Mathematical Problems in Engineering

Volume 2017, Article ID 7139157, 16 pages

https://doi.org/10.1155/2017/7139157

## Optimization of Pesticide Spraying Tasks via Multi-UAVs Using Genetic Algorithm

^{1}School of Management, Hefei University of Technology, Hefei 230009, China^{2}Key Laboratory of Process Optimization & Intelligent Decision-Making, Ministry of Education, Hefei 230009, China

Correspondence should be addressed to He Luo; nc.ude.tufh@ehoul

Received 20 April 2017; Revised 23 August 2017; Accepted 1 October 2017; Published 12 November 2017

Academic Editor: Dylan F. Jones

Copyright © 2017 He Luo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Task allocation is the key factor in the spraying pesticides process using unmanned aerial vehicles (UAVs), and maximizing the effects of pesticide spraying is the goal of optimizing UAV pesticide spraying. In this study, we first introduce each UAV’s kinematic constraint and extend the Euclidean distance between fields to the Dubins path distance. We then analyze the two factors affecting the pesticide spraying effects, which are the type of pesticides and the temperature during the pesticide spraying. The time window of the pesticide spraying is dynamically generated according to the temperature and is introduced to the pesticide spraying efficacy function. Finally, according to the extensions, we propose a team orienteering problem with variable time windows and variable profits model. We propose the genetic algorithm to solve the above model and give the methods of encoding, crossover, and mutation in the algorithm. The experimental results show that this model and its solution method have clear advantages over the common manual allocation strategy and can provide the same results as those of the enumeration method in small-scale scenarios. In addition, the results also show that the algorithm parameter can affect the solution, and we provide the optimal parameters configuration for the algorithm.

#### 1. Introduction

With the development of artificial intelligence and unmanned processes in agriculture, UAVs have rapidly become an important platform in agricultural aviation operations due to their high efficiency, low labor intensity, and low comprehensive cost, and they have been widely applied in precision seeding, vegetation testing, pesticide spraying, and other agricultural aviation operations [1, 2]. The current operation of UAVs to carry out agricultural aviation work is mainly through manual remote control. Therefore, the actual results of the work are closely related to the skill level of the operator. The workload of the operator increases with the increased number of UAVs and tasks. This may cause high missing and repetition rates in operations. Thus, with the minimum of human intervention, using UAVs to complete agricultural aviation operations tasks autonomously has attracted widespread attention.

Pesticide spraying requires multi-UAVs to do blanket spraying on multiple farmlands. Such tasks not only need to ensure that all crops in the farmland are sprayed but also need to assign a specific task sequence and flight trajectory for each UAV in order for them to fly between multiple farmlands. For pesticide spraying assignments, without considering the UAV flight trajectory inside the farmland, each farmland can be abstracted as a task point, and the process of UAV visiting all task points can be described as the traveling salesman problem (TSP). However, due to UAV flight distance and the factors of farm size, number, and distribution, it is difficult for the UAV to traverse all the farmlands in a single flight. Therefore, we regard the assignment of pesticide spraying tasks as a team orienteering problem (TOP). With this problem, as the efficacy of pesticide spraying is mainly related to the temperature during spraying [3], the time window of spraying each farmland varies. Meanwhile, the UAV spraying strategy within a farmland can affect the UAV flight distance and change the time for the UAV to reach the next farmland, thus affecting the efficacy of the pesticide spraying for the next farmland. Therefore, the farmland cannot be abstracted as a task point in the pesticide spraying task assignment process. We need to consider not only the task allocation scheme and order of execution but also the UAV flight trajectory inside and outside of the farmland.

Thus, the optimization of multi-UAV pesticide spraying assignments studied in this paper can be described as requiring that multi-UAVs must spray pesticides for multiple farmlands in the time window and that each farmland can be sprayed by only one UAV. Under a variety of factors, we selected the appropriate farmland from the candidate farmlands and assigned the UAV for pesticide spraying, the spraying order, and spraying method to maximize the efficacy of the pesticide spraying [4]. To this end, we proposed a Dubins team orienteering problem- (DTOP-) variable time windows- (VTW-) variable profits (VP) model. Compared to the regular TOP, this model facilitated research on the following three aspects of pesticide spraying.

First, this model extends the TOP model to the DTOP model by taking into account the impact of the each UAV’s turning radius on UAV flight time. In the process of UAV pesticide spraying, it is necessary to adjust the flight direction at the edge of the farmland according to the minimum turning radius in order to achieve the spray coverage for the farmland. Meantime, the UAVs are subject to kinematics and dynamics in the course of flight. At this time, UAV flight distance is not described by the Euclidean distance in the regular TOP but by the length of the Dubins path [5]. Therefore, the distance between the two farmlands discussed in this paper is described in terms of Dubins path length. Because there are many changes in the points where the UAV enters and leaves the farmland, there are multiple Dubins paths between the two farmlands, which is described herein as the DTOP model.

Second, in the regular team orienteering problem with time windows (TOPTW), the time window is generally divided into two categories: one is from the perspective of the points of interests (POIs) to generate multiple fixed target access time windows based on the opening hours [6]; the other is from the perspective of the customers to generate multiple fixed available time windows based on each customer’s time slots [7]. Obviously, the creation of these time windows is static or predetermined. However, the time window in which the farmland can be sprayed with pesticides is affected by the temperature, and the resulting time window would change with the temperature. 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. Therefore, we determined the time window based on the dynamics of the temperature and extended the DTOP model to the DTOP-VTW model.

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. Therefore, we calculated the spraying efficacy based on the dynamics of temperature in this paper and extended the DTOP-VTW model further to the DTOP-VTW-VP model.

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. Therefore, we used the GA to solve the DTOP-VTW-VP in this study.

The rest of this paper is organized as follows. The researches related to this topic are reviewed and analyzed in Section 2. In Section 3, the DTOP-VTW-VP model under the impact of temperature is proposed. 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.

#### 2. Related Work

TOP is an extension of the orienteering problem (OP). The OP is also referred to as the selective traveling salesman problem (STSP) [15]. So, when the objective function in the STSP is only a profit and there are targets that are not visited, the STSP is the TOP [16]. 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]. However, for UAV task allocation problems, due to the constraints of the UAV’s kinematic constraints, the distance between targets visited by UAVs is no longer Euclidean distance but rather Dubins path length. 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]. According to the number of time windows, the TOP can be further divided into single time window (TOP-TW) or multiple time windows (TOP-MTW) problems, which are NP-hard problems [37]. 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]. In the TOP-MTW, each vertex can have multiple fixed time windows. 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]. The second category is to classify the time windows according to the customers’ time slots, such as those based on the fact that different visitors have different amounts of free time during the trip to generate multiple fixed time windows [7]. 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. OPVP is a special case of the optional TSP (STSP), and it is also an NP-hard problem [8]. 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. believed that the GA’s results for 60% of the benchmark instances were better than those from other heuristic algorithms [12] and proved that using GA to solve the TOP within the acceptable time can produce good results. 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].

#### 3. Problem Description and Formulation

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.e., the profit in the model) is also affected by the temperature. At the same time, the UAV’s performance, size of the farmlands, trajectories of pesticide spraying, and flight trajectories of UAVs between farmlands all have impacts on the results of the allocated tasks. 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.

##### 3.1. UAVs

Considerdenotes the set of UAVs performing the spraying tasks, and each UAV can carry only one type of pesticide. 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.(2)UAVs fly at the same cruising speed and same cruising altitude, so that the impacts of these factors on the spray effect are not considered.(3)The impacts of the external environments on a UAV’s flight trajectory are not considered.(4)UAVs can carry the pesticides required to carry out the task, but the amount of fuel carried is limited.

##### 3.2. Farmlands

Setas the starting and ending points of UAVs,as the rectangular farmlands to be sprayed with pesticides, and as a rectangle with an area of . The set of UAV beginning point, the farmlands, and ending point are When overlays spraying pesticides for , a UAV’s entering point to the farmland is , its exit point is , and it is assumed that the UAV can only leave after it has completely sprayed the entire farmland. At the same time, each farmland can only be sprayed, at most, once.

##### 3.3. Time Window

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, whenUAVs do not have any time window to do pesticide spraying; whenonly 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.