithm [ 50 ], tabu search algorithm [ 51 ], simulated annealing algorithm [ 11 , 41 ], ant colony algorithm [ 38 ], and so forth are usually used to solve the regular TOP or extended TOP models. Through analyzing the results of solving 24 standard TOP benchmark instances using a heuristic algorithm, Ferreira et al.
When solving the OP with time windows [ 52 ] or OP with stochastic profits [ 47 ], the GA results have advantages over those from other heuristic algorithms. Meanwhile, in practical application processes such as UAVs task allocation [ 53 ] and mission planning [ 54 ], the GA not only requires less calculating time [ 14 ] but also gives better results [ 13 ].
The characteristics of the pesticide spraying task make the number and length of time windows in which the farmland can be sprayed affected by the temperature, and the efficacy of spraying pesticide i. In this regard, this section describes in detail the TOP proposed for UAVs to carry out pesticide spraying tasks, with variable profits and variable time windows under the impact of the ambient temperature.
During the flight, all of the UAVs have the same minimum turning radius and flight speed and carry a nozzle with a spray radius of. Considering the characteristics of UAVs performing pesticide spraying, we make the following assumptions: 1 UAVs have the ability to automatically avoid obstacles.
In the face of a collision, UAVs can use the control strategy of self-circumvention, and the resulting path deviation relative to the length of the total flight trajectory is very small and negligible. Set as the starting and ending points of UAVs, as the rectangular farmlands to be sprayed with pesticides, and as a rectangle with an area of. At the same time, each farmland can only be sprayed, at most, once.
The temperature range over which pesticide spraying can achieve a satisfactory level of efficacy is limited; therefore, only one or several time snippets can be used in a day. The time snippets are defined as the time windows in which pesticide spraying tasks can be carried out. The temperature range in which the farmland can be sprayed with pesticides generates time windows for a UAV to carry out the tasks, where and represent the beginning and ending times in the time window for the UAV to spray pesticides on the farmlands.
In general, the temperature in a day usually changes from low to high and then from high to low. This pattern can be approximated as a quadratic function distribution or a normal distribution.
The temperature range in which the farmland can be sprayed with pesticides can further generate three types of time windows, as shown in Figure 1 , when UAVs do not have any time window to do pesticide spraying; when only one time window can be generated to carry out the tasks; when only two time windows can be generated.
Thus, for the pesticide spraying assignment problems described in this paper, the number of time windows may be 1 or 2. When spraying pesticides, UAVs not only conduct the covered spray inside the farmland but also need to fly among different farmlands to complete the pesticide spraying tasks.
Therefore, there are two types of flight trajectories: that inside the farmland and that between farmlands. Within , UAVs fly along Dubins paths under kinematic constraints and conduct the covered pesticide spray using back-and-forth path strategy.
During this process, enters from point in time window. At this time, the time that it takes to spray farmland is. The back-and-forth path strategy is one of the most optimal strategies for UAVs to conduct covered pesticide spray on rectangular farmlands. It is also the most convenient control strategy to implement and is widely applied to tasks for covered areas [ 55 ]. There are two implementations of this strategy, namely, parallel track search and creeping line search [ 56 ].
For example, in Figure 2 a , there are two ways to execute a covered scan of the rectangular area. One is the parallel track search [as shown in Figure 2 b ] and the other is the creeping line search [as shown in Figure 2 c ]. Although the UAV can enter the farm from any point on the edge, the entering point with the shortest flight trajectory for UAVs to cover a rectangular area is the point that has a distance of to the apex of the farmland, according to literature [ 55 ].
We have chosen eight points, on the edges of the rectangle with a distance of to the four vertexes of the rectangle as the entry points for the UAVs to enter the rectangular area.
For a UAV to have the shortest covered flight trajectory in a rectangular region, it must enter the rectangular area from one of the eight entry points. After spraying pesticides for farmland in time window , the UAV needs to spray for. Thus, the UAV must fly along the Dubins path between and [ 57 ]. Meanwhile, the UAV does not spray pesticides when flying between farmlands. According to the generating principle of a Dubins path [ 31 ], the UAV starts from of , turns right denoted by R , and flies along the arc path and then along a straight linear path denoted by S , and finally turns left denoted by L and takes the arc flight path to arrive at of.
Owing to the back-and-forth path strategy used inside the farmland, the angles of the UAV entering or leaving the farmland must be perpendicular to the edge of the farmland. Although the lengths of the flight trajectory and flight duration in Figure 3 a are greater than those in Figure 3 b , this does not mean the result of pesticide spraying by the UAV is worse. The overall optimal result is affected by flight trajectories both inside and outside of the farmland.
Since the spraying equipment, spraying method, and other hardware and software conditions have been determined, a task profit in this paper is defined by the efficacy of pesticide spraying by a UAV on a farmland. When a UAV sprays pesticides on farmlands, the spraying efficacy is closely related to temperature and temperature will differ with changes in time.
Therefore, the efficacy of a UAV spraying pesticides on farmland within time window can be described by the efficacy function over time. While using different type of pesticides, may manifest as a concave function, convex function, normal distribution, or linear decreasing function. Suppose that begins to spray pesticides on farmland with an area of at in time window , and it takes for to spray the entire farmland in time window.
The task profit can be defined as. UAVs and farmlands are given, and each farmland has time windows. Note that and are the opening time and closing time in time window for UAV to spray pesticides on farmland. Two decision variables are used: if UAV completes the pesticide spraying task for farmland within time window and 0 otherwise; if UAV begins to fly between and within time window and 0 otherwise.
The objective function of optimization is defined as the maximization of the total task profits for all the UAVs. The objective function 10 maximizes the total tasks profits for all the UAVs. Constraint 11 ensures that all routes have the same starting point and ending point, and the number of routes is the sum of UAV routes in each time window. Constraint 12 ensures the connectivity of the route. Constraint 13 shows that the spraying time is needed when visiting farmlands. Constraint 14 ensures that each farmland can be visited at most once in all of time windows.
Constraint 15 ensures that the time of spraying must be within time window. Constraint 17 defines the decision variables. In the algorithm, the initial population is generated by amount of chromosomes which represent different solutions of this problem, and it is updated by three operators: selection, crossover, and mutation.
The process continues until the satisfactory solution is obtained or the maximum iteration is reached. The specific process is shown in Figure 4. Chromosome encoding represents a selected solution to the problem. Therefore, the chromosome corresponds to the serial number of the farmland, UAV and entry point of the farmland, and the starting time of the UAV visiting the first farmland in each route.
The chromosome shown in Table 1 describes a potential solution for and to spray pesticides on seven farmlands in two time windows. In the first time window [, ], first enters from at and then enters from and finally returns to the starting point; first enters from at and then enters from and finally returns to the starting point.
Similarly, in the second time window [, ], enters from at and returns to the starting point after completing the tasks; enters from at and returns to the starting point after completing the tasks.
In this scenario, is not sprayed with pesticides. In the algorithm, the objective function in formulation 10 is defined as the fitness function. The chromosome selection operation among population is carried out through the roulette wheel method. Therefore, the fitter the chromosome becomes, the higher probability it has to be selected to be included into the parental population. Through the crossover operator, the offspring can inherit the relatively good genes from the parent.
Single-point crossover and multipoint crossover are some common approaches of crossover. Those crossover approaches are all integral operations of part gene in parent chromosomes. That is, for the selected part of parent chromosomes, the gene bits are either entirely replaced with new chromosome structures or are not replaced.
According to the chromosome encoding, the same UAV must have the identical starting time in the chromosome. However, in these common approaches, the offspring chromosomes can hardly satisfy this constraint. A new crossover operator has been proposed. That is, the crossover site is randomly decided and the genes are replaced in the two parent chromosomes. The fourth row of the gene represents the starting time of the UAV visiting the first farmland via the same route.
Therefore, in the crossed offspring chromosomes, it is also necessary to replace the fourth row of the other genes involved in the same route with the new time. The crossover example is illustrated in Figure 5.
For offspring A, the second column of parent B is copied to the fifth column of parent A, and since the starting time of visiting the first farmland is adjusted to 8. Similarly, another offspring chromosome, B, can be obtained. Mutation operator is done to prevent the GA from falling into the local optimum.
In our study, the chromosome mutations include four ways, which are the mutation of the first row of farmland order and the single-point mutations of the second, third, and fourth rows of the chromosomes, that is, the mutation of the visiting UAV, entry point, and entry time. According to the mutation probability, these four kinds of mutation may not occur, or one or more than one kinds may occur.
The mutation example is illustrated in Figure 6 ; the second and third rows of chromosome A mutated. The second row and second column of chromosome A mutated from 0 to 2. This mutation represents the fact that will visit , which would not be visited originally. The third row and third column mutated from 5 to 7. This mutation represents the fact that the entry point of visiting farmland was replaced by. To ensure the consistency of the starting time of visiting the first farmland in the first time window, on the basis of the above mutation, the second column and fourth row of chromosome B were updated to 9.
All of the results are the average results of the same experiment that was run ten times. The relevant parameters in the experiment are defined as follows. In the experiment, we used two UAVs, and , with different maximum flight time.
The heading angle of taking-off and heading angle of returning are all 0. The detailed UAV configurations are shown in Table 2. The targets described in regular TOP and related TOP extended models are all point targets, and thereby the corresponding benchmark instances [ 4 , 16 ] are also generated for the point targets. At present, there are no benchmark instances for the problem studied and reported in this paper. Therefore, we randomly generated four instances according to the number of farmlands, labeled , , , and.
All of the farmlands in the instances are rectangles, as shown by the shaded areas in Figure 7. We have chosen four pesticides with different potencies in this study. Table 3 shows the temperature range required for the different pesticides to be effective and the spraying time windows generated according to the temperature curve shown in Figure 8 on the day the pesticides were sprayed.
When spraying pesticides by UAVs in real life, the UAVs are operated mainly by operators who assign spraying tasks manually. In this experiment, we used two UAVs of the type to carry out pesticide spraying task on the six farmlands in area , the thirty farmlands in area , and the fifty farmlands in area under the environment , respectively.
The UAVs needed to return to after completing the tasks. In this experiment, the crossover probability is 0. The population size is for and for and. Currently, there are two typical manual assignments.
The first is to spray in the order of the serial number of the farmlands FCFS and the second is to divide the environmental areas based on the number of UAVs AA and route each UAV to spray the multiple farmlands in its assigned area in an orderly manner.
According to the different approaches used to partition the area, this AA strategy can be further subdivided into horizontal partitioning HAA and vertical partitioning strategy VAA. At this time, can only spray farmlands 1-D, 2, and 3; can only spray farmlands 1-U, 4, and 5; and no UAV sprays farmlands 6-U and 6-D, shown in Figure 9 a ; when using the VAA strategy, farmland 4 is divided into two parts, namely, farmlands 4-L and 4-R.
At this time, can only spray farmlands 1, 2, and 3; can only spray farmlands 4-L and 6; and there is no UAV to spray farmlands 4-R and 5, shown in Figure 9 b. Furthermore, on the basis of the HAA and VAA partitioning strategies, it is not necessary to use the simple sequential spraying method to spray pesticides after the partitioning. The experimental results for the six strategies are listed in Table 4.
In actual operations, the FCFS strategy is very simple and convenient but ignores the many factors that influence the efficacy of pesticide spraying, causing low spraying efficiency or even the need to respray pesticides on farmlands. When making a simple partition of area according to the number of UAVs, the efficacy of pesticide spraying exhibits a certain degree of improvement from the FCFS strategy, although after the partitioning the FCFS strategy is still used for pesticide spraying.
For example, the profit of VAA strategies is 8. However, if we use the GA to solve for the assignment after partitioning, the spraying efficacy can be further improved. The average profit of In the ten experiments, the deviation from the lowest profit of The stability of the solution, therefore, is reasonable.
Similarly, for areas and with larger number of farmlands, the solutions of the GA are also better than other strategies. This experiment is to analyze the quality of the solution of the problem obtained by the GA. In small-scale scenario, enumeration method is a suitable method to obtain optimal solution of this problem.
So, we choose enumeration method to obtain the optimal solution first, and then the solution obtained by the GA at the same scale is compared to it. In both methods, the time was discretized to 0. Firstly, we considered the case of two UAVs of type spraying pesticides on two farmlands of area in the environment.
The two UAVs returned to after completing the tasks. Furthermore, we considered spraying six farmlands of area under the same conditions. In the experiment, the crossover probability is 0. The population size is for and for. As shown in Table 5 , the results obtained by the GA are the same as the enumeration results. Firstly, we considered the case of two UAVs of type spraying pesticide on 30 farmlands in area under the environment.
The two UAVs needed to return to after completing the tasks. In this experiment, the population size is , the number of iterations is , and the different combinations of crossover and mutation probabilities are tested.
The results of the experiment are shown in Table 6. Furthermore, we considered the case of two UAVs of type spraying pesticide on 50 farmlands in area under environment. In this experiment, the population size is , the number of iterations is , and different combinations of crossover and mutation probabilities are tested. The experimental results are shown in Table 7. As can be seen from the above results, in the case of different numbers of farmlands, different temperature curves, different time windows, and different pesticide efficacy profit functions, using the GA to solve the DTOP-VTW-VP is more stable in general.
However, the demonstrated effects with different crossover and mutation probabilities are moderately different Figure When the crossover probability is fixed, the profit of the experiment increases in general with the increase of mutation probability.
The profit reaches the maximum when the crossover probability is 0. When the mutation probability is fixed, the profit of the experiment also shows an upward trend as a whole with the increase of crossover probability. The model extends the distance between two points in the regular TOP to the Dubins path distance and dynamically determines time windows of pesticide spraying according to the temperature conditions at the time the UAVs perform the tasks. Based on these factors, maximizing the profit of pesticide spraying was set as the optimization target of the model, thus avoiding the problem that in the practical application process the pesticide spraying result is not satisfied after the pesticide spraying task is completed.
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