A Multiregional Agricultural Machinery Scheduling Method Based on Hybrid Particle Swarm Optimization Algorithm

Abstract

The reasonable scheduling of agricultural machinery can avoid their purposeless flow during the operational service and reduce the scheduling cost of agricultural machinery service centers. In this research, a multiregional agricultural machinery scheduling model with a time window was established considering the timeliness of agricultural machinery operation. This model was divided into two stages: In the first stage, regions were divided through the Voronoi diagram, and farmlands were distributed to intraregional service centers. In the second stage, the model was solved using the hybrid particle swarm optimization (HPSO). The algorithm improves the performance of the algorithm by introducing a crossover, mutation, and particle elimination mechanism, and by using a linear differential to reduce the inertia weight and trigonometric function learning factor. Next, the accuracy and effectiveness of the algorithm are verified by different experimental samples. The results show that the algorithm can effectively reduce the scheduling cost, and has the advantages of strong global optimization ability, high stability, and fast convergence speed. Subsequent algorithm comparison proves that HPSO has better performance in different situations, can effectively solve the scheduling problem, and provides a reasonable scheduling scheme for multiarea and multifarmland operations.

1. Introduction

Modernized agriculture is an important constituent part of national and international competitiveness. The application of modern information technologies such as agricultural IoT [1] contributes to a higher intelligence level than traditional agriculture. Agricultural mechanization, a significant mark of modernized agriculture, has been realized in developed countries. However, many developing countries stay at the transformational stage from manpower & animal power operation to mechanized operation. Agricultural machinery means an expensive input for small households in developing countries, and not all peasant households can afford large tractors, harvesters, etc.

In some countries such as China, the land area of peasant households is usually very small, so if agricultural machines are purchased by each household, the utilization rate of agricultural machinery will be too low, thus leading to a waste of resources. To solve the abovementioned problems, some countries have tracked out the operational service mode of agricultural machinery, i.e., agricultural machinery service demanders such as ordinary households release their own operational requirements, while agricultural machinery service centers such as agricultural machinery cooperatives carry out operational services. In this process, agricultural machinery service centers should allocate appropriate agricultural machines to numerous farmland operation orders with demands already released, plan the optimal operation route, and complete the operation scheduling of agricultural machinery. Due to the interregional unbalanced development of agricultural resources, establishing a reasonable agricultural machinery scheduling plan will be conducive to resource integration [2], improve the operation efficiency, and reduce the operation cost.

In many countries, great importance has been attached to the operational services of agricultural machinery. However, the development progress varies from country to country. In China, the agricultural machinery operation service industry is developing rapidly and has received strong support from the government. In 2020, the operational service area of agricultural machinery cooperatives in China reached 56,509,000 hectares, and the cross-regional operational service area reached 19,900,000 hectares. In many provinces, agricultural machinery dispatching related services have been built [3]. Beidou navigation and other technologies have been applied to agricultural machinery operation services, which has also spawned a lot of research on agricultural machinery intelligent dispatching platform [4,5]. The development situations in relevant countries are listed in

Table 1. Agricultural machinery operation service development.

Country	Application	Features	Literature Source
Bangladesh	The small-scale farmers who own the machines, typically hire out their services to others for additional income	People (as a group or individual) who can afford a machine would invest in it, while the marginal farmers would get the service on a rental basis. Three different types of ownership models have been practiced in the country i.e., group ownership, cooperative ownership, and individual ownership. This ensures the appropriate use of farm machines and increases the farmers’ access to these machines	[6,7]
Ghana	The majority of farmers within the local farming system who are using agricultural machinery either through ownership or through a service rental market	The machinery use is not uniform across the northern savannah areas of Ghana but varies by location. The high level of tractor use in most districts, despite a low stock of tractors. The 83 percent of tractor users hired in machinery services for plowing. This confirms the extensive use of the hiring market by farmers who do not own their own machinery.	[8]
Myanmar	A vibrant informal private sector dominates the outsourcing services market in Myanmar.	Rental of small machines (e.g., two-wheel tractors and threshers) is offered mainly by farm households with medium or large landholdings, within the immediate vicinity of their villages. Large farms are the main outsourcing providers for large machinery, and often acquire machines principally to rent out to others. Four-wheel tractor owners usually service farms in the surrounding area, whereas combine harvester owners often provide services both locally and in more distant locations.	[9]
Ethiopia	Smallholders are increasingly using mechanization, especially in the wheat sector. Commercial mechanization service providers are rapidly emerging that provide plowing, harrowing, and harvesting services for a fee.	The level of ownership and use of machines for agricultural activities by smallholders is low. Mechanization is more prevalent in the case of wheat. The majority of farmers who employ agricultural machinery in their farming rely on rental services. Very few farmers use their own tractor, combine harvester, or thresher. Own machines only make up 19 percent of machine use for land preparation, 22 percent for harvesting, and 2 percent for threshing.	[10]
Thailand	The small-scale growers will get mechanical harvester services from the medium and larger growers, or third-party inbound logistics providers who own mechanical harvesters.	If the mills could manage the needs of small-scale growers in terms of time period and a number of mechanical harvesters required and the availability of mechanical harvesters belonging to third-party inbound logistics providers or medium and large-scale growers more efficiently, the cost of sugar production by small-scale growers and the mills would be lower. The medium and large-scale growers or third-party inbound logistics providers (i.e., agents), would not only get paid on time as contracted but could also earn more revenue from sharing their available mechanical harvesters with others.	[11]
Nigeria, India	Uber for tractors, which aims to enable farmers to access tractor hire services in a way that is similar to the Uber service for ride-hailing.	Uber for tractor models can reduce transaction costs for service providers, in particular the owners of several tractors, by enabling the monitoring of tractors and operators through GPS devices. Farmers who access services rely on ‘‘analog” solutions—booking agents and phone calls—rather than a smartphone app to request services.	[12,13]
China	It has formed a supply system of agricultural machinery operation services based on agricultural machinery households, including agricultural machinery households, agricultural machinery professional cooperatives, agricultural machinery operation service companies, land trusteeship service organizations, etc.	The service mode of agricultural machinery operation has evolved from single to diversified. At present, agricultural machinery operation service modes include order operation mode, trust operation mode, contract operation mode, cross-area operation mode, and other types.	[3]
Japan	Local machinery service providers conduct business in the agricultural field as custom-hire services.	The cost of farm operations could be reduced to almost half by custom hiring of the machinery services. Customized services for different types of crops and accurate services through GNSS.	[14]
France	Large farms purchase machines by themselves, and small- and medium-sized farms have purchased or organized agricultural machinery cooperatives in a cooperative manner	According to the amount paid by the members, there are times of free use of agricultural machinery. If the number of times exceeds a certain number, they need to pay the rental fee. The agricultural machinery and equipment have integrated modern microelectronic technology, control technology, and information technology. Field automatic navigation systems and ma-chine vision systems have begun to be applied to tractors and self-propelled agricultural machinery.	[15]
United States	A professional organization of grain combine harvesters have appeared. The lessor of agricultural machinery signs a contract with the farmer to provide professional services for the farm according to the contract	Self-purchasing and self-use models are currently the mainstream of small agricultural machinery. Farmers can also rent agricultural machinery to other farmers for use. Most of the machinery leased by the company is large-scale agricultural machinery; small-scale agricultural machinery rental services are not provided. Agriculture in the United States is a highly mechanized, large-scale, automated, specialized, and precise operation.	[15]
Korea	The agricultural association directly operates the agricultural machinery service center.	The agricultural association provides a series of services such as sowing, planting, harvesting and packaging.	[16]

.

The development of agricultural machinery operation services has led to the emergence of agricultural machinery scheduling. Agricultural machinery scheduling can help agricultural machinery operation services reduce operating costs and improve agricultural machinery resource utilization and operating efficiency. In the existing studies regarding agricultural machinery scheduling, many models and algorithms have been established, especially with the application of intelligent algorithms such as simulated annealing algorithm [17], ant colony algorithm [18], tabu search algorithm [19,20], genetic algorithm [21], and particle swarm optimization (PSO) algorithm [22], considerable research progress has been achieved with regard to agricultural machinery scheduling.

Agricultural machinery operation has a high timeliness requirement. In the operation season, agricultural machinery should be scheduled to complete the task in time, which is conducive to improving the efficiency of agricultural production, avoiding the loss of farmers due to time factors, reducing the cost of agricultural machinery operation, and reducing the waste of agricultural machinery resources. A lot of scheduling schemes and models have been proposed by scholars. For instance, Basnet et al. established a scheduling model from the service company to multiple farmland operation points for agricultural machinery operational services, aiming at the shortest operation time of each farm [19]. Foulds investigated the harvesting scheduling problem of multiple farmland operation points, with the objective of minimizing the total operation time [23]. Edwards et al. constructed a route planning model targeting the shortest distance [24]. Orfanou et al. explored an agricultural machinery scheduling model based on owner selection [25]. Ali and Seyyedhasani et al. established a vehicle scheduling model with capacity limitations [26,27].

Cao et al. planned machinery group operation tasks using the ant colony algorithm to reach the shortest driving distance of agricultural machinery [28]. Li et al. used the graphics transformation method to transform the agricultural machinery scheduling problem into a multi-traveling-salesman problem with a time window [29]. Zhang solved the agricultural machinery scheduling problem through the table-manipulation method and gave a scheduling scheme leading to the minimum cost [30]. Wang established an agricultural machinery scheduling model with a time window via genetic algorithm [31]. Wu and Wang et al. probed a time window-based mathematical spatial–temporal scheduling model of agricultural machinery resources and solved it through a heuristic algorithm [32,33]. Wu and Zhang et al. established an agricultural machinery emergency scheduling model and a cross-regional operation model [34,35]. Although the above research has made some progress in different aspects, for the agricultural machinery scheduling problem the consideration is not comprehensive, and the application and improvement of the algorithm is still relatively basic.

Generally, the multi-depot vehicle routing problem with time window (MDVRPTW) is decomposed into a single-depot VRPTW [36]. The Vehicle Routing Problem (VRP) is a nondeterministic polynomial(NP)-hard problem [37], and MDVRPTW is an advanced version of the VRP problem, adding more constraints, making the solution of the original NP-complex vehicle scheduling problem even more difficult. Then single depots are solved through the HPSO algorithm [38], and the results are verified.

The PSO algorithm is an intelligent bionic algorithm that simulates bird swarm foraging. It has been applied in many fields of agricultural production. For example, Meng et al. applied PSO algorithm to agricultural precision fertilization control [39]. Cheng et al. used improved particle swarm optimization algorithm to improve the speed-regulation characteristics of agricultural machinery CVT [40]. In the field of agricultural machinery scheduling, PSO has also made some progress [41,42]. In this research, a multiregional scheduling model with time window was constructed by fully considering the shortest distance and the operation cost of service centers [43] based on the characteristics of multi-service center scheduling. following the [44,45] multi-stage solution idea, the Voronoi diagram is used to divide the region [46,47,48]. Then, the hybrid particle swarm optimization algorithm is designed to solve the single-region scheduling problem, and the optimal scheduling scheme is obtained and verified by experiments. This paper first explains the research on agricultural machinery operation service scheduling and agricultural machinery scheduling in various countries, then designs models and algorithms to solve the scheduling problem, and comprehensively improves the scheduling algorithm, hoping to bring some inspiration to relevant researchers.

2. Materials and Methods

2.1. Problem Description

In this study, the multiarea multimachine operation scheduling problem is described as: M cooperative centers dispatch agricultural machinery to N farmlands for operation. Each cooperative center has sufficient agricultural machinery, agricultural machinery has different parameters, and different types of agricultural machinery have different starting costs, maximum operating capacity, and maximum driving distance. Before the study, the following conditions are first declared:

(1)

Each piece of agricultural machinery departs from their respective service center, and finally returns to this service center.

(2)

Each farmland can be served only by one piece of agricultural machinery for the agricultural machinery operation.

(3)

To arrive at the farmland, each piece of agricultural machinery should meet the time window constraint. In this research, the soft time window constraint was used; if the machinery arrived earlier than the operation time, a waiting cost would be generated. If it reached later than the operation time, a penalty cost would be incurred.

(4)

Each operation route should not exceed the maximum driving distance of agricultural machinery.

(5)

The operational quantity of each farmland should not exceed the operational capability of vehicles.

(6)

The quantity of agricultural machinery used should not exceed the total quantity of agricultural machinery.

(7)

The model optimization aims to minimize the operation cost of agricultural machinery.

2.2. Mathematical Model

First, the following mathematical symbols were defined:

m denotes the cooperative center set, m = {1, 2, …, M}; n denotes the set of job nodes, n = {1, 2, …, N}; s denotes the set of agricultural machinery, s = {1, 2, …, K}; the maximum operating capacity set of agricultural machinery, Q = {Q₁, Q₂, …, Q_K}; d represents the maximum travel distance set of agricultural machinery, D = {D₁, D₂, …, D_k}; C₀ represents the starting cost set of agricultural machinery, C₀ = {C₀₁, C₀₂, …, C_0K}. The mathematical model of MDVRPTW is determined with the minimum transportation cost as the objective function, and the parameter definition is shown in

Table 2. Glossary.

Parameter	Meaning
Z	Scheduling cost
C0k	Agricultural machinery k start-up cost
C1, C2, C3	Driving cost, waiting cost, punishment cost
dij	The distance from i to j
xijk	Agricultural machinery k travels from i to j
Pk	Agricultural machinery k time cost
qj	The task amount of plot j
Qk	Operation of agricultural machinery k
Dk	The maximum driving distance of agricultural machinery k
wki	Agricultural machinery k reaches the waiting time of node i earlier than the service time.
ETi, LTi	The earliest working time of node i, the latest working time of node i
Ti	The job time of the i th node
tki	The time when agricultural machinery k reaches node i
V	Running speed

.

Scheduling objective:

$$MinZ=\sum _{k=1}^K{C}_{0k}+C_1\sum _{i=1}^{N+M}\sum _{j=1}^{N+M}\sum _{k=1}^K{d}_{ij}{x}_{ijk}+\sum _{k=1}^K{P}_k$$

(1)

Constraint conditions:

$$\sum _{i=1}^M\sum _{j=1}^N\sum _{k=1}^K{x}_{ijk}=K$$

(2)

$$\sum _{i=1}^N{x}_{ijk}=\sum _{i=1}^N{x}_{jik};\left(k\in s,i,j\in n\right)$$

(3)

$$\sum _{i=1}^{N+M}\sum _{j=1}^{N+M}{x}_{ijk}{q}_j\le Q_k;(k\in s)$$

(4)

$$\sum _{k=1}^K\sum _{j=1}^N{x}_{ijk}=1;(i\in n)$$

(5)

$$\sum _{i=1}^{N+M}\sum _{j=1}^{N+M}{x}_{ijk}{d}_{ij}\le D_k;(k\in s)$$

(6)

$$\sum _{i=1}^K{Q}_i\ge \sum _{j=1}^N{q}_j$$

(7)

$$w_{ki}=\mathit{max}(0,E{T}_i-t_{ki});(i\in n,k\in s)$$

(8)

$$t_{ij}=\frac{d_{ij}}{V};(i,j\in n\cup m)$$

(9)

$$t_{ki}=t_{kj}+w_{ki}+T_j+t_{ij};(i,j\in n\cup m,k\in s)$$

(10)

$$P_k=\left\{\begin{array}{c}{C}_2\left(E{T}_i-t_{ki}\right);\\ C_3\left(t_{ki}-L{T}_i\right);\\ 0,others;\end{array}\right.(i\in n,k\in s)$$

(11)

$$x_{ijk}=\left\{\begin{array}{c}1, Agricultural machinery k travels from i to j\hfill \\ 0, Agricultural machinery k does not travel from i to j\end{array}\right.$$

(12)

The Equation (1) represents the minimum total scheduling cost as the objective function, which includes three parts. The first part is the starting cost of all agricultural machinery, the second part is the driving cost of all agricultural machinery, and the third part is the time cost of all agricultural machinery. The sum of the three is the scheduling cost, and the other Equations are the constraints. Equation (2) means that all agricultural machinery departs from service centers and returns after completing all operation tasks. Equation (3) represents the equal numbers of agricultural machinery coming in and out of farmlands. Equation (4) indicates that the operational capability of each piece of agricultural machinery does not exceed its maximum operational capability limit. As revealed by Equation (5), each farmland is served by a piece of agricultural machinery only once. Equation (6) means that each piece of agricultural machinery does not exceed the maximum driving distance. Equation (7) restrics by the total number of pieces of agricultural machinery; the operation capacity of all agricultural machinery is greater than or equal to the task volume of farmland operation points. Equation (8) is a waiting time constraint. Equation (9) represents an operation time constraint. Equation (10) is a time constraint, namely, time to arrive at the present farmland = time to arrive at the previous farmland + waiting time + operation time of the previous farmland + operation time from the previous farmland to the present farmland. Equation (11) stands for a penalty cost constraint: If a piece of agricultural machinery arrives at the farmland earlier than the earliest operation time, a waiting cost is generated. A penalty cost will be generated if it arrives later than the latest operation time. Equation (12) is a decision variable, reflecting whether the agricultural machinery k drives from the farmland I to the farmland j.

2.3. Regional Classification of Voronoi Diagram

A Voronoi diagram consists of one group of continuous polygons formed by a perpendicular bisector of the straight line connecting two adjacent points. A Voronoi diagram has the following features:

(1)

Only one discrete point data is contained within the Voronoi diagram.

(2)

In the Voronoi diagram, the distance from a point to the corresponding discrete point is the shortest.

(3)

The distances from a point on the edge of polygon to discrete points at two sides are equal.

Since the MDVRPTW problem is usually solved by decomposing it into multiple VRPSTW problems. The present mainstream method is clustering algorithm, which is a classical algorithm featured by conciseness and high speed, but it is also subjected to some defects: the established data points must be known, namely, the coordinates of each farmland need to be known. The service center to which each farmland is allocated can be identified using a Voronoi diagram via graphical interfaces. In addition, the clustering algorithm needs to traverse all data points and find the center the closest to discrete data points by calculating distances, while what is generated by the Voronoi diagram is a graphical result. Moreover, the Voronoi diagram can obtain the visualized result faster regarding to which center a farmland belongs (a local problem).

Compared to the clustering algorithm, the Voronoi diagram simplifies the steps and accelerates the early-stage algorithm convergence rate after regional planning, thus performing better in local operations. For example, to plan the route to arrive at one farmland, it is only necessary to judge to which region it belongs, and then the local HPSO algorithm is adopted, which makes it more convenient for the local calculation.

In this research, regional classification was performed using a Voronoi diagram based on the algorithm operation efficiency and the features of the Voronoi diagram, which is expressed as follows: Given a discrete point set P = {P₁, P₂, …, P_i, …, P_M} on the plane, the space is segmented into M regions using Voronoi diagram, and each discrete point corresponds to one region R_(Pi), which is defined as below:

$$R_{\left(Pi\right)}=\left\{x|\left|P_i-x\right|\le \left|P_j-x\right|,\forall j\ne i\right\},1\le i\le n,1\le j\le n\}$$

(13)

Equation (13) means that the distance of each point within the region V_(Pi) to P_i is smaller than the distance to other discrete points. With M service centers as discrete points and N as farmlands, the regional division based on the Voronoi diagram with the maximum sample size (M = 10, N = 120) is displayed in Figure 1, where blue points stand for service centers and green ones represent farmlands. It can be seen from Figure 1 that the distance from the farmlands within the region R_(Pi) to P_i is the shortest, so the priority should be given to farmlands inside R_(Pi) during operation, which contributes to a shorter scheduling route and a lower cost.

The regional division steps through the Voronoi diagram are as follows:

Step 1: The M service centers and N farmlands are determined, the distance from farmlands to service centers is calculated, and the boundaries are determined.

Step 2: Voronoi diagram-based regional division is implemented.

Step 3: The ray method is used to judge whether a farmland is located within the region of the Voronoi diagram. With the farmland N_i as the endpoint, a ray is drawn towards a random direction, and the number of interaction points between this ray and the polygon is calculated. A λ value is set, and N is within the polygon if it is an odd number and λ > 0; N is outside the polygon if it is an even number and λ < 0.

Step 4: The farmlands within the polygon are attributed to the corresponding service centers, thus forming M point sets.

2.4. HPSO Algorithm

As an optimization algorithm based on the swarm intelligence theory, the PSO algorithm searches for the optimal solution by simulating the foraging process of bird flocks and continuously evolving the collaborative and competitive relations in bird flocks. A standard particle swam flies, at a certain speed, within a D-dimensional particle swarm space consisting of pop_s particles; the particle I is expressed as X_i = (X_i1, X_i2, …, X_id), the corresponding velocity is v_i = (v_i1, v_i2, …, v_id), the individual optimal solution searched by particles is pbest_i = (pbest_i1, pbest_i2, …, pbest_id), and the group optimal solution searched by the particle swarm is gbest_i = (gbest_i1, gbest_i2, …, gbest_id).When the particle i proceeds to the k + 1 (th) iteration, the d-dimensional velocity and position are updated through Equations (14) and (15), respectively:

$$v_{id}^{k+1}=w{v}_{id}^k+c_1{r}_1(pbes{t}_{id}^k-x_{id}^k)+c_2{r}_2(gbes{t}_{gd}^k-x_{gd}^k)$$

(14)

$$x_{id}^{k+1}=x_{id}^k+v_{id}^{k+1}$$

(15)

where w, c_1, and c₂ represent the inertia weight, self-learning factor, and social learning factor, respectively. r₁ and r₂ are random numbers within (0, 1). Aiming at the improvement strategy of particle swarm optimization algorithm, many scholars have made many attempts in different fields. Zheng et al. improved the tractor field operation emission reduction problem by optimizing the inertia factor [49]. Chen et al. introduced the advantages of combining particle swarm optimization and a genetic algorithm in solving distributed planning problems [50]. In this paper, factor optimization, particle elimination, and algorithms are combined for the improvement direction of a particle swarm optimization algorithm. For the factor optimization link, linear differential decline inertia weights [51] were used in this research. In the early algorithm operation stage, w declined slowly, so the global search ability of the algorithm was strengthened. In the later phase, w declined at an accelerated speed, thus enhancing the local search ability of the algorithm. Meanwhile, c₁ was firstly great and then decreased while c₂ was firstly small and then increased, since a greater c₁ meant a strong global search ability, and a greater c₂ indicated a stronger local search ability. In the meantime, the Equation became nonlinear because of trigonometric function learning factor [52]. In most results, a faster convergence rate and a higher convergence precision were manifested.

At the same time, considering the convergence of the algorithm, the particle swarm is added to the elimination mechanism, because the larger particle population can enrich the solution of the particle, but also reduce the operation speed. At the beginning of the algorithm, more particles far away from the current optimal position are eliminated. Considering that the number of particles is an integer, the particles are eliminated according to the formula, and the expression is approximately a curve. Establish w, c₁, c₂, and pops expressions, where the parameter definitions are shown in Table 3.

Table 3. Equation parameters.

Parameter	Meaning
wmax, wmin, wk, w	Maximum inertia weight, minimum inertia weight, current inertia weight, initial inertia weight
tmax, t	Maximum number of iterations, current number of iterations
c1,c2	Self-learning factor, social learning factor
σ,δ	Constant 2, constant 0.5
popmax, popmin	Maximum particle number, minimum particle number.
pop	Current particle number

Table 3. Equation parameters.

Parameter	Meaning
wmax, wmin, wk, w	Maximum inertia weight, minimum inertia weight, current inertia weight, initial inertia weight
tmax, t	Maximum number of iterations, current number of iterations
c1,c2	Self-learning factor, social learning factor
σ,δ	Constant 2, constant 0.5
popmax, popmin	Maximum particle number, minimum particle number.
pop	Current particle number

$$\frac{d\omega }{dt}=\frac{2(w_{\mathit{max}}-w_{min})}{t_{max}^2}\times t$$

(16)

$$w_k=w_{max}-\frac{(w_{max}-w_{min})}{t_{max}^2}\times t^2$$

(17)

$$c_1=\partial \times \mathit{sin}((1-\frac{t}{t_{max}})\times \frac{\pi }{2})+\delta$$

(18)

$$c_2=\partial \times \mathit{cos}((1-\frac{t}{t_{max}})\times \frac{\pi }{2})+\delta$$

(19)

$$pop={pop}_{max}-\frac{({pop}_{max}-{pop}_{min})}{t_{max}^2}\times (2t_{max}t-t^2)$$

(20)

In this research, the conceptions of crossover and mutation were introduced based on the standard PSO algorithm according to the fact that the particle swarm would continuously learn the individual optimal solution and the group optimal solution in order to strengthen the global search ability of the algorithm and implement the agricultural machinery scheduling algorithm based on the HPSO algorithm. The algorithm flowchart is exhibited in Figure 2.

Algorithm steps:

Step 1: The particle swarm is initialized. The number of particles in the particle swarm is set to be pop_s. pop_i is the i(th) particle (i = 1, 2, …, pop_s). The number of iterations is set as T_max.

Step 2: Each particle in pop is evaluated. f_i denotes the fitness of particles. The individual optimal value and the group optimal value are updated. The fitness of particles can be obtained by Equations (1) and (21).

$$f_i=1/Z$$

(21)

Step 3: Crossover is introduced to select the random positive sequential position, and roulette wheel selection is performed to accept the optimal solution of individual particles, the present optimal solution Pbest_i, and the global optimal solution Gbest_i at the probability of w/(w + c₁ + c₂), c₁/(w + c₁ + c₂) and c₂/(w + c₁ + c₂) respectively. Only one is determined as the parent for the sequential crossover operation with particles (Figure 3). parent₁ represents a particle and parent₂ stands for the selection result at one probability. After sequential crossover, two offspring, child₁ and child_2, are generated, followed by their fitness evaluation. The offspring with the highest fitness is selected as new particles.

Step 4: The mutation method is introduced and single-point mutation is adopted. If the particles after mutation show a better fitness than old particles, replacement should be performed, or otherwise it should not be implemented.

Step 5: The poor particles are eliminated, and the particles whose particle position is far from the current optimal particle, that is, the fitness is far lower than the current optimal particle, are eliminated.

Step 6: The fitness of each particle in pop is evaluated, followed by repeated iterations. Judge whether the present number of iterations reaches T_max; if yes, output the optimal solution, or otherwise return to Step 2.

3. Results and Discussion

To verify the scheduling capability of the algorithm, the data of four samples were selected as experimental samples for solving. The largest one among the first three sampled data points was taken, and farmland positions were randomly generated. The algorithm operated under the following environment: AMD Ryzen 7 5700G 3.8GHz (processor) (AMD, Silicon Valley, NC, USA), 16 G (memory), Windows11 (operation system), and Python Python 3.10.0 (programming language). In the data of the largest experimental sample, M = 10 and N = 120. The data of some farmlands are listed in

Table 4. Basic information on some farmlands.

Farmland No.	X-Coordinate	Y-Coordinate	The Earliest Operation Time (h)	The Latest Operation Time (h)	Operation Duration (h)	Operation Area (hm2)
1	47	69	6	18	0.2	1.6
2	88	70	6	17	0.2	1.6
3	91	6	8	18	0.2	1.8
4	86	47	9	19	0.4	1.4
5	96	85	6	18	0.2	1.7
6	33	81	5	18	0.5	1.6
7	40	68	5	17	0.4	3.1
8	62	13	6	18	0.2	2.2
9	11	23	8	19	0.2	1.8
…	…	…	…	…	…	…
120	79	1	5	18	0.2	2

.

3.1. Regional Division Based on Voronoi Diagram

The regions where farmlands belonged were judged using the ray method via VScode and Python. The regional division results of the largest sample are displayed in

Table 5. Regional classification results.

Center No.	Farmland No.
1	45, 51, 58, 61, 63, 64, 65, 67, 69, 112, 116
2	7, 18, 20, 27, 48, 50, 80, 82, 84, 92, 94, 96, 97, 102
3	11, 16, 31, 35, 41, 42, 49, 52, 53, 55, 56, 59, 60, 95, 113
4	3, 21, 29, 30, 34, 37, 43, 75, 81, 86, 89, 91, 98, 103, 108, 109, 110, 119, 120
5	17, 68, 71, 78, 101, 114
6	8, 33, 38, 54, 88, 99
7	9, 14, 19, 46, 62, 77, 79, 93
8	6, 10, 22, 24, 25, 26, 39, 66, 70, 72, 73, 83, 85, 100, 115, 117, 118
9	1, 13, 23, 32, 36, 44, 47, 76, 104, 105, 111
10	2, 4, 5, 12, 15, 28, 40, 57, 74, 87, 90, 106, 107

.

3.2. Algorithm Results and Analysis

The initial parameters of the HPSO algorithm were set as follows: The maximum number of particles pop_max = 100, the minimum number of particles pop_min = 50, and the number of iterations was T_max = 100. A total of 10 floating points within (0.4, 0.9) were randomly selected via Python with the maximum value of w_max and the minimum value of w_min, and the first generation of w was an extracted random number. The iterative changes of w, c₁, c_2, and pop_s are exhibited in Figure 4. It could be seen that in the early algorithm operation phase, w declined slowly, but the decrease was accelerated in the later phase. c₁ decreased nonlinearly while c₂ increased nonlinearly with the number of iterations. The number of particles is reduced at a faster rate in the early stage and basically maintained in the later stage.

The starting cost of agricultural machinery is the natural number within 25–35 (yuan.vehicle⁻¹), agricultural machinery operation cost within unit distance as C₁ = 3 (yuan.km⁻¹), waiting cost within unit time as C₂ = 2 (yuan.min⁻¹), and penalty cost within unit time as C₃ = 30 (yuan.min⁻¹). The maximum operating capacity is within 15–20 (hm²), the maximum driving distance is within 150–200 (km), and the total number of agricultural machinery pieces owned is K_m = 20 (vehicle). The algorithm operation was implemented via Python code for ten times, the optimal result was taken, and the scheduling route of the largest sample is displayed in

Table 6. Data information of scheduling route.

Center No.	Scheduling Route	Number of Agricultural Machinery (Vehicle)	Optimal Cost (Yuan)	Average Cost (Yuan)	Optimal Distance (km)	Average Distance (km)
1	112→64→69→116→67→45, 61→63→58→51→65	2	431.40	437.42	123.00	125.21
2	20→48→92→18→82→94→7→102→27, 96→80→50→97→84	2	481.30	487.16	140.60	142.30
3	42→31→55→35→59→53→113→60→41, 11→95→56→52→16→49	2	427.20	429.53	119.30	121.42
4	103→89→3→91→120→37→109, 75→43→110→34→81→30, 29→119→98→86→21→108	3	645.20	649.03	181.60	184.12
5	114→71→101→17→68→78	1	156.00	156.75	38.80	38.81
6	33→88→38→8→54→99	1	187.70	190.56	52.30	52.39
7	62→9→77→46→79→19, 14→93	2	338.50	338.59	101.80	101.83
8	6→73→26→22→85→115→118→10→70→25→66, 100→39→24→83→117→72	2	533.40	541.87	156.70	157.51
9	23→47→104→76, 1→44→36→13→111→105→32	2	382.40	384.69	113.20	113.60
10	57→74→107→5→90→40, 2→106→12→28→4→87→15	2	466.10	470.10	120.20	126.20

.

Through the experimental samples, the total average cost of the scheduling is 4085.70 yuan, the total optimal cost is 4049.20 yuan, the optimal driving distance is 1147.5 km, the average driving distance is 1164.2 km, and the total number of agricultural machinery pieces required is 19. The average time of the algorithm is 16.67 s, and the optimal time of the algorithm is 15.78 s. The operation route obtained by calculating all samples is shown in Figure 5. It can be seen that all farmlands belong to the cooperative center closest to themselves. All routes start by the cooperative center closest to the field. The operation roadmap of four samples that can meet the requirements of farmers‘ orders is shown in Figure 5.

In order to analyze the performance of the HPSO algorithm, this paper uses the standard particle swarm optimization (PSO), genetic algorithm (GA), improved particle swarm GPSO, particle swarm, genetic algorithm combined with PSO-GA four algorithms, and HPSO run ten times as a comparison. By comparison, the performance of the hybrid particle swarm optimization algorithm is evaluated, and the superiority of each algorithm under different sample conditions is verified. The results are shown in

Table 7. Comparison results of three algorithms under different sample conditions.

Number of Service Centers	Number of Farmlands	Optimal Cost (Yuan)
		PSO	GA	HPSO	GPSO	PSO-GA
1	10	1439.60	1434.40	1416.80	1425.10	1425.10
3	30	1999.30	2002.20	1990.40	1990.40	1994.50
5	60	2709.00	2669.10	2644.60	2673.80	2652.8
10	120	4105.40	4118.20	4049.20	4081.20	4058.30

, when the number of centers is 1. The optimal cost obtained by using HPSO is 1.58% lower than that of PSO with the worst effect. When the number of centers is 3, the effect of using HPSO is equivalent to that of GPSO, but the optimal cost is reduced by 0.39% compared with the worst GA. When the number of centers is 5, HPSO is the best, and the optimal cost is 1.37% lower than the worst result PSO. When the number of centers is 10, the optimal cost obtained by using HPSO is 1.68% lower than that obtained by GA. It can be seen that the results of the hybrid particle swarm optimization algorithm are not very obvious when the number of samples is small, but in the case of large amounts of data, the performance of the superiority is more and more prominent.

The iterative changes of the optimal costs obtained by the three algorithms under four sample conditions are displayed in Figure 6. It could be observed that if no suitable starting point was chosen, the standard PSO algorithm could easily fall into the local optimum, which, however, was improved by the HPSO algorithm. Moreover, the HPSO algorithm reaches the best effect among the five algorithms, with a fast convergence rate and favorable global optimization ability.

3.3. System Development

The agricultural machinery order operation program is developed on the WeChat applet to realize information interaction, and the data is uploaded to the background system for scheduling calculation. The developed agricultural machinery order operation program is shown in Figure 7, the text information on the map is the local information identifier carried by the Tencent map layer.

On the Visual Studio Code 2019 platform, use the Python language and select the Django framework to build an agricultural machinery scheduling system combined with the agricultural machinery scheduling model. The main functions of the system include agricultural machinery management, route planning, task management, and background task management. Among them, the real-time scheduling roadmap implemented by the algorithm is shown in Figure 8, the feasibility of the application of the scheduling algorithm is verified by the implementation of the system design.

This section verifies the feasibility of the algorithm through algorithm comparison and system design. From the perspective of algorithm comparison results, the hybrid particle swarm optimization algorithm performs better and more stable than other algorithms. From the perspective of system implementation, this method design can effectively help farmers solve scheduling problems.

4. Conclusions

In this research, a multiregional scheduling model with a time window was established for agricultural machinery operational services. After the regional division based on a Voronoi diagram, the model was solved using the HPSO algorithm with the minimum operation cost of agricultural machinery as the optimization objective. In addition, through a variety of algorithms as a comparison, the results show that after combining the Tyson polygon, the early convergence speed of the scheduling model is accelerated and the route planning ability of the algorithm is enhanced. The use of a hybrid particle swarm optimization algorithm makes the standard particle swarm optimization algorithm jump out of the shortcomings of the easy-to-fall-into local optimum, combines the advantages of faster convergence speed of particle swarm optimization, and improves the ability to find the global optimal solution. Experiments show that this method can effectively solve the multicooperative agricultural machinery scheduling problem with time windows, and the algorithm is more excellent when the sample data set is larger. In the largest sample used in the experiment, the optimal operation cost of scheduling calculation is 4049.20 yuan, which is 1.68% lower than the worst scheduling algorithm in the comparison. This algorithm can effectively save the time cost of agricultural machinery scheduling and meet the solution of multiregional agricultural machinery scheduling problem. This shows that it can improve operational efficiency and economic benefits for the relevant practitioners. For researchers, this study’s unique model and the use of a Voronoi diagram and a faster computational efficiency method, combined with the optimized hybrid particle swarm optimization algorithm, enhance the efficiency of the algorithm, and may be able to bring a little inspiration.