1. Introduction
China, a significant consumer and producer of vegetables [
1], faces complexity in the vegetable production process, with a limited availability of intelligent machines suitable for this task [
2,
3]. Manual harvesting, in particular, consumes a large amount of labor [
4]. Farmers need to transport vegetables to nearby collection stations, which currently rely on collaborative work between harvesters and harvest-aid vehicles. While large agricultural technology companies have developed vegetable harvesters for specific crops, such as chili peppers, kale, and carrots, their application on a large scale remains limited due to high costs. In this context, scientific research institutions, such as the Beijing Academy of Agricultural and Forestry Sciences, have studied intelligent harvest-aid vehicles equipped with weight sensors and GPS modules.
The cost of an intelligent harvest-aid vehicle is significantly higher compared to that of a regular loading vehicle. On large farms, there can be numerous crops that require harvesting during the harvest season. Equipping each harvester with multiple harvest-aid vehicles for harvesting is not feasible. Sequential crop harvesting could involve multiple harvest-aid vehicles for each harvester, but this approach may cause delays in crop harvesting and result in losses. Therefore, it is necessary to study the collaborative scheduling of harvest-aid vehicles and harvesting operations. These vehicles offer flexibility, compactness, and improved efficiency by reducing non-productive labor time for farmers. However, the scheduling of harvest-aid vehicles still relies on manual processes, leading to challenges such as empty running and blocking during large-scale harvesting tasks [
5]. Unlike other industrial scenarios, agricultural environments are influenced by agronomic factors and crop conditions, requiring further exploration of digital modeling techniques [
6]. Furthermore, there is a lack of standardization in the driving path and collaborative manner of vegetable harvesting and transportation.
To address these issues, this paper proposes a cooperative method for vegetable harvesting and transportation, considering factors such as harvester speed, efficiency, driving path, and load capacity. The objective is to minimize the waiting time for harvesters, ensure the conflict-free scheduling of harvest-aid vehicles, and optimize the efficiency of harvester collaboration [
7]. By studying the collaboration between harvest-aid vehicles and harvesters (CHVH), this research aims to provide reference solutions for improving vegetable harvesting efficiency and promoting intelligent development in vegetable production.
In the field of agriculture, the cooperative scheduling of agricultural machinery has primarily focused on food crops [
8,
9]. Liu et al. addressed the issue of low efficiency in the continuous multitasking of agricultural machinery in hilly regions by proposing a scheduling model with the objective of minimizing the total scheduling time. To tackle this problem, they introduced an improved version of the non-dominated sorting genetic algorithm III [
8]. Kan et al. proposed a random task allocation method, known as Next-Best Action Planning, specifically for grape picking. This approach captured the action constraints of agricultural machinery in an agricultural environment through SAG. By considering resource and energy budgets, the algorithm simultaneously determined the best sampling location and the optimal time to return to the collection station [
10,
11].
Additionally, Peng et al. focused on the dynamic scheduling of multiple robots in strawberry harvesting. They established a transportation function that comprehensively considered vehicle speed, unloading time, and harvesting time [
12]. Their modeling efforts led to a near-optimal scheme for handling requests, providing an upper-efficiency limit for scheduling algorithms. Furthermore, Yang et al. integrated agricultural machinery resources using blockchain technology and established matching functions based on factors such as weather, road conditions, cost, and benefit. They employed genetic algorithms to supervise and schedule agricultural machinery [
13]. In terms of harvesting, the failure rate of agricultural machinery tends to increase significantly. To address the uncertain needs of machinery maintenance services, Hu et al. proposed a scheduling model that minimized scheduling distance and the fixed costs of scheduled agricultural machinery [
14]. Pan et al. proposed a deep reinforcement-learning-based method for scheduling multiple agricultural machineries, aiming to minimize crop losses and completion time in emergency scenarios. The experimental results demonstrated that this method outperformed existing approaches, with an average improvement of 26.7% in computation rate and 21.9% in completion time, while also significantly enhancing computational efficiency [
15].
Large-scale autonomous robot networks have found widespread applications in logistics and various industries, including logistics warehouses [
16,
17,
18], unmanned container terminals [
19], and intelligent transportation systems [
20,
21,
22]. As a notable example, JingDong, a Chinese enterprise, operates the “Asia No.1” unmanned warehouse, which is a fully automated facility that handles the entire process of harvesting, storage, packaging, and order selection. To optimize robot scheduling in intelligent warehouses, Ma et al. considered the part segmentation caused by constraints such as part arrival time and mobile robot energy. They conducted a study using a variable neighborhood-search-based nondominated sorting genetic algorithm to obtain Pareto solutions that maximized chemical station satisfaction and minimized energy consumption. This approach enabled the generation of multiple high-quality mobile robot scheduling schemes, leading to improved workstation satisfaction and reduced energy consumption [
23]. In a similar vein, Liu et al. proposed a multi-objective task allocation function that simultaneously optimized time cost and potential path conflicts for large-scale autonomous robot networks. They utilized a generalized conflict graph to transform the multi-objective function into a linear programming example, resulting in reduced robot congestion and improved system operation efficiency [
24]. Furthermore, Ham studied the real-time scheduling of production stations and transfer robots in flexible workshops, employing constraint programming to minimize the maximum completion time of tasks in FJSP+ robot scheduling [
25]. Gultekin et al. proposed second-order cone programming and a heuristic algorithm to find Pareto solutions based on factors such as robot order, movement speed, energy consumption, and system production speed. This approach allowed for the generation of a robot scheduling plan with a small number of machines within a reasonable computational time [
26]. Additionally, a novel middleware was proposed for compiling and building multi-robot collaborative applications [
27].
The research on agricultural machinery scheduling primarily centers around resource allocation and path planning for agricultural machinery in the field, with a specific focus on harvesting grain crops or strawberries. Furthermore, the constraints and objective functions associated with scheduling multiple machines in an industrial context differ significantly from those in agriculture. Consequently, directly applying existing methods to the agricultural setting poses challenges. The core issue in this paper is how to integrate vegetable planting agronomy into the development of a collaborative scheduling model for multiple harvesters and harvest-aid vehicles in dynamic environments. While multiple robot scheduling has been successfully applied in industrial production, theoretical studies can be further improved based on actual production effects [
28]. In the agricultural field, research on the cooperative operation of agricultural machinery is relatively recent, with limited consideration given to different crops, various agricultural machineries, and diverse agronomic needs. However, some studies have focused on cooperative operation scheduling for harvesting, primarily centered around static scheduling. Once certain conditions are met, cooperative operation requests are issued, and harvest-aid vehicles are directed by the decision-making center to predetermined locations. It is important to recognize that both the harvester and harvest-aid vehicles are dynamic during the harvesting process, and factors such as planting agronomics and machine size introduce constraints to the path and arrival time. Therefore, this paper aims to address the cooperative scheduling of agricultural machinery in the context of vegetable harvesting and transportation.
3. Discrete Multi-Objective Jaya Algorithm
In 2016, Rao proposed the Jaya algorithm as an efficient and easy-to-run method for solving complex optimization problems [
29]. This algorithm requires only a small number of control parameters, namely the predefined population size and the maximum number of iterations. During the optimization process,
Xmn represents the value of the
n-th dimensional variable of the
m-th solution in the population.
Xbest and
Xworst denote the optimal and worst solutions in the population, respectively. The Jaya algorithm utilizes two random numbers,
ra1,j and
ra2,j, generated in the
j-th dimensional space, where
ra1,j and
ra2,j ∈ [0, 1].
ra1,j and
ra2,j are random numbers within the range of [0, 1], introducing randomness to the algorithm to help it escape local optima and enhance its global search.
The Jaya algorithm starts by forming a random initial population, and then iteratively adjusts the individuals to approach the optimal solution and move away from the worst solution, aiming to find the best solution. During the iteration process, the algorithm dynamically adjusts the positions of candidate solutions based on the performance of the current population, so there is no need for additional control parameters to adjust the search strategy of the algorithm. Compared to some other optimization algorithms, the performance of the Jaya algorithm is usually less sensitive to parameters. The update method is as follows [
30]:
In the traditional Jaya algorithm, if the F (X′mn) corresponding to the solution X′mn is better, then X′mn is accepted and replaces the original solution; otherwise, the original solution is maintained. The term ra1,j (Xbest − |Xmn|) indicates that the solution is close to the optimal solution, and − ra2,j (Xworst − |Xmn|) suggests that the solution is far from the worst solution. It is worth noting that the traditional Jaya algorithm is primarily used for addressing continuous single-objective optimization functions. To adapt it for discrete harvest-aid vehicle scheduling models, this paper introduces a discrete multi-objective Jaya algorithm. The main steps of this algorithm include encoding, decoding, initialization, individual updating, and Pareto sorting.
- (1)
Encoding and decoding
Firstly, the solutions for collaborative requests and harvest-aid vehicle scheduling are encoded as shown in
Figure 3. Each solution is represented by a matrix that contains the scheduled harvest-aid vehicle ID. The encoding length is equal to the total number of vehicles. For example, if the code for collaborative requests is [1, 5, 4, 2, 6, 3], and
X = [2, 3, 1, 4, 5], it indicates that harvester 1 is assisted by harvest-aid vehicle 2, harvester 5 is assisted by harvest-aid vehicle 3, and so on.
Next, when a harvest-aid vehicle arrives at the location of harvester j before it stops and waits, the latest harvest-aid vehicle is selected. On the other hand, if the vehicle arrives after the harvester has stopped, the earliest harvest-aid vehicle that reaches the location of harvester j is chosen. By following this scheduling rule to respond to cooperative transport requests, the maximum utilization efficiency of harvest-aid vehicles can be achieved, and the coordination time can be minimized.
Decoding is the process of determining the minimum scheduling cost and generating a specific coordination scheme for harvesting and transportation based on the encoding. In the decoding phase, when a vehicle completes a round of vegetable harvesting and unloading, it becomes idle at the collection station. If the next task for the harvest-aid vehicle is to assist harvester j, the vehicle needs to reach the expected position.
- (2)
Pareto optimization
In multi-objective optimization, it becomes challenging to determine the optimal and worst solutions, since there are multiple objectives to consider. To effectively address this, a combination of non-dominated ranking and a crowding mechanism is used for Pareto optimization. The solution group can be ranked based on the dominance theory. The rule is as follows:
For two solution schemes,
Xe and
Xq, if
Xe is not inferior to
Xq in all objectives and is strictly superior to
Xq in at least one objective, then
Xe dominates
Xq, indicating that
Xe has a higher rank. The solution with a higher rank is considered to be superior to other solutions. In the case of two solutions with the same rank, the solution with a higher crowding distance is considered to be superior to the others.
where
K is the number of objective functions,
q is the
j-th particle in the population, and
fq+1k and
fq−1k represent the
k-th objective function values of the two positions adjacent to
q. Furthermore,
fmaxk and
fmink represent the maximum and minimum values of the
k-th objective function in the population, respectively. The optimal solution is determined based on the highest level and the highest degree of crowding, while the worst solution is determined accordingly. Once these two solutions are identified, the individuals in the new population will be updated.
- (3)
Pareto optimization
For individual updates, the existing Jaya algorithm’s individual update method is suitable for continuous variables and cannot be directly applied to discrete optimization problems. To effectively update individuals, specific update methods are designed for different situations:
(a) If Xmn is identical to Xworst or Xbest, a new individual X′mn is regenerated.
(b) If
Xmn is not exactly the same as
Xworst or
Xbest, based on the principles of the Jaya algorithm, the code in
Xmn that matches
Xworst is deleted and replaced with the code at the corresponding position in
Xbest, thereby generating a new individual
X′
mn. However, considering the time constraints of vehicle scheduling, if the replacement code at the corresponding location in
Xbest is the same as another location code, the location code is regenerated. The individual update methods in this mode are illustrated in
Figure 4.
The current optimal and worst solutions in the population play crucial roles in guiding individuals towards a global optimal solution. However, if these solutions fall into a local extreme region, this can result in search stagnation within the population and hinder the attainment of a better global optimal solution [
31]. To address this issue, this paper explores the discrete Jaya algorithm based on an opposition-based learning mechanism. By incorporating opposition-based learning into the current optimal and worst solutions, the algorithm increases the likelihood of escaping from local extreme value regions. It is important to note that opposition-based learning is only performed on the current optimal and worst individuals, rather than all individuals individually [
32]. This approach helps to prevent the Jaya algorithm from falling into local optimization while maintaining convergence speed.
Assuming that
Xbest (
iter) represents the individual of the
iter-th population, the opposition-based learning process is as follows:
where
Nr represents the total number of scheduling requests.
X′
best and
X′
worst represent the inverse individuals of the optimal solution and the worst solution, respectively. It is important to note that some harvest-aid vehicles are already engaged in work and cannot be rescheduled. For the sake of simplifying the opposition-based learning process, this paper performs opposition-based learning based on the coding order in
Xbest, rather than opposition-based learning of the code itself. The process of opposition-based learning is illustrated in
Figure 5.
The opposition-based learning mechanism described above primarily focuses on enhancing the local development ability of the algorithm. To further improve the solution accuracy and stability while enhancing the global exploration ability, long-term memory libraries are designed to increase the diversity of solutions [
33]. This memory library establishes an elite solution memory with a size of
H, denoted as
fh = {
fh1,
fh2, …,
fhH}. The solution
fhH represents the
H-th solution stored in the memory.
X∗m is a candidate solution obtained through iterative local search. To update the memory,
X∗m is compared with
Xm-H, where
Xm-H corresponds to
fhimodH in the memory. If
X∗m is superior, the objective function value of
X∗m replaces
fhimodH in the memory, and the current solution is replaced by
X∗m for the next iteration. The complete iteration process is illustrated by Algorithm 1.
Algorithm 1 Discrete Multi-objective scheduling algorithm |
1: Initialize parameters Maxiter, D, Nr, H |
2: Initialize population X |
3: Decode and compute the objective function f1(X) |
4: Decode and compute the objective function f2(X) |
5: According to Pareto sort, calculate the initial best solution Xbest and the worst solution Xworst |
6: Set t = 1 |
7: while iter ≤ Maxiter do |
8: According to the Equation (9), Pareto sort is used to select the current best solution X′best and the worst solution X′worst |
9: According to the opposition-based learning mechanism, obtain the inverse solution Xr′best and Xr′worst of X′best and X′worst |
10: if X′best dominates Xbest or Xworst dominates X′worst then |
11: Xbest = X′best or Xworst = X′worst |
12: else |
13: Xbest = Xbest or Xworst = Xworst |
14: end if |
15: if Xbest dominates Xiter-H then |
16: Xiter-H ⇐Xbest |
17: else |
18: Xbest ⇐ Xiter-H |
19: end if |
20: Renew individuals and generate new population according to Section 3 |
21: end while |
22: Output Pareto frontier if the termination condition is met, otherwise go to 7 |
4. Experiment and Analysis
To validate the effectiveness of the collaborative scheduling approach proposed in this paper, experiments were conducted using a realistic scenario. Simulation verification was performed for the multi-harvest-aid vehicle scheduling. An overall farm was divided into different vegetable fields/greenhouses, as shown in
Figure 6, with each workplace having its own entrance. Two collection stations were located at the center of the farm, where multiple harvest-aid vehicles and harvesters collaborated in the harvesting process. The vegetables in the fields were planted on beds, ranging in length from 100 to 250 m. Harvesting was performed in a backhoe manner, with the furrow heading direction being randomly generated within the range of (0, 180°].
To evaluate the effectiveness of the method proposed in this paper, three examples were designed based on actual farm data. From each group of data,
z harvest-aid vehicles were randomly selected from the agricultural machinery station as the initial idle harvest-aid vehicles currently owned by the decision-making center, and their locations and unloading sequences were randomly generated. At the same time,
p harvest-aid vehicles that were about to be fully loaded were randomly generated, and the information for each vehicle included the harvester it was currently assisting, the current load, and whether it needed to be replaced. The farm consisted of
g vegetable blocks and
s vegetable greenhouses. Considering the different harvesting efficiencies of different vegetables, the task parameters are summarized in
Table 2.
Additionally, the relevant parameters of the harvest-aid vehicles are presented in
Table 3. It is worth noting that all the harvest-aid vehicles had the same structure.
4.1. Algorithm Performance Verification
The parameter settings have a significant impact on the performance of the algorithm. Let
g = 7,
s = 5,
p = 6, and
z = 10. The maximum iteration times of the algorithm were set as
itermax = 1000, the population size was set as
pop = 80, and the optimal solution adopted the generalized opposition-based learning approach with a memory library
H = {1, 5, 10, 15, 20, 25}. The experiment was repeated 30 times with the same parameters and examples. The experimental environment used an Intel (R) Core (TM) i7-9700H processor with 16 GB of memory. The average value of objective function
f1 and the average value of objective function
f2 are shown in
Figure 7 and
Figure 8.
Each group of data were subjected to an S-W normal distribution test. It can be concluded that all 30 experimental results obtained at
H = {1, 10, 15, 20, 25} followed a normal distribution. The experimental results had a certain degree of stability, except when
H = 10.
Figure 7 and
Figure 8 show that, under different
H lengths, the 30 repeated experiments showed that the average maximum waiting time of the harvester and the scheduling time were smaller when
H = 15, and the box type was narrower, indicating a more stable distribution. Therefore, this paper sets
H = 15.
Partial opposition-based learning extends multiple solutions by making inverse changes to partial positions of the solution. On the other hand, generalized opposition-based learning expands diversity by performing opposition-based learning on each position of the solution. In order to verify the learning mechanism applicable to the collaboration model in this paper, both partial opposition-based learning and generalized opposition-based learning were implemented. As shown in
Figure 9, when the algorithm applied partial opposition-based learning, the average maximum waiting time of the harvester and the scheduling time of the harvest-aid vehicle were minimized, and the data distribution remained stable. Therefore, this paper compares algorithm performance and analyzes collaborative scheduling schemes using partial opposition-based learning.
4.2. The Comparison of Algorithm Performance
In the context of harvest-aid vehicle scheduling, this paper compares three commonly used algorithms as follows: NSGA-II (second-generation non-dominated sorting genetic algorithm), modified multi-objective particle swarm optimization (MMOPSO), and the traditional Jaya algorithm.
NSGA-II proposes a fast non-dominated sorting genetic algorithm that reduces computational complexity. It combines the parent population with the offspring population, allowing the next generation’s population to be selected from a double space. This ensures that all the best individuals are preserved. By introducing elite strategies, NSGA-II prevents the discarding of excellent individuals during evolution, thereby improving optimization accuracy. Moreover, it utilizes crowding degree and crowding degree comparison operators, promoting the uniform expansion of individuals in the quasi-Pareto domain to the entire Pareto domain and ensuring population diversity.
On the other hand, the MMOPSO algorithm designs two search strategies based on particle swarm optimization to update the particle speed. It also shares elite information in external archives through an evolutionary search strategy that simulates binary crossover and polynomial mutation.
The convergence curves of the four algorithms with the same parameters are shown in
Figure 10. Convergence means that, as the algorithm is executed, it will eventually reach a stable state or solution. This is the foundation for algorithms to obtain meaningful results.
Figure 10a depicts the iterations of objective function 1 with different algorithms, while
Figure 10b shows the iterations of objective function 2. It is evident that the collaboration between harvest-aid vehicles and harvesters (CHVH) exhibits a faster convergence speed and can obtain better solutions.
z represents the number of harvest-aid vehicles in the agricultural machinery station as the initial idle harvest-aid vehicles currently owned by the decision-making center.
p represents the number of harvest-aid vehicles about to be fully loaded.
g represents the number of vegetable blocks and
s represents the number of vegetable greenhouses. In fact, all four parameters are determined by the scale and configuration of the farm.
Table 4 shows a comparison between the HVH algorithm and other algorithms on farms of different sizes. AM represents the solution obtained by the CHVH algorithm. The scheduling efficiency was analyzed by calculating the gap between the solution obtained by other algorithms and AM. It can be seen that our CHVH algorithm was also superior to the other three commonly used heuristic algorithms. Among them, NSGA-II was superior to the MMOPSO algorithm, but there was still a gap between the CHVH algorithm and the multi-objective genetic algorithm. The algorithm in this paper had the shortest solution time and a higher efficiency. Reducing the non-productive waiting time of the harvester can more flexibly improve the scheduling efficiency and reduce the loss cost of the harvester. The discrete Jaya algorithm had a relatively short running time, but still lagged behind the CHVH algorithm in terms of the accuracy of the solution. The discrete multi-objective Jaya algorithm is of great significance in effectively reducing the operating costs of systems and achieving intelligent vegetable harvesting and transportation.
To further analyze the distribution differences of the non-dominated solutions obtained by the CHVH algorithms, the non-dominated solutions were sorted after 1000 iterations. Consequently, the optimal non-dominated solution sets of the two algorithms were obtained, as shown in
Figure 11. It can be seen that the Pareto front of the Jaya algorithm and the improved algorithm in this paper were clearly nonlinear. The solution obtained by CHVH demonstrated a wider distribution range and better diversity. Additionally, the scheduling scheme outperformed the comparison algorithms.
5. Discussion
This paper addresses the collaborative optimization of scheduling for harvesting operations and harvest-aid vehicles on farms, with the goal of minimizing the total non-productive waiting time and coordination costs. The study takes into account both field and greenhouse scenarios, designing a multi-machine collaborative scheduling model focused on intelligent harvest-aid vehicles. The dynamic scheduling approach adopts a priority strategy based on the closest relative distance between the harvest-aid vehicle and the harvester, considering the static harvesting path and current vehicle status. When the harvest-aid vehicle reaches its load threshold, the system calculates the time for an idle vehicle to join the operation and the waiting time for the harvester, optimizing the collaboration between the harvester and the harvest-aid vehicle.
By comparing the CHVH algorithm with NSGA-II, MMOPSO, and JAYA, the results show that, for small farms, CHVH reduces the total non-productive time by 5.1%, 28.2%, and 4.9%, respectively, and lowers the coordination costs by 32.0%, 57.8%, and 31.5%. For large farms, it reduces the total non-productive time by 0.7%, 31.5%, and 5.3%, while reducing the coordination costs by 31.5%, 39.2%, and 27.5%.
This study converts the scheduling scheme into digital encoding, designing a JAYA algorithm for discrete multi-objective problems. A search strategy combining an opposition-based learning mechanism with a long-term memory library balances global and local searches. MMOPSO shares elite information through an external library, while JAYA directly uses the global best and worst solutions. JAYA’s straightforward approach is theoretically more efficient, especially in quickly finding the global optimum. Although all four algorithms are multi-objective optimization algorithms, the non-dominated sorting and crowding calculation of NSGA-II are effective, but they also introduce additional computational complexity and sensitivity to parameters. There is no better performance in solving multi-machine scheduling problems under complex and multi-constraint conditions.
For farms of different scales, the average running time of our algorithm is 10.2 s, which is the shortest among the four algorithms. The MMOPSO algorithm combines particle swarm optimization and an evolutionary strategy, and its particle updates depend on the relative position and velocity between particles, which may result in a lower search efficiency than the JAYA algorithm for complex problems. NSGA-II retains superior individuals through non-dominated sorting and an elitist strategy. In contrast, the JAYA algorithm generates new solutions by directly utilizing the current global best and worst solutions. It leverages the two most promising and representative extreme points in the search space, thus providing a higher efficiency.
This paper aims to shorten total non-production waiting time and coordination costs, and designs a multi-machine collaborative scheduling algorithm. This provides a reference plan for the collaboration of harvesting and transportation. The experiment shows that the research algorithm can schedule auxiliary transport vehicles on farms of different scales. However, this study does not take into account obstacles, auxiliary transport vehicle energy, and path planning on farms. This requires improving the intelligence level of auxiliary transport vehicles and increasing the algorithm constraints, which is also one of our future research directions.