Research on Traversal Path Planning and Collaborative Scheduling for Corn Harvesting and Transportation in Hilly Areas Based on Dijkstra’s Algorithm and Improved Harris Hawk Optimization

Abstract

This study addresses the challenges of long traversal paths, low efficiency, high fuel consumption, and costs in the collaborative harvesting of corn by harvesters and grain transport vehicles in hilly areas. A path-planning and collaborative scheduling method is proposed, combining Dijkstra’s algorithm with the Improved Harris Hawk Optimization (IHHO) algorithm. A field model based on Digital Elevation Model (DEM) data is created for full coverage path planning, reducing traversal path length. A field transfer road network is established, and Dijkstra’s algorithm is used to calculate distances between fields. A multi-objective collaborative scheduling model is then developed to minimize fuel consumption, scheduling costs, and time. The IHHO algorithm enhances search performance by introducing quantum initialization to improve the initial population, integrating the slime mold algorithm for better exploration, and applying an average differential mutation strategy and nonlinear energy factor updates to strengthen both global and local search. Non-dominated sorting and crowding distance techniques are incorporated to enhance solution diversity and quality. The results show that compared to traditional HHO and HHO algorithms, the IHHO algorithm reduces average scheduling costs by 4.2% and 14.5%, scheduling time by 4.5% and 8.1%, and fuel consumption by 3.5% and 3.2%, respectively. This approach effectively reduces transfer path costs, saves energy, and improves operational efficiency, providing valuable insights for path planning and collaborative scheduling in multi-field harvesting and transportation in hilly areas.

1. Introduction

Path planning is crucial for the collaborative operation of multiple agricultural machines, as a scientifically designed and efficient path-planning method can significantly improve operational efficiency and reduce costs [1]. However, despite China being one of the major corn-producing regions in hilly areas, the development of agricultural mechanization in these areas has been slow due to the uneven terrain, scattered farmlands, and difficult road conditions. As a result, corn planting in hilly regions primarily consists of small-scale farming and slope land cultivation [2]. To ensure the adequate harvesting and timely transportation of corn, it is necessary to implement rational multi-machine collaborative path planning to ensure that the corn harvester can efficiently cover the farmland and the transport vehicles can promptly deliver the grain, thereby improving both the yield and quality of corn. However, in hilly areas, difficult road access and uneven terrain can cause delays and higher labor costs, reducing the efficiency of harvesting and transportation and impacting economic benefits. Additionally, the extended traversal paths, the urgency of timely harvesting, and the need for timely transportation further exacerbate these challenges [3]. This has led to increased demand for the optimal configuration and scheduling of harvesters and transport vehicles, necessitating more efficient path optimization strategies. Furthermore, the fuel consumption of agricultural machinery has become a widely discussed issue in both academic and industrial circles [4]. Finding ways to achieve energy savings, reduce consumption, and reduce cost during the harvesting and transportation processes in hilly areas through reasonable and efficient path optimization has become an urgent problem that needs to be addressed in the agricultural sector.

The research on multi-machine collaborative scheduling mainly focuses on field resource scheduling and path planning. Firstly, for the field path-planning problem, reference [5] achieved path planning for autonomous robots through Simultaneous Localization and Mapping (SLAM) and deep learning (DL) with artificial vision, and the results demonstrated that the intelligent algorithm has high accuracy. Reference [6] combines the D*Lite algorithm with the Dynamic Window Approach (DWA) using a two-layer graph and feasible domain strategy, resulting in a good optimization effect in terms of global planning time and time cost. Reference [7] proposes a new global and local fusion path-planning algorithm, effectively addressing the issues of excessive path redundancy and too many turning points. Secondly, in terms of multi-machine scheduling. Reference [8] proposes agricultural machine scheduling and path-planning algorithms based on simulated annealing, addressing the path-planning problems for both homogeneous and heterogeneous agricultural machines. References [9,10] convert the multi-machine collaborative scheduling problem into a vehicle routing problem, using improved tabu search algorithms to determine reasonable agricultural machine operation path planning and scheduling schemes. Reference [11] introduces a joint optimization problem consisting of vehicle routing and assignment problems and uses mixed-integer programming to optimize wheat harvesting and transportation paths. Additionally, reference [12] investigates the multi-machine collaborative scheduling problem for multiple farmlands. For the scheduling of harvesting and transportation operations, reference [13] applies NSGA-III and ant colony algorithms to achieve collaborative scheduling of harvesters and transport vehicles, effectively reducing transfer distances. Reference [14] focuses on path optimization for harvester and transport vehicle operations in the field, significantly reducing loading and unloading times, though it does not comprehensively address the complex constraints of multi-machine operations in the field.

In summary, existing research primarily focuses on coverage path planning for a single large farmland plot or traversal path planning for multiple plots. However, research on multi-machine, multi-field traversal path planning in hilly areas is relatively scarce. Moreover, the full coverage operational path planning for corn harvesters and the scheduling path problems for transport vehicles are complex combinatorial optimization problems, which require consideration of key factors such as fuel consumption, time windows, capacity, and path costs [15]. Farmlands in hilly areas are densely distributed and influenced by slopes. While single-machine path planning for multiple plots is relatively straightforward, multi-machine collaborative path planning faces challenges due to frequent transfers between fields. Therefore, this study aims to explore a multi-machine field traversal path-planning method suitable for hilly areas.

First, a field model is constructed based on Digital Elevation Model (DEM) data to achieve full coverage path planning for the corn harvester. To address the issue of unclear connectivity during the harvesting and transportation process, a road network for field transfers is established. Additionally, a traversal algorithm combining Dijkstra’s algorithm and the Improved Harris Hawk Optimization (IHHO) is proposed. Dijkstra’s algorithm is used to find the shortest path between nodes in the field road network graph. Furthermore, to improve harvesting and transportation efficiency and identify the optimal traversal path, a multi-machine collaborative scheduling model is developed, with the IHHO algorithm employed to solve this model.

To overcome the issues of local optima and premature convergence often encountered in the HHO algorithm, initialization is enhanced using a good point set and quantum computing. The update strategy for the nonlinear energy factor is also improved to increase search efficiency. Moreover, the exploration phase is integrated with the slime mold algorithm to enhance the global search capability. To improve local search ability and diversity, an average differential mutation strategy is introduced, preventing premature convergence and avoiding local optima. Additionally, non-dominated sorting and crowding distance calculations are employed to balance multiple objective functions, maintaining the diversity of the population.

Finally, simulation testing and statistical analysis are conducted using actual farmland data to evaluate and improve the algorithm. The results demonstrate that the Dijkstra and IHHO algorithms offer significant advantages in solving multi-machine collaborative field path-planning problems. The proposed approach is highly effective in improving harvesting and transportation efficiency, reducing costs, and enhancing energy savings, thus providing substantial practical application value.

2. Harvester Field Operation Environment Model

2.1. Full Coverage Environment Modeling

The hilly areas are characterized by irregularly shaped, densely distributed, and unordered farmland plots, with a three-dimensional terrain that poses significant challenges for collaborative harvesting and transportation operations, as well as for full coverage path planning. To achieve offline and visualized full coverage path planning and collaborative harvesting tasks, this study employs a geometric approach [16] for environmental modeling. The operational environment of the harvester and transport vehicle is mapped into geometric space using points, lines, and surfaces to describe the environmental features.

Digital Elevation Model (DEM) data for 20 farmlands near coordinates 31°33′25.15″ N, 104°33′40.51″ E were obtained through electronic mapping. The DEM data were then vector-clipped using ArcGIS (pro 3.3) software to extract the latitude and longitude data for the farmland areas to be worked on. To accurately depict the field boundaries, contour points of the farmland plots were marked one by one. Using Matlab, the latitude and longitude coordinates were converted to UTM coordinates, and the farmland boundaries were subjected to erosion processing. The processed boundary coordinates were returned and represented as polygons. The final field boundary image is shown in Figure 1, where fields are numbered 1 to 20 in sequence.

2.2. Full Coverage Path Planning for Harvester Field Operations

Considering the three-dimensional terrain environment of hilly areas and the operating conditions of autonomous corn harvesters, a reciprocating coverage method [17] is chosen for full coverage path planning. This method is easy to implement and offers the advantage of a high operational coverage rate. Additionally, since tracked agricultural machinery is commonly used for corn harvesting in hilly regions, a right-angle turning approach is adopted.

To maximize energy efficiency and reduce consumption, it is necessary to minimize harvesting duration and path length. Therefore, while ensuring coverage, the number of turns should be minimized [18]. To achieve this, the operation is performed in a manner parallel to the longest side of the farmland plot [19]. Finally, the optimal coverage path’s entry and exit coordinates [20] are output, with the full coverage path diagram displayed in Figure 2. In the diagram, “en” represents the entry point, and “ex” represents the exit point.

3. Multi-Field Connection Path Planning Based on Dijkstra’s Algorithm

3.1. Design of Field Transfer Road Network Graph

Path planning for agricultural machines within a single farmland plot is relatively straightforward. However, due to the lack of clear connectivity between fields, continuous driving routes cannot be established between plots, requiring repeated searches for suitable transfer points. This increases harvesting and transportation time and costs. Therefore, to establish a connection between fields, key nodes such as turning points, entry/exit points, and the locations of agricultural machinery storage facilities are set at the field boundaries in the headland areas, facilitating the selection of nodes and transfers by the machinery. Using this approach, a road network graph between multiple fields is constructed [21], as shown in Figure 3.

The key nodes obtained using the above method for connecting multiple fields may not always represent the shortest path. To address this issue, this study adopts a method of adding additional key nodes between adjacent key nodes through linear interpolation, thereby ensuring a more continuous and accurate path representation, which allows Dijkstra’s algorithm to compute the optimal path more effectively. Given that the transport vehicle must repeatedly cooperate with the harvester for loading and unloading in the field, it is specified in this study that the harvester and transport vehicle can only find the shortest path through Dijkstra’s algorithm.

3.2. Solving the Shortest Path Using Dijkstra’s Algorithm

To further reduce the redundancy of the traversal path and effectively improve harvesting efficiency, this study employs Dijkstra’s algorithm [22], a graph theory method used to find the shortest path, to solve the shortest path problem between key nodes across all fields. Based on a greedy strategy, the algorithm traverses to the neighboring node that is closest and has not been visited, continuing until all nodes have been visited. This process ultimately yields the shortest path and distance between any two nodes, thereby determining the optimal path that the harvester or transport vehicle should take.

Using the established road network graph, the distances between nodes and their adjacency relationships are known. As a result, the entire road network can be represented as an undirected graph with weights, as shown in Figure 4. Additionally, a distance matrix is used to store the adjacency relationships and distance data between nodes, as illustrated in Figure 5, where “Inf” represents an infinite distance between nodes that are not directly connected.

3.3. Iterative Calculation

The values in the matrix are updated through multiple iterations. In each iteration, a vertex V_j with the current shortest distance is selected from the set of vertices V–S, which have not yet been added to the set of determined shortest paths, and added to the set S. First, the set S = {0}, and the initial values of the distance array dist[] are set as dist[i] = edge[0][i] (representing the direct distance from the starting point to other vertices), where i = 1, 2, …, n − 1, and dist[0] = 0. The initial values of the path array path[] are set as path[i] = −1 (indicating that the predecessor vertex has not been determined yet). In each iteration, the vertex V_j with the shortest path is selected, and the path lengths from this vertex to other vertices not yet in set S are updated by comparison. If a shorter path is found from the current vertex to another vertex, the shortest path length for that vertex is updated, and the path array path[] is modified to record the predecessor node for each vertex, as shown in Figure 6.

The path p from the source node 0 to node u is decomposed as 0~(P₁)~x → y~(P₂)~u, where node y is the first node on path p that does not belong to the set S, and node x (where x ∈ S) is the predecessor of node y on path p. Nodes x and y are distinct, although it is possible that x = 0 or y = u, and path P₂ may or may not re-enter S.

4. Collaborative Harvesting and Transport Scheduling Model

4.1. Problem Description

In the actual agricultural harvesting and transportation process, the agricultural machinery equipment available in the machinery storage and the farmland plots to be worked on are typically not unique. A reasonable matching of agricultural machinery to farmland plots can significantly improve harvesting and transportation efficiency. This study transforms the problem of determining the optimal traversal order of harvesters across multiple fields into a Traveling Salesman Problem (TSP) and the task of coordinating transport vehicles for loading and unloading grain with the harvester into a vehicle routing problem (VRP). The scheduling path in the fields is also solved using the Dijkstra method outlined in Chapter 3. The collaborative operation of harvesters and transport vehicles to complete the harvesting and transportation task is regarded as a collaborative scheduling problem. Therefore, this chapter focuses on the collaborative scheduling problem of different capacity harvesters and transport vehicles under time window constraints, as shown in Figure 7.

4.2. Grain Loading and Unloading Process

The traditional grain unloading method, where the harvester stops harvesting when the grain tank is full and proceeds to the transport vehicle parking area on the tractor path to unload, not only affects harvesting efficiency but also increases non-operating time for the harvester, especially when harvesting in large, contiguous, and dense farmlands. To maximize harvesting efficiency and ensure continuity, it is necessary to reduce the downtime of the harvester and predict unloading points when the grain tank is full. A collaborative unloading mode is proposed, where the transport vehicle moves to the unloading point.

However, considering the dynamic complexity of the path and the real-time nature of harvesting and transportation, this study ignores the transport vehicle’s loading and unloading path within the field. The unloading point is defined as the nearest key node from the harvester when it is fully loaded. When the harvester requires unloading, it sends a response instruction to the transport vehicle. To achieve the shortest scheduling path, the optimal responding transport vehicle is selected, and the shortest path to the unloading point is determined using Dijkstra’s algorithm. After unloading, the transport vehicle waits for the next response or returns to the machinery storage, while the harvester continues harvesting until all tasks are completed, after which it returns to the machinery storage.

4.3. Assumptions and Conditions

Based on the above problem description, various factors affecting the operations of the harvester and transport vehicle are considered, and a collaborative scheduling model with time window and capacity constraints is established. The optimization objectives are to minimize scheduling costs, minimize scheduling time, and minimize fuel consumption.

Let the farmland set be F = {f₁, f₂, …, f_k}, the harvester set be H = {h₁, h₂, …, h_k}, the transport vehicle set be G = {g₁, g₂, …, g_k}, and the task set be P = {p₁, p₂, p₃, p₄}, where p₁ represents transport, p₂ represents loading and unloading, p₃ represents harvesting operations, and p₄ represents return. Here, k represents the maximum number, and A represents the machinery storage. The problem can be described as follows: multiple harvesters h and multiple transport vehicles g need to complete multiple tasks p. For the sake of clarity, the following mathematical symbols are defined, as shown in

Table 1. Symbols and meaning.

Symbols	Meaning
Estart	The fuel consumption during cold start: This refers to the fuel consumption of the harvester and grain transport vehicle when the engine has been turned off for a period of time, and the temperature has dropped below the normal operating temperature at the time of ignition.
Estart1	Total startup fuel consumption: This refers to the fuel consumption during the process in which the engine reaches a stable minimum idle speed after starting.
Estart2	Total idle fuel consumption: This refers to the fuel consumption during the process from parking to starting the movement of the vehicle.
Estart(h), Estart(g),	Harvester starting fuel consumption, grain transport vehicle starting fuel consumption.
Eidle(h), Eidle(g),	Harvester idling fuel consumption, grain transport vehicle idling fuel consumption.
tstart(h), tstart(g),	Harvester start time, grain transport vehicle start time.
tidle(h), tidle(g)	Harvester idle time, grain transport vehicle idle time.
Eflied	Fuel consumption of harvesters during field operations.
Ework	Fuel consumption during harvesting in the field.
Eturn	Fuel consumption during turning in the field.
Ehf	Fuel consumption of harvesters h during harvesting.
lhf	The number of rows harvested by harvester h in field f.
Df	Distance per row in farmland f.
Eh	Fuel consumption of harvester h during empty runs.
qhf	The number of turns made by harvester h within a farm field f.
Dh	Turning distance of harvester h.
dij	Distance from node i to node j.
Eschedue	Total scheduling fuel consumption: fuel consumption generated by the movement of the harvester and the grain transport vehicle in the field.
Esh Esg	Fuel consumption for scheduling of harvester h. Fuel consumption for scheduling of grain transport vehicle g.
dhf-en	Distance from the entrance of the field to the starting point of harvesting for harvester h.
dhf-ex	Distance from the end point of harvesting to the exit of the field for harvester h.
Eallot	Empty run fuel consumption: fuel consumption generated by the harvester from the entrance of the field to the starting position of harvesting and from the end position of harvesting to the exit of the field.
Cd	Distance cost between the harvester h and the grain transport vehicle g.
Cf	Fixed costs (scheduling cost costs).
Cp	Total penalty cost for violating time windows.
dc	Unit distance travel cost.
Ch,Cg	Scheduling costs for harvester h and grain transport vehicle g.
Pthg	Penalty cost for violating time windows.
Th	Maximum time Th taken to complete all tasks among all harvesters.
Tg	Maximum time Tg taken to complete all tasks among all grain transport vehicles.
Uh,Lg	Unloading time and loading time, where Uh = Lg.
Vh,Vg	Movement speeds of harvester h and grain transport vehicle g.
th	Harvesting time of harvester h for field f.
Hv,Gv	Capacity of harvesters, capacity of grain transport vehicles.
qf	The amount of farmland f to be harvested.
Di	Planting density of farmland f.
Ueh	Grain unloading efficiency of harvester h.
Cw,Cl	Cost of waiting per unit of time, cost of penalization per unit of time.
Bi,Ei	Time allowed for the start of the task on the farmland f and the latest time.
thgf	The moment the harvester or grain transport vehicle arrives on the farmland f.
Xijh,Xijg	Decision variable: a binary variable indicating whether harvester h or grain transport vehicle g travels from node i to node j.

.

The following assumptions need to be made prior to the study:

(1)

Each farmland can be accessed by only one harvester and at least zero grain transport vehicles.

(2)

Both the harvester and grain transport vehicles depart from the machinery depot, returning to the depot after all tasks are completed, with the coordinates of the depot known.

(3)

The capacity and travel speed of both the harvester and grain transport vehicles are known.

(4)

The coordinates, time windows, and harvesting requirements for each farmland are known.

(5)

The maximum load capacity of the harvester and grain transport vehicle cannot exceed their respective capacities, and the grain transport vehicle must return to the machinery depot once fully loaded.

(6)

The grain transport vehicle can only move to key nodes to assist the harvester with loading and unloading grain.

(7)

The nearest key node to the position where the harvester is fully loaded is the unloading point.

(8)

The time, distance, and fuel consumption associated with the movement of the grain transport vehicle within the field are ignored.

(9)

It is assumed that there are no machinery failures or obstacles in the field during operations.

4.4. Collaborative Scheduling Model

Objective function:

$$\left\{\begin{array}{c}\mathrm{m}\mathrm{i}\mathrm{n}E=E_{start}+E_{flied}+E_{schedue}+E_{allot}\\ \mathrm{m}\mathrm{i}\mathrm{n}C={C_d+C}_f+C_p\\ \mathrm{m}\mathrm{i}\mathrm{n}T=\max (T_h,T_g)\end{array}\right.$$

(1)

In Equation (1), the fuel consumption E includes the cold start fuel consumption E_start, field harvesting fuel consumption E_field, scheduling fuel consumption E_schedule, and idle fuel consumption E_allot. The scheduling cost consists of the distance cost C_d, fixed cost C_f, and penalty cost for violating time windows C_p. The scheduling time refers to the maximum time taken by any harvester or grain transport vehicle, denoted as T_h or T_g.

Among them, the cold start fuel consumption includes startup fuel consumption and idle fuel consumption, as shown in Equation (2), with the calculation formula in Equation (3). Field harvesting fuel consumption includes both harvesting fuel consumption and turning fuel consumption, as shown in Equation (4), with the calculation detailed in Equation (5). Scheduling fuel consumption is shown in Equation (6), and idle fuel consumption is given in Equation (7). The scheduling cost is represented by Equation (8). The scheduling time is shown in Equation (9), where the scheduling time for the harvester includes the travel time, unloading time, and harvesting time. The scheduling time for the grain transport vehicle includes the travel time and loading time.

$$E_{start}=E_{start1}+E_{start2}$$

(2)

$$\left\{\begin{array}{c}{E}_{start1}={\displaystyle \sum _{h=1}^H} (E_{start\left(h\right)}{t}_{start\left(h\right)})+{\displaystyle \sum _{g=1}^G} \left(E_{start\left(g\right)}{t}_{start\left(g\right)}\right)\\ E_{start2}={\displaystyle \sum _{h=1}^H} (E_{idle\left(h\right)}{t}_{idle\left(h\right)})+{\displaystyle \sum _{g=1}^G} \left(E_{idle\left(g\right)}{t}_{idle\left(g\right)}\right)\end{array}\right.$$

(3)

$$E_{flied}=E_{work}+E_{turn}$$

(4)

$$\left\{\begin{array}{c}{E}_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}}={\displaystyle \sum _{f=1}^F} {\displaystyle \sum _{h=1}^H} {x_{ijh}E}_{hf}{l}_{hf}{D}_f\\ E_{\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{n}}={\displaystyle \sum _{f=1}^F} {\displaystyle \sum _{h=1}^H} E_{\mathrm{h}}{q}_{hf}{D}_h\end{array}\right.$$

(5)

$$E_{\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{e}}=\sum _{i=1}^F \sum _{j=1}^F \sum _{h=1}^H x_{ijh}{d}_{ij}{E}_{sh}+\sum _{i=1}^F \sum _{j=1}^F \sum _{g=0}^G x_{ijg}{d}_{ij}{E}_{sg}$$

(6)

$$E_{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{t}}={\sum }_{f=1}^F {\sum }_{h=1}^H x_{ijh}\lceil {\displaystyle \frac{l_{hf}}{l_{hf}+1}}\rceil E_{\mathrm{h}}(d_{hf-en}+d_{hf-ex})$$

(7)

$$\left\{\begin{array}{c}{C}_d=({\displaystyle \sum _{i=1}^F} {\displaystyle \sum _{j=1}^F} {\displaystyle \sum _{h=1}^H} x_{ijh}{d}_{ij}+{\displaystyle \sum _{i=1}^F} {\displaystyle \sum _{j=1}^F} {\displaystyle \sum _{g=0}^G} x_{ijg}{d}_{ij})d_{\mathrm{c}}\\ C_f={\displaystyle \sum _{h=1}^H} C_h+{\displaystyle \sum _{g=1}^G} C_{g }\\ C_p={\displaystyle \sum _{h=1}^H}{P}_{thg}+{\displaystyle \sum _{g=1}^G}{P}_{thg}\end{array}\right.$$

(8)

$$\left\{\begin{array}{c}{T}_h=\max ({\displaystyle \frac{{\sum }_{i=1}^{F+A} {\sum }_{j=1}^{F+A} x_{ijh}{d}_{ij}}{V_h}}+{\displaystyle \sum _{h=1}^H} U_h+{\displaystyle \sum _{h=1}^H} t_h)\\ T_t=\max ({\displaystyle \frac{{\sum }_{i=1}^{F+A} {\sum }_{j=1}^{F+A} x_{ijg}{d}_{ij}}{V_g}}+{\displaystyle \sum _{g=1}^G} L_g)\end{array}\right.$$

(9)

Constraints:

$$\left\{\begin{array}{c}{\displaystyle \sum _{i=1}^F} {\displaystyle \sum _{h=1}^H}{x}_{ijh}=1\\ {\displaystyle \sum _{i=1}^F} {\displaystyle \sum _{g=1}^G}{x}_{ijg}\ge 0\end{array}\right.$$

(10)

$$\left\{\begin{array}{c}{\displaystyle \sum _{h=1}^H} {\displaystyle \sum _{i\in \left\{A\right\}\cup F}} {\displaystyle \sum _{j\in F\cup \left\{A\right\}}} d_{ij}\cdot x_{ijh}={\displaystyle \sum _{h=1}^H} {\displaystyle \sum _{i\in F}} {\displaystyle \sum _{j\in F}} d_{ij}\cdot x_{ijh}\\ {\displaystyle \sum _{g=1}^G} {\displaystyle \sum _{i\in \left\{A\right\}\cup F}} {\displaystyle \sum _{j\in F\cup \left\{A\right\}}} d_{ij}\cdot x_{ijg}={\displaystyle \sum _{g=1}^G} {\displaystyle \sum _{i\in F}} {\displaystyle \sum _{j\in F}} d_{ij}\cdot x_{ijg}\end{array}\right.$$

(11)

$$\left\{\begin{array}{c}{\displaystyle \sum _{f=1}^F} q_f{x}_{ijh}\le H_v\\ {\displaystyle \sum _{f=1}^F} q_f{x}_{ijg}\le G_v \end{array}\right.$$

(12)

$$U_h={L_g=D}_i{H}_v/U_{eh}$$

(13)

$$P_{thg}=\left\{\begin{array}{c} C_w\left(B_i-t_{hgf}\right);\\ C_l\left(t_{hgf}-E_i\right);\\ 0,\mathrm{others};\end{array}\right.$$

(14)

$$\left\{\begin{array}{c}{X}_{ijh}=\left\{\begin{array}{c}1,\text{Harvester h travels from node i to node j}\\ 0,\text{Harvester h does not travel from node i to node j}\end{array}\right.\\ X_{ijg}=\left\{\begin{array}{c}1,\text{Grain transport vehicle g travels from node i to node j}\\ 0,\text{Grain transport vehicle g does not travel from node i to node j}\end{array}\right.\end{array}\right.$$

(15)

Equation (10) indicates that each farmland can be accessed by only one harvester and at least zero grain transport vehicles. Equation (11) specifies that all harvesters and grain transport vehicles depart from the machinery depot and return to the depot at the end of their tasks. Equation (12) states that the maximum load of the harvester and grain transport vehicle cannot exceed their respective capacities. Equation (13) implies that the unloading time and loading time are equal. Equation (14) represents the soft time window constraints. Equation (15) defines the decision variables X_ijh and X_ijg.

4.5. Response Strategy

The grain transport vehicles employ a responsive approach, with the harvester as the responsive entity. The response time U_t is determined based on factors such as the current capacity of the harvester, the time H_ht to reach full load, and the harvesting and unloading efficiency. The method calculates the time and distance for the responding grain transport vehicle to reach the unloading point and determines whether it involves idle time. After sorting, the optimal grain transport vehicle is selected to proceed to the unloading point. This approach is of significant practical importance for reducing the waiting time of the harvester and improving the efficiency of harvesting and transportation.

$$H_{ht}=U_t{h}_v/H_v$$

(16)

$$F_h=G_t+H_{ht}$$

(17)

In Equation (16), H_ht represents the time required for the harvester to complete the remaining harvesting tasks, U_t denotes the time when the harvester reaches the unloading state, and h_v indicates the current harvested capacity of the harvester. In Equation (17), F_h represents the time at which the harvester completes the harvesting operation for farmland f, and G_t denotes the travel time for grain transport vehicle g to reach farmland f.

To consider the real-time data required for the scheduling of grain transport vehicles, real-time transmission devices (e.g., sensors) are assumed to be used in the actual harvesting and transportation tasks. These devices provide feedback and corrections to acquire real-time data such as the harvested amount and the travel time to the response location. This study assumes that sensors for empty grain bins, full grain bins, and response times have been installed.

5. Collaborative Scheduling and Transportation Planning Based on IHHO

5.1. Model Solution Algorithm Design

The problem addressed in this chapter involves the collaborative scheduling of harvesters and grain transport vehicles, as well as the shortest path problem in the field, which belongs to the NP-Hard category in complex environments. The Harris Hawk Optimization (HHO) algorithm, a newly proposed metaheuristic algorithm in recent years [23], is utilized for solving this problem. This algorithm is inspired by the cooperative behavior and surprise attack hunting style of Harris hawks during prey capture. The optimization process of the HHO algorithm consists of three stages: exploration, exploration-exploitation transition, and exploitation. Its powerful local search capability has been widely applied to solve various optimization problems. The technical approach is shown in Figure 8, with the improvements indicated in red font.

5.2. Model Solution Process

Based on the collaborative scheduling model, the specific improvements to the HHO algorithm and solution process are as follows:

(1)

Parameter setting: Basic parameters for farmlands, machinery storage, harvesters, and grain transport vehicles are set, along with the relevant parameters for the HHO algorithm and slime mold algorithm.

(2)

Encoding and decoding: A real-number encoding scheme is used. The state of each hawk is represented by a string of floating-point numbers between 0 and 1, where each number represents the fitness value for a particular farmland. The fitness value changes based on the distance between the hawk and its prey, as well as the current state of both the hawk and the prey. The decoding process converts the hawk’s state into a specific farmland allocation plan.

(3)

Best point set (BPS) and quantum computing initialization: In the traditional HHO algorithm, the initial population is generated randomly. However, this initialization process can lead to uneven population distribution, reducing population diversity, which significantly affects the quality of the initial solution and convergence speed. Therefore, the BPS method [24] is employed to generate effective and uniformly distributed points, thereby improving global search efficiency. Theoretical research [25] shows that a weighted sum of n best points leads to a smaller error than using any other set of n points, particularly in complex search spaces. Let G_s represent a unit cube in the s-dimensional Euclidean space, and if g ∈ G_s, as shown in Equation (18), then P_n(k) denotes the best point set, with g being the best point.

$$P_n\left(\mathrm{k}\right)=\left\{\left(\left\{g_1^{\left(n\right)}\times k\right\},\left\{g_2^{\left(n\right)}\times k\right\},\cdots ,\left\{g_s^{\left(n\right)}\times k\right\}\right),1⩽k⩽n\right\}$$

(18)

$$\phi \left(n\right)=C(g,\epsilon )n^{-1+\epsilon }$$

(19)

Equation (19) represents the deviation, where C(g,ε) is a constant that depends only on g and ϵ(ϵ > 0). Let g = {2cos(2πk/p)}, where 1 ≤ k ≤ s, and p is the smallest prime number such that (p − 3/2) ≥ s.

Based on the initial population obtained from the BPS initialization, quantum computing [26] is further employed to generate additional initial populations. This process improves the distribution quality of the initial population within the search space. In this case, quantum bits (qubits) can simultaneously exist in a superposition of two quantum states:

$$|\phi ⟩=\alpha |0⟩+\beta |1⟩$$

(20)

In Equation (20), the quantum state ∣$\phi$⟩ is a superposition between the ∣0⟩ and ∣1⟩ states, with α and β representing the probability amplitudes of the quantum states ∣0⟩ and ∣1⟩, respectively. These amplitudes satisfy the normalization condition [27], as shown in Equation (21), where ∣α∣² corresponds to the probability of the quantum state being ∣0⟩, and ∣β∣² corresponds to the probability of the quantum state being ∣1⟩.

$${\left|\alpha \right|}^2+{\left|\beta \right|}^2=1$$

(21)

By superimposing multiple quantum bits, a larger state space can be generated. For example, the state of two quantum bits is shown in Equation (22). The Hadamard gate is then applied to the quantum bits [28] to transform them into a superposition state, as shown in Equation (23).

$$\left|\phi \right.⟩={\alpha }_{00}\left|00\right.⟩+{\alpha }_{01}|01⟩+{\alpha }_{10}|10⟩+{\alpha }_{11}|11⟩$$

(22)

$$\left\{\begin{array}{c}H\left|0\right.⟩={\displaystyle \frac{1}{\sqrt{2}}}(|0⟩+|1⟩)\\ H\left|1\right.⟩={\displaystyle \frac{1}{\sqrt{2}}}(|0⟩-|1⟩)\end{array}\right.$$

(23)

(4)

Objective function calculation: The HHO algorithm is typically a single-objective optimization algorithm, while the problem addressed in this study is a complex, multi-objective optimization problem, necessitating an extension of the HHO algorithm. However, multi-objective functions often conflict with one another, requiring the identification of a set of good solutions within the solution space. Therefore, a non-dominated sorting approach combined with a crowding distance mechanism to maintain diversity is employed. First, non-dominated solutions are calculated, followed by non-dominated sorting, and the non-dominated ranking of all solutions is computed. The crowding distance mechanism is then used to maintain diversity between solutions, ensuring that the diversity of solutions is effectively preserved during the optimization process. The crowding distance is defined as:

$$\left\{\begin{array}{c}D\left(i,j\right)={\displaystyle \frac{f_j(i+1)-f_j(i-1)}{f_j^{max}-f_j^{min}}}\\ D\left(i\right)={\displaystyle \sum _{j=1}^n} D(i,j)\end{array}\right.$$

(24)

In Equation (24), D(i,j) represents the distance of individual i in objective j; f_j^max and f_j^min are the maximum and minimum values of objective j; f_j(i + 1) and f_j(i − 1) are the neighboring values of individual i in objective j. D(i) represents the crowding distance of individual i, which is the sum of the distances in all objectives. The crowding distance for boundary points in each front is set to infinity.

(5)

Fitness calculation: The fitness value of each Harris hawk individual is calculated, the best fitness value is recorded, and the position of the Harris hawk with the optimal fitness is set as the prey position.

(6)

Exploration phase with fused slime mold algorithm (SMA): To address the issue where HHO can easily get trapped in local optima when solving complex problems, the multiple search mechanism of the SMA is incorporated into the exploration phase of HHO to enhance its global search ability [29]. First, the exploration phase of HHO conducts a broad search for high-quality solutions in the solution space, as shown in Equation (25). Then, a local search is performed on the high-quality solutions using SMA’s approach strategies, including the proximity strategy (Equation (27)), encircling strategy (Equation (30)), and acquiring strategy. Specifically, the multiple search mechanism of SMA adjusts different search patterns based on fitness values, while also separating a portion of the organic matter to explore other domains. The oscillation effect between vb (simulating the interaction process of individual slime mold information) and vc (simulating the slime mold’s retention of its own information) causes the search direction of the slime mold to become more divergent, thus increasing the possibility of global exploration. The multiple exploration mechanism endows the algorithm with strong global optimization capabilities. Additionally, when rand < z (where z represents the probability that the slime mold separates individual agents to search for other food sources), a portion of the slime mold population is randomly generated in the solution space, simulating the behavior of slime mold individuals separating and exploring other potential food sources.

$$X(t+1)=\left\{\begin{array}{cc}{(X}_{prey}\left(t\right)-X_m\left(t\right))-{rand}_3(lb+{rand}_4(ub-lb)),& q<0.5\\ X_{rand}\left(t\right)-{rand}_1\left|U_{rand}\left(t\right)-2ran{d}_{2}X\left(t\right)\right|,& q\ge 0.5\end{array}\right.$$

(25)

$$X_m\left(t\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N X_i\left(t\right)$$

(26)

Equation (25) represents the exploration phase of the HHO algorithm, where X(t + 1) denotes the position vector of the hawk in the next iteration t + 1, and t is the current iteration number. X_prey(t) represents the prey’s position, while X_m denotes the average position of the Harris hawk population at the current iteration, as shown in Equation (26). X_i(t) represents the position of each hawk i at iteration t, and N denotes the total number of hawks. X(t) is the current position vector of the hawk, and X_rand(t) represents a randomly selected individual from the population at generation t. The terms rand₁, rand₂, rand₃, rand₄ and q are random numbers in the range (0,1), and lb and ub denote the lower and upper bounds.

$$X(t+1)=\left\{\begin{array}{cc}{X}_b\left(t\right)+vb\cdot \left[W\cdot \left(X_A\right(t)-X_B(t\left)\right)\right],& r_1<p\\ vc\cdot X\left(t\right),& r_1\ge p\end{array}\right.$$

(27)

Equation (27) describes the approaching strategy, which simulates the slime mold’s behavior of searching for food sources based on air concentration. In this case, X_b(t) represents the position of the individual with the highest concentration (i.e., the highest fitness); vb and vc are oscillation parameters, where vb is a random number in the range [−a, a], with a = arctanh(1 − t/Max_t), and Max_t is the maximum iteration number. vc is a parameter that linearly decreases from 1 to 0 over time. W is the weight factor of the slime mold, corresponding to its mass, as shown in Equation (28). X_A(t) and X_B(t) are two randomly selected individuals in the slime mold population, and r₁ is a random number in the range [0, 1]. P is a condition parameter that controls the update of the slime mold position, calculated as shown in Equation (29), where S(t) denotes the fitness value of the individual, and DF represents the best fitness value found across all iterations.

$$W\left(Sindex\left(i\right)\right)=\left\{\begin{array}{c}1+r_2\cdot log\left({\displaystyle \frac{bF-S\left(i\right)}{bF-wF}}+1\right),condition\\ 1-r_2\cdot log\left({\displaystyle \frac{bF-S\left(i\right)}{bF-wF}}+1\right),others\end{array}\right.$$

(28)

Sindex represents the ascending order of population fitness values; r₂ is used to simulate the uncertainty of venous contraction and is a random number in the range of [0, 1]. Condition refers to the slime mold individuals with the top half of the fitness ranking in S(t). bF and wF denote the best and worst fitness values during the current iteration, respectively. Log is used to dampen the rate of change in values to make the contraction frequency change more gradual.

$$p=tanh\left|S\right(t)-DF|$$

(29)

$$X(t+1)=\left\{\begin{array}{cc}{rand}_5\cdot (ub-lb)+lb,& rand<z\\ X_b\left(t\right)+vb\cdot [W\cdot (X_A\left(t\right)-X_B\left(t\right)\left)\right],& r_1<p\\ vc\cdot X\left(t\right),& r_1\ge p\end{array}\right.$$

(30)

Equation (30) describes the encircling strategy that simulates the venous structure of the slime mold during the search process. Here, z is set to 0.03, and rand₅ is a random value in the range of [0, 1].

The final attraction strategy simulates how the food source attracts the slime mold, causing oscillations that alter the cytoplasmic flow within the venous network. As a result, the slime mold moves closer to the food source. The value of vb oscillates within the range of [−a, a], while vc oscillates within [−1, 1]. Both values gradually decrease to 0 as the iteration progresses.

(7)

Exploration and exploitation transition phase: A nonlinear prey escape energy update strategy is adopted. The key parameter that controls the exploration and exploitation phases (the global to local search transition) of the HHO algorithm is the prey escape energy El, as shown in Equation (31). If ∣El∣ ≥ 1, the algorithm enters the global search phase, i.e., the exploration phase. If ∣El∣ < 1, the algorithm enters the local search phase, i.e., the exploitation phase. The energy factor E is updated in a linear fashion from 2 to 0. Figure 9 illustrates the change in E over 1000 iterations. This linear decrease causes the algorithm to get trapped in local search during the later stages of iteration. To address this, a nonlinear energy factor update strategy, based on the experience in [30], is adopted, as shown in Equation (32). This strategy allows the algorithm to maintain the possibility of global search while performing local search in the later iterations. The parameter δ is set to 3.5. Figure 10 shows the variation in El over 1000 iterations. The improved El changes slowly in the early stages of the algorithm, enhancing the global search performance of HHO, while in the later stages, the reduction rate of El accelerates, improving the local search performance and search efficiency of the algorithm.

$$El=E\times E_0$$

(31)

$$E=2\times (1-{(t/\text{Max\_t})}^{\delta })$$

(32)

$$E_0=2 \ast {rand}_6-1$$

(33)

In Equation (33), E₀ represents the initial energy of the prey, and rand₆ is a random number in the range of [0, 1).

(8)

Development strategy: After locating the prey, the Harris hawk initiates an attack during the development phase by encircling the prey, waiting for an opportunity to strike. However, the actual hunting process is complex. For instance, a prey surrounded by the hawk may escape the encirclement, prompting the hawk to adjust its actions based on the prey’s behavior. Consequently, the HHO algorithm employs four strategies to mimic the hunting behavior of the Harris hawk. These strategies are soft encirclement, hard encirclement, progressive rapid dive with soft encirclement, and progressive rapid dive with hard encirclement.

Let r₃ denote the escape probability of the prey, which is a random number between 0 and 1. When r₃ ≥ 0.5, it indicates that the prey has not successfully escaped, and vice versa. The hunting strategy is determined by combining the escape energy ∣E∣ and the escape probability r₃.

• Soft encirclement when 0.5 ≤ ∣E∣ < 1 and r₃ ≥ 0.5

When the prey still possesses escape energy and attempts to escape by random jumps from the encirclement, the hawk adopts soft encirclement to exhaust the prey, enabling the hawk to ambush it. The position update formula is as follows:

$$X\left(t+1\right)=\Delta X\left(t\right)-E\left|J{X}_{prey}\left(t\right)-X\left(t\right)\right|$$

(34)

$$\Delta X\left(t\right)=X_{prey}\left(t\right)-X\left(t\right)$$

(35)

In Equation (34), ΔX(t) represents the difference between the prey’s position and the hawk’s position at iteration t, as described in Equation (35). J = 2(1 − r₄) represents the intensity of the random jumps made by the prey during its escape, where r₄ is a random number between 0 and 1. The value of J changes randomly in each iteration, simulating the inherent movement of the prey.

b.

Hard encirclement when ∣E∣ < 0.5 and r₃ ≥ 0.5

The prey has no energy to escape and no opportunity to flee. The Harris hawk uses a hard boundary to capture the prey for a final assault. Its updated formula is given by Equation (36).

$$X\left(t+1\right)=X_{prey}\left(t\right)-E\left|\Delta X\left(t\right)\right|$$

(36)

c.

Progressive rapid dive with soft encirclement when 0.5 ≤ ∣E∣ < 1 and r₃ < 0.5

When the prey has the potential to escape from the encirclement and possesses sufficient energy to avoid capture, the hawk adopts a progressive rapid dive with soft encirclement. The hawk adjusts its position gradually based on the deceptive behavior of the prey (e.g., Lévy flights) to capture it effectively. This is implemented in two strategies, with the second strategy used if the first one fails:

$$U\left(t+1\right)=\left\{\begin{array}{c}Y=X_{prey}\left(t\right)-E\left|J{X}_{prey}\left(t\right)-X\left(t\right)\right| ifF\left(Y\right)<F\left(X\left(t\right)\right)\\ Z=Y+S\times L{e}^{\prime }\nu y\left(D\right)ifF\left(Z\right)<F\left(X\left(t\right)\right)\end{array}\right.$$

(37)

In Equation (37), F() is the fitness function, D is the problem’s dimensionality, and S is a random vector between 0 and 1. The term Le′νy represents the Lévy flight strategy, given by:

$$L{e}^{\prime }\nu y\left(x\right)=0.01\times {\displaystyle \frac{u\times \sigma }{{\left|v\right|}^{\frac{1}{\beta }}}},\sigma ={\left({\displaystyle \frac{\Gamma (1+\beta )\times sin\left(\frac{\pi \beta }{2}\right)}{\Gamma \left(\frac{1+\beta }{2}\right)\times \beta \times 2^{\left(\frac{\beta -1}{2}\right)}}}\right)}^{{\displaystyle \frac{1}{\beta }}}$$

(38)

In Equation (38), u and v are random numbers between 0 and 1, and β is set to 1.5.

d.

Progressive rapid dive with hard encirclement when ∣E∣ < 0.5 and r₃ < 0.5

The prey is exhausted but still has a chance to escape. The Harris hawks use a progressive rapid dive to harden the encirclement of the prey. This strategy’s position update formula for the hawk is similar to the one used in the progressive rapid dive for soft encirclement. In this case, the Harris hawk group attempts to reduce the average distance between their positions and the target prey. The update formula is given in Equation (39), and the formula for X_m(t) is provided in Equation (26).

$$X\left(t+1\right)=\left\{\begin{array}{c}Y=X_{prey}\left(t\right)-E\left|J{X}_{prey}\left(t\right)-X_m\left(t\right)\right| ifF\left(Y\right)<F\left(X\left(t\right)\right)\\ Z=Y+S\times L{e}^{\prime }\nu y\left(D\right)ifF\left(Z\right)<F\left(X\left(t\right)\right)\end{array}\right.$$

(39)

(9)

Average difference mutation strategy: Due to the convergence of Harris hawks in the population towards the optimal individual during the iterative process, if there are local optima, Harris hawks tend to cluster around them as the number of iterations increases, which can lead to a reduction in population diversity. To avoid the algorithm getting trapped in local optima and prevent premature convergence, an average difference mutation [31] is introduced to adjust the individual and global best solutions.

First, define the vector population in a D-dimensional search space as x_i = [x_i,1, x_i,2, …, x_i,D], where i = 1, 2, …, Popsize, with Popsize representing the population size. Two random vectors, x_r1 and x_r2, are selected, and new vectors are generated using their average, as shown in Equation (40):

$$\left\{\begin{array}{c}{x}_{c1}={\displaystyle \frac{{\mathrm{x}}_{r1}+{\mathrm{x}}_{r2}}{2}}\\ x_{c2}={\displaystyle \frac{x_{r1}+x_{best}}{2}}\end{array}\right.$$

(40)

where x_best represents the current best solution.

Next, two strategies are employed depending on the iteration stage. The rules are as follows: during the first two-thirds of the iterations, if FES ≤ 0.6 × MAX_FES or rand ≤ pm, the first mutation is executed. For the remaining iterations, the second mutation is applied with a certain probability to enhance the search. A mutation vector vi is obtained in each iteration, as described in Equation (41):

$${\mathrm{v}}_i=\left\{\begin{array}{c}{\mathrm{x}}_{c1}+F\left({\mathrm{x}}_{c1}-{\mathrm{x}}_i\right)+F\left({\mathrm{x}}_{c2}-{\mathrm{x}}_i\right)\\ {\mathrm{x}}_{best}+F\left({\mathrm{x}}_{c1}-{\mathrm{x}}_i\right)+F\left({\mathrm{x}}_{c2}-{\mathrm{x}}_i\right) \end{array}\right.$$

(41)

where FES represents the number of function evaluations, and MAX_FES denotes the maximum number of function evaluations. pm is an internal mutation switch parameter that controls exploration and exploitation during the later iterations, with a value of 0.6. F is a scaling factor, with the mutation range for the first variation being [0.2, 0.25], and for the second mutation, rand(−,+) × 0.5, where rand(−,+) represents a random number between −1 and 1, aiding in balanced movement.

This mutation strategy helps the algorithm to search locally during the early stages and shift towards global exploration near the best solution in the later stages, which assists in escaping local optima.

(10)

Updating objective function and fitness values. After updating the objective function and fitness values, the individual with the maximum fitness is identified, and the fitness value is returned.

(11)

Iterative evolution. The algorithm checks whether it has reached the specified number of iterations. If so, the algorithm terminates, and the optimal result is decoded. Otherwise, the algorithm continues iterating. The pseudocode for the IHHO algorithm is shown in Algorithm 1.

Algorithm 1. IHHO algorithm pseudocode

Input: The population size Popsize and maximum number of iterations Max_t

Output: Prey location and fitness values

Initialize the population using Equations (18) and (20)–(23) for the Canonical point set and quantum computation.

While termination criteria have not been satisfied do

Calculate the fitness values of hawks

Set Xprey as the location of prey (best location)

$\hspace{1em}\mathbf{For} (\mathrm{each} \mathrm{hawk} (X_i$))) do

Update the initial energy E0 and jump strength J

Update the E using Equation (32)

if (∣E ≥ 1∣) then     ->Exploratory phase of the fusion mucilage algorithm

Update the location vector using Equations (25), (27) and (30).

if (∣E∣ < 1) then            ->Exploitation phase

if (0.5 ≤ ∣E∣ < 1 and r3 ≥ 0.5) then     ->Soft besiege

Update the location vector using Equation (34)

else if (∣E∣ < 0.5 and r3 ≥ 0.5) then     ->Hard besiege

Update the location vector using Equation (36)

else if (0.5 ≤ ∣E∣ < 1 and r3 < 0.5) then    ->Soft besiege with progressive rapid dives

Update the location vector using Equation (37)

else if (∣E∣ < 0.5 and r3 < 0.5) then    ->Hard besiege with progressive rapid dives

Update the location vector using Equation (39)

Equation (41) was used for the mean difference variance strategy.

Return Xprey

6. Example Studies

6.1. Experimental Data

To validate the subsequent analysis, the experiments were conducted using the MATLAB 2022b platform, which ran on a computer equipped with an Intel i5-8250U CPU (1.60 GHz) and 8 GB of RAM, with the operating system being Windows 10. For the algorithmic part, the parameter settings for the IHHO algorithm are provided in

Table 2. Algorithm parameter settings.

Algorithm Parameter Setting
Max_t ← 50	% Number of iterations
Popsize ← 20	% Population size
ub ← 10	% Upper bound
lb ← −10	% Lower bound
dim ← 30	% Dimension
nq ← 4	% Number of quantum bits

. The experimental parameters are listed in

Table 3. Experimental parameter settings.

Experimental Parameter Setting
Vh ← 10	% Harvester travel speed
Vg ← 10	% Grain transport vehicle travel speed
Hv ← [1000, 1700]	% Harvesters capacity
Gv ← 4000	% Grain transport vehicle capacity
Ueh ← [50/3, 85/3]	% Harvester unit unloading efficiency(m3/min)
gn ← 10	% Harvest per unit area
hc ← [300, 400]	% Harvester transfer, maintenance and fuel costs
gc ← 600	% Transfers, maintenance and fuel costs for grain transport vehicles
dc ← 1	% Cost per unit distance traveled
Cw ← 1	% Cost of waiting per unit of time
Cl ← 1	% Unit time penalty cost

. The basic information of the corn harvester and grain transport vehicles is shown in

Table 4. Basic information of agricultural machinery.

Name	Volume (kg)	Model	Maximum Usage
Wode 4LZ-8.0EZ(Q) crawler corn harvester	1700	4LZ-8.0EZ(Q) (China Jining Fengtuo Machinery Technology Co., Jining, China)	5
Huishou 4YZP-2D crawler two row corn harvester	1000	4YZP-2D (China Shenyinong (Shandong) Agricultural Equipment Co., Weifang, China)	5
Jinan jinwang JW-4t crawler grain transport vehicle	4000	JW-4t (China Jining Jinwang Machinery Equipment Co., Jining, China)	5

, which includes model, capacity, power, and maximum number of units, among other details. It is assumed that the purchase cost of agricultural machinery is not considered in this experiment. The field information is presented in

Table 5. Basic field information.

Fields No.	Area/m2	Harvest to Be Harvested /m3	Time Window
A	/	/	[0, 1000]
1	1073	536.5	[242, 580]
2	1450	725	[160, 859]
3	722	361	[613, 771]
4	1963	981.5	[381, 651]
5	1724	862	[273, 625]
6	1485	742.5	[645, 809]
7	1448	724	[506, 950]
8	1182	591	[562, 900]
9	1360	680	[280, 920]
10	915	457.5	[659, 740]
11	1190	595	[94, 885]
12	1921	960.5	[366, 555]
13	2117	1058.5	[288, 655]
14	1638	819	[441, 875]
15	2040	1020	[605, 925]
16	1194	597	[513, 777]
17	844	422	[563, 790]
18	818	409	[815, 1111]
19	896	448	[340, 888]
20	1415	707.5	[886, 999]

, which includes the field area, harvestable quantity, and time window. The agricultural machinery depot is denoted by the letter “A” and is located at latitude 31°33′25.15″ N and longitude 104°33′40.51″ E.

6.2. Experimental Results

The IHHO algorithm was applied to conduct the solving experiments. The working paths of corn harvesters 1, 2, and 3 in the field are shown in Figure 11, Figure 12, and Figure 13, respectively. The scheduling paths of grain transporters 1, 2, and 3 in the field are illustrated in Figure 14, Figure 15 and Figure 16. The Gantt chart for the coordination between the corn harvesters and grain transporters is shown in Figure 17. The agricultural machinery depot is represented by black dots labeled with the letter “A”, and the black circular dots indicate the key nodes closest to the actual unloading points. The shortest paths for each piece of agricultural machinery may overlap, as Dijkstra’s algorithm could generate identical routes for different machines. Different colors are used to represent each transfer route.

To verify the superiority and applicability of the IHHO algorithm, comparative experiments were conducted with the traditional HHO algorithm and the HHO algorithm presented in the literature [32]. The Gantt charts for the HHO algorithm and the one from the literature [32] are shown in Figure 18 and Figure 19, respectively. The route plan comparison is presented in

Table 6. Route plan comparison.

	Harvester Routes	Grain Transport Vehicle Routes
IHHO	H1(V1200): A->1->11->6->17->18->19->12->16->A H2(V1200): A->13->20->14->15->A H3(V1700): A->3->2->7->8->4->9->5->10->A	G1: A->3->11->8->20->19->12->A G2: A->1->7->17->18->14->A->10->16->A G3: A->2->13->6->4->A->9->5->15->A
HHO	H1(V1700): A->8->13->14->20->10->5->9->4->A H2(V1700): A->7->12->11->15->A H3(V1200): A->3->2->1->6->16->17->18->19->A	G1: A->3->12->6->17->19->A G2: A->8->13->14->A->10->5->A G3: A->7->1->20->15->A->4->A G4: A->2->11->16->18->9->A
HHO algorithm in the literature [32]	H1(V1700): A->1->11->6->16->13->A H2(V1200): A->4->8->14->20->15->10->5->9->A H3(V1200): A->3->2->7->12->19->18->17->A	G1: A->3->7->16->17->5->A G2: A->1->8->14->15->A G3: A->2->12->20->A->9->A G4: A->4->6->18->10->A G5: A->11->19->13->A

. In the figures, the letter “H” represents a harvester, and the letter “G” represents a grain transporter. The numerical suffix indicates the machine number. The values in parentheses for the harvesters, V₁₂₀₀ and V₁₇₀₀, denote the harvester’s capacity of 1.2 cubic meters and 1.7 cubic meters, respectively.

To analyze the performance of the improved algorithm in terms of cost, time, and fuel consumption, experiments were conducted with 50 iterations for each of the three algorithms. The comparison of iteration curves is shown in Figure 20, where the scheduling cost comparison, scheduling time comparison, and fuel consumption comparison are shown in Figure 20a–c, respectively. The corresponding data comparisons are shown in

Table 7. Comparison of cost, time, and fuel consumption data.

	Scheduling Costs (CNY)	Scheduling Time (min)	Fuel Consumption (L)
IHHO	4819	147	208.8
HHO	5012	157	217.9
HHO algorithm in the literature [32]	5633	160	215.1

.

As analyzed from Figure 20, compared to the traditional HHO and the HHO algorithm in the literature [32], the IHHO algorithm in this study provides better solutions in terms of scheduling cost, scheduling time, and fuel consumption reduction. Moreover, the IHHO algorithm exhibits the fastest convergence speed in solving the three objective functions. This indicates that the proposed algorithm exhibits certain advantages in solving accuracy and optimization capability. The final iteration data in Table 7 show that, compared to the HHO and the algorithm in the literature [32], the IHHO algorithm reduces scheduling costs by 3.9% and 14.5%, respectively, scheduling time by 6.4% and 8.1%, and fuel consumption by 4.2% and 2.9%, with the lowest solution time. Therefore, the IHHO algorithm proposed in this study is capable of solving the path planning and coordination scheduling problem for harvest and transportation in hilly areas, achieving satisfactory solution quality. However, a single experiment cannot effectively prove the algorithm’s stability and robustness. Therefore, multiple experiments are necessary to verify whether the IHHO algorithm is suitable for solving multi-machine field path planning and coordinated scheduling problems in hilly regions.

7. Statistical Analysis of the IHHO Algorithm

To comprehensively evaluate the performance of the IHHO algorithm, 50 independent experiments were conducted for each of the three algorithms. The experimental results for the IHHO, HHO, and the algorithm from the literature [32] are presented separately in the Appendix A 

Table A1. Results of 50 experiments with the IHHO algorithm.

No.	Scheduling Costs (CNY)	Scheduling Time (min)	Fuel Consumption (L)	No.	Scheduling Costs (CNY)	Scheduling Time (min)	Fuel Consumption (L)
1	4819	147	208.8	26	4827	150	207.8
2	4851	146	209.5	27	4848	149	209.2
3	4855	147	208.4	28	4782	145	208.9
4	4819	144	207.9	29	4796	144	208.3
5	4801	146	209.3	30	4806	144	207.3
6	4828	150	209.6	31	4859	147	209.5
7	4794	149	209.3	32	4831	144	209.1
8	4838	150	209	33	4800	149	209.3
9	4805	146	208.8	34	4815	149	209.6
10	4852	144	209	35	4806	144	207.3
11	4824	148	208.3	36	4828	150	208.7
12	4800	148	208.3	37	4792	147	209.5
13	4792	148	208.7	38	4785	148	207.7
14	4841	149	209.1	39	4803	149	207.8
15	4855	146	208.2	40	4855	150	208.2
16	4847	146	208.1	41	4781	148	207.9
17	4839	150	207.3	42	4818	150	207.8
18	4846	144	208.4	43	4827	150	208.2
19	4830	147	207.8	44	4841	150	208.1
20	4828	145	208.4	45	4859	148	208.2
21	4859	149	208.3	46	4794	148	207.9
22	4864	150	208.2	47	4787	146	209.2
23	4796	144	207.6	48	4815	145	209.5
24	4837	145	208.6	49	4842	149	207.9
25	4804	149	208.8	50	4842	148	207.8

,

Table A2. Results of 50 experiments with the HHO algorithm.

No.	Scheduling Costs (CNY)	Scheduling Time (min)	Fuel Consumption (L)	No.	Scheduling Costs (CNY)	Scheduling Time (min)	Fuel Consumption (L)
1	5012	157	217.9	26	5026	155	215.6
2	4991	156	215.6	27	4994	157	214.2
3	4966	155	218.4	28	4996	149	215.8
4	5060	149	214.4	29	5071	158	215.9
5	4983	159	214.7	30	4993	149	215
6	4963	155	218.1	31	4962	159	215.3
7	5113	149	218	32	5043	154	215.4
8	4987	152	215	33	5053	149	217.1
9	5012	152	217.7	34	5021	151	218.8
10	5067	149	215.6	35	5113	153	217.7
11	4996	158	217.5	36	5082	150	214.7
12	5044	158	216.4	37	5007	156	218.3
13	5120	158	218.8	38	5073	155	217.5
14	5120	159	214.1	39	4964	156	218.4
15	4963	155	218.6	40	5049	150	215.6
16	5121	158	219.2	41	5096	160	216.2
17	5099	149	218.3	42	5090	156	215.1
18	5028	154	215.3	43	5093	158	215.1
19	5114	152	214.2	44	5006	160	218.5
20	4961	154	216.5	45	5064	158	217.8
21	5034	159	217.2	46	5110	149	214.5
22	5001	153	214.6	47	5097	160	216.1
23	5041	149	214.1	48	5072	158	215.6
24	4978	156	214.7	49	5036	151	215
25	5062	160	219.1	50	4961	154	218.6

and

Table A3. Results of 50 experiments with the algorithm from the literature [32].

No.	Scheduling Costs (CNY)	Scheduling Time (min)	Fuel Consumption (L)	No.	Scheduling Costs (CNY)	Scheduling Time (min)	Fuel Consumption (L)
1	5633	160	215.1	26	5606	160	216.2
2	5541	162	216.7	27	5702	157	217.3
3	5738	159	217.1	28	5631	158	216.2
4	5619	162	214.7	29	5706	161	214
5	5674	162	215	30	5643	160	214.9
6	5699	159	214.5	31	5676	160	214.4
7	5663	161	217.1	32	5589	161	216
8	5642	161	215.1	33	5711	157	216.8
9	5614	158	217.4	34	5739	157	215.5
10	5743	162	216.5	35	5737	161	215.9
11	5603	162	216.7	36	5755	163	214.7
12	5608	157	216.7	37	5754	158	216.3
13	5678	157	217.2	38	5743	161	215.8
14	5619	161	214.8	39	5541	162	214.8
15	5674	159	213.8	40	5675	160	216.5
16	5608	164	214.9	41	5710	162	214.3
17	5547	163	214.4	42	5715	163	214.7
18	5667	160	216.7	43	5634	162	215.9
19	5672	157	214.7	44	5633	162	214.5
20	5623	161	213.9	45	5675	159	217.3
21	5650	164	214.1	46	5639	157	215
22	5578	157	215.2	47	5560	164	215
23	5717	157	214.4	48	5559	161	217.4
24	5713	163	216.9	49	5587	157	214.5
25	5632	164	215.2	50	5577	158	217.2

.

The box plot comparisons of cost, time, and fuel consumption for 50 experimental runs are presented and analyzed separately, as shown in Figure 21, Figure 22 and Figure 23.

First, by comparing the positions of the boxes (data distribution) and the lengths (the interquartile range, i.e., the distance between Q3 and Q1) of the three objective functions’ boxplots [33], it is observed that the IHHO algorithm exhibits less data fluctuation and is more concentrated compared to the HHO algorithm and the algorithm from the literature [32]. This phenomenon indicates that the IHHO algorithm demonstrates better robustness. Next, by examining the minimum value (Min), maximum value (Max), and the median (Median), it becomes evident that the IHHO algorithm presents significantly better data. Furthermore, the Kruskal–Wallis ANOVA test was conducted, and the corresponding p-values [34] were found to be less than 0.05, indicating a significant difference between the three algorithms. Finally, by comparing the interquartile range (IQR) of the three objective functions, it is observed that the IHHO algorithm has fewer outliers in terms of optimal cost, time, and fuel consumption [35].

Through calculations, it was found that the IHHO algorithm reduced the average scheduling cost by 4.2% compared to the HHO algorithm and by 14.5% compared to the algorithm in the literature [32]. In terms of average scheduling time, the reductions were 4.5% compared to the HHO and 8.1% compared to the algorithm in the literature [32]. Regarding average fuel consumption, the IHHO algorithm reduced the consumption by 3.5% and 3.2%, respectively. Overall, it can be concluded that the IHHO algorithm proposed in this study outperforms the HHO algorithm and the algorithm from the literature [32], demonstrating superior performance and stability.

8. Conclusions

This paper investigates field path-planning methods for harvesters and grain transporters with varying capacities, suitable for hilly regions. First, considering the irregular and dense three-dimensional environmental characteristics of the fields in such regions, and addressing the issue of excessive traversal path lengths during corn harvesting, an environmental model for full coverage of multiple fields by corn harvesters and a field road network are established. The Dijkstra algorithm is then employed to solve the shortest path between fields. Subsequently, to tackle the issue of low efficiency in field collection and transportation, high fuel consumption, and high scheduling costs in multiple fields, a multi-objective collaborative scheduling model is formulated. An Improved Harris Hawk Optimization (IHHO) algorithm is designed to solve this model. Finally, compared with the traditional HHO and the path-planning solutions obtained by the HHO algorithm in the literature, the proposed method yields more optimal paths and effectively reduces fuel consumption, costs, and improves efficiency. The main conclusions of this study are as follows:

(1) To address the issue of excessive traversal paths for corn harvesting in the complex environment of hilly regions, a field road network for transfers is designed. Based on the Dijkstra algorithm, the node distances between any two fields are obtained, and this distance matrix is then input into the IHHO algorithm for optimization, ultimately resulting in the optimized harvesting sequence and grain transport routes.

(2) A multi-objective cooperative scheduling model is developed and solved using an improved IHHO algorithm. The algorithm improves the initial population distribution quality by introducing a candidate set and quantum computing initialization. The slime mold algorithm is incorporated during the exploration phase to enhance the global search ability. A non-linear energy factor (E) update method is applied to improve local search performance. Additionally, an average difference mutation strategy is introduced to help the algorithm escape local optima. Non-dominated sorting and crowding distance calculations are also integrated to optimize the solution set, enhancing both solution quality and diversity.

(3) Experimental results and statistical analysis show that the IHHO algorithm reduces the average scheduling cost by 4.2% compared to HHO and 14.5% compared to the method in the literature [32]. The average scheduling time is reduced by 4.5% and 8.1%, and average fuel consumption is decreased by 3.5% and 3.2%, respectively. These findings demonstrate the improved optimization capability and performance of the modified HHO algorithm. The enhanced HHO algorithm presented in this paper effectively addresses the requirements for collection and transportation operations in hilly and mountainous regions, thereby offering valuable support for the rational planning of the entire collaborative scheduling process.

(4) Outlook: The proposed IHHO algorithm, while tailored for hilly regions, holds great potential for adaptation to other terrains. By adjusting the environmental model and optimizing the road network design, the algorithm can be extended to flat, mountainous, or even other types of irregular terrains. Further research could explore the algorithm’s applicability in various agricultural scenarios, including urban farming or large-scale industrial operations, to evaluate its versatility and scalability across different settings.