Construction of Orchard Agricultural Machinery Dispatching Model Based on Improved Beetle Optimization Algorithm

Abstract

In order to enhance orchard agricultural efficiency and lower fruit production expenses, we propose a BL-DBO (Beetle Optimization Algorithm introducing Bernoulli mapping and Lévy flights) to solve the agricultural machinery dispatching model within the orchard area. First, we analyze the agricultural machinery dispatching problem in the orchard area and establish its mathematical model with the objective of minimizing dispatching costs as a constraint. To tackle the problems of uneven individual position distribution and the risk of becoming stuck in local optimal solutions in the traditional DBO algorithm, we introduce Bernoulli mapping during the initialization phase of the DBO. This method ensures a uniform distribution of the initialized population. Furthermore, during the iterative process of the algorithm, we incorporated the Lévy flight approach into the positional update equations for beetles involved in breeding, foraging, and theft activities within the DBO. This helps the beetles escape from local optimal solutions. Finally, we conduct experiments based on location information of Shunping Shunnong Orchard and fruit trees in Shijiazhuang. The results indicate that, compared to dispatching using human experience and the traditional DBO algorithm, the dispatching results generated by the BL-DBO not only reduce the number of agricultural machinery purchases but also decrease the energy loss from non-working distances of the machinery, effectively saving fruit production costs.

1. Introduction

Within the broader scope of land redistribution, extensive planting has fostered an environment conducive to the smart advancement of agricultural equipment [1,2]. The use of GPS-based navigation systems, such as the BDS (Beidou Navigation Satellite System), Glonass, and Galileo systems, is on the rise in precision farming practices [3]. Navigation systems are comprised of modules for perceiving information, measuring positioning and attitude, planning paths, and controlling tracking [4]. Path planning stands out as a crucial aid in deploying autonomous driving systems within engineering contexts [5]. By leveraging computer technology to map out the operational routes of agricultural machinery for extensive farmlands with established boundary data, it offers vital guidance for self-driving agricultural machinery and direction for human driving techniques [6,7].

Currently, the cultivation model of orchards in China is rapidly developing towards dwarf and dense planting. Compared to traditional tree-type cultivation models, the new cultivation model is more conducive to the mechanization and intelligence of orchard management. The issue of agricultural machinery path planning within a single orchard has received widespread attention from researchers, who have developed various solutions [8]. For example, in 2008, Bochtis proposed the B-patterns model, which represents the coverage operation of agricultural machinery over farmland as a directed graph traversal problem, transforming the optimal path problem into finding a sequence traversal that minimizes the total non-working distance [9]. In 2015, Bochtis and others applied the improved B-patterns model to orchard environments, reducing the non-working time of traversal paths by 10.7% to 32.4% [10]. Back in 2024, our group suggested an enhanced path-planning algorithm for orchard mowers, utilizing a refined ant colony algorithm capable of significantly lowering both operating duration and fuel usage relative to conventional ant colony algorithms [11].

Unlike large fields, orchards in China are predominantly distributed in hilly and mountainous areas. Although farm owners can manage large areas of land, due to topographical constraints, these lands cannot form a regular orchard area but are instead dispersed into many small plots [12,13]. Typically, farm owners do not equip each small area with a complete set of management machinery, as this would make production costs prohibitively high. Instead, they establish a facility to store agricultural machinery and transport the equipment to a small plot when work is needed [14]. In this model, relying solely on manual scheduling within the area cannot balance the supply and demand of agricultural machinery or achieve reasonable coordination, which can lead to missed orchard management work times or resource waste [15]. To address these issues, it is necessary to design a reasonable scheduling model for agricultural machinery suitable for orchard areas. A reasonable scheduling model can help agricultural machinery avoid a significant amount of non-working time, reduce the turnover of machinery between different small orchards, and lower unnecessary energy consumption while saving operational time.

The DBO (Dung Beetle Optimizer) is an emerging optimization algorithm proposed in recent years, designed to mimic the natural behavior of dung beetles to solve complex optimization problems [16]. Dung beetles are known in nature for their unique methods of food acquisition and reproduction, as they roll dung balls to attract mates and provide nutrients for their offspring [17]. This process involves not only searching for the best food sources but also reflects the dung beetle’s ability to explore and exploit its environment [18]. The DBO accurately mimics various physical interactions within the dung beetle community, encompassing both the attraction between individuals and their respective targets, as well as the near and far-reaching allure among individuals [19]. The DBO persistently adjusts the location of each dung beetle to mimic their flight patterns, encouraging evolutionary progress toward improved solutions while preserving the population’s overall composition [20]. In contrast to alternative swarm intelligence methods, the DBO more precisely mirrors the evolving dynamics in the dung beetle community and demonstrates quicker convergence. Similar to various swarm intelligence optimization methods, the DBO is also plagued by disparities in worldwide exploration and local exploitation skills, potentially resulting in local peaks and diminished overall exploration capacities [20,21,22,23,24,25,26].

In summary, this paper proposes an optimized Dung Beetle Optimizer algorithm to construct a scheduling model for orchard areas. The remaining sections of this paper are organized as follows: Section 2 establishes the scheduling model for agricultural machinery in orchard areas. Section 3 introduces the improved BL-DBO, demonstrating its superiority through ablation experiments, as well as presenting the experimental base of this study. In Section 4, we compare the BL-DBO with other traditional algorithms and examine the outcomes of this comparison. Section 5 provides a summary of this research and outlines our plans for future studies.

2. Problem Statement and Model Development

2.1. Problem Statement

In the problem of cross-regional scheduling of agricultural machinery, the scheduling cost issue is often analyzed by drawing an analogy to the optimization production cost (OPC) problem. Under the premise of ensuring scheduling operations and work efficiency, the scheduling cost within the region is considered to consist of the agricultural machinery purchase costs, path scheduling costs, and operational costs. For multi-machine scheduling problems, the situation is often quite complex. Therefore, to emphasize the scheduling cost, we make the following reasonable assumptions:

• The scheduling of each agricultural machine starts from the storage facility, and after completing all operational tasks, it returns to the facility;
• The operations of each agricultural machine are independent of each other;
• The scheduling considers only the same type of agricultural machinery fleet, and the path costs for each machinery unit are the same;
• The scheduling costs are calculated based on distance, without considering the operating costs of the agricultural machinery.

The scheduling cost model proposed in this paper for orchard areas starts from a system-wide management perspective, aiming to minimize scheduling costs. This means minimizing the sum of agricultural machinery purchase costs, path costs for scheduling within the region, and operational costs. First, we consider the agricultural machinery purchase costs required to meet scheduling demands within the region. Additionally, due to the complex topography of hilly and mountainous areas, this model also takes into account the path costs and operational costs during the scheduling process. Moreover, the number of agricultural machines required in a region is not fixed; it fluctuates with changes in demand and related information about farmland and machinery, leading to variations in agricultural machinery purchase costs.

2.2. Model Development

According to the above analysis, the scheduling model is as follows:

$$F_1=\min ({\displaystyle \sum _{A=1}^A(S_1+S_2)}+{\displaystyle \sum _{A=1}^A{\displaystyle \sum _{T=1}^T{S}_T}})$$

(1)

$$F_2=\min ({\displaystyle \sum _{j=1}^{n}L(A,T_j)})\beta +{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^{n}L(A,T_j)\eta +{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^{n}L(A,T_j}}{T}_k)}}\theta )$$

(2)

$$S_T=E_T\times e$$

(3)

$$S_2=F_2\times q$$

(4)

$$\beta =\left\{\begin{array}{c}0,(T_j\ne T_1)\\ 1,(T_j=T_1)\end{array}\right.$$

(5)

$$\eta =\left\{\begin{array}{c}0,(T_j\ne T_n)\\ 1,(T_j=T_n)\end{array}\right.$$

(6)

$$\theta =\left\{\begin{array}{c}0,(k-j\ne 1)\\ 1,(k-j=1)\end{array}\right.$$

(7)

In the equation, F₁ represents the minimization of the scheduling cost function, which includes agricultural machinery purchase costs, scheduling costs within the region, and operational costs; A represents the agricultural machinery identification number; S₁ denotes the purchase cost of a single agricultural machine within the region during actual scheduling; S₂ represents the path cost of agricultural machinery scheduling in the region; T represents the task identification number; S_T denotes the operational cost of agricultural machine A in a small park area; F₂ indicates the minimum value of the regional scheduling path function; L(A, T₁) represents the distance from the storage facility to the first small park area that requires operation by agricultural machine A; L(A, T_jT_k) represents the distance from task T_j to task T_k, where j, k ∈ {1, …, n}; L(A, T_n) signifies the distance from the last task back to the storage facility; E_T represents the area of the small park area corresponding to task T; e denotes the operational cost per unit area; q represents the unit distance cost of agricultural machinery scheduling; Equation (5) indicates that β takes the value of 1 when T_j is the first operational task, otherwise it takes the value of 0; Equation (6) indicates that η takes the value of 1 when T_j is the last operational task, otherwise it takes the value of 0; and Equation (7) indicates that θ takes the value of 1 when task T_j is the task preceding task T_k, otherwise it takes the value of 0.

The constraints are as follows:

$$\forall p\mathrm{in}1\cdot \cdot \cdot P,({\displaystyle \sum _{A=1}^A{Q}_{mp}\times {\mathrm{in}}_A})\le SEPEp$$

(8)

$$\forall a\mathrm{in}1\cdot \cdot \cdot A,{\mathrm{in}}_A\ge D_a$$

(9)

$$L(A,T)=R\sqrt{{(L{E}_A-L{E}_T)}^2+\cos ({\displaystyle \frac{L{E}_A+L{E}_T}{2}}){(L{N}_A-L{N}_T)}^2}$$

(10)

In Equation (8), Q_mp represents the demand quantity for agricultural machinery scheduling; SEPEp denotes the maximum number of operators, where p is the number of operators and p indicates the index from 1 to P; a represents the index from 1 to A; in_a refers to the agricultural machinery scheduled within the region; and D_m is the demand quantity for agricultural machinery. In Equation (10), L(A,T) indicates the distance for agricultural machine A scheduled to the operational task area T, where R is the radius of the Earth, LE_A and LE_T are the latitude and longitude of the agricultural machine, and LE_T and LN_T are the latitude and longitude of the farmland.

3. Materials and Methods

3.1. Dung Beetle Optimization Algorithm

The scheduling problem of agricultural machinery within a region is classified as an NP-hard problem, and using exact algorithms to solve large-scale instances can result in long running times, making it difficult to find optimal solutions. Heuristic algorithms are constructed based on intuitive experience, providing feasible solutions to the problem at hand within acceptable costs (such as computation time, space usage, etc.). Although heuristic algorithms can yield reasonable feasible solutions in large-scale sorting problems, the DBO, as an emerging heuristic algorithm, has been successfully applied to solve various types of NP-hard problems. The DBO incorporates five distinct update rules. Each dung beetle swarm comprises five different types of agents: rolling beetles, dancing beetles, breeding beetles, foraging beetles, and stealing beetles [27].

The term ‘rolling dung beetles’ denotes the act of beetles shaping feces into round forms and relocating them to a secure hiding spot. Utilizing celestial signals like the sun, moon, and polarized light, dung beetles are capable of maneuvering, enabling their balls to move linearly. Alterations in the environment lead to shifts in the location of dung beetles, and their rolling actions can be described as:

$$x_i(t+1)=x_i(t)+\sigma \cdot k\cdot x_i(t-1)+q\cdot \Delta x$$

(11)

$$\Delta x=\left|x_i(t)-X^W\right|$$

(12)

In the equation, t represents the present iteration number, and x_i(t) is the position of the i-th dung beetle during the t-th iteration. K ∈ (0, 0.2] is a constant that represents the deflection coefficient, and q ∈ (0, 1] is a random number. Σ = ±1, where 1 indicates no deviation and −1 indicates a deviation from the original direction. In this paper, to simulate the complex environment of the real world, a probabilistic method is used to set it to 1 or −1. Δx represents the environmental alterations. X^W denotes the most unfavorable situation within the present population.

If a dung beetle comes across a barrier and is unable to advance, it adapts its course by dancing, thus discovering a different route. Currently, the update of position is governed by the tangent function, with the formula outlined below:

$$x_i(t+1)=x_i(t)+\sigma \cdot k\cdot x_i(t-1)+q\cdot \Delta x$$

(13)

In the equation, tan(θ) signifies the deflection angle. Given the periodicity of the tangent function, we solely consider its values within the range [0, π]. The notation ∣x_i(t) − x_i(t − 1)∣ denotes the positional difference of the i-th dung beetle between the t-th and t − 1-th iterations. Consequently, the positional update of the dancing dung beetle is heavily reliant on both current and past information. It is crucial to highlight that when θ equals 0, π/2, π, or 2π (π being included as a limit case where tan(θ) trends towards infinity but is practically undefined in a discrete context), tan(θ) is undefined, and hence, the dung beetle’s position remains unchanged.

Naturally, dung beetles transport dung balls to secure zones, ensuring a safe habitat for their young. The approach for choosing boundaries in this region is outlined as follows:

$$\left\{\begin{array}{c}L{b}^{\ast }=\max (X^{\ast }\cdot (1-R),Lb)\\ U{b}^{\ast }=\min (X^{\ast }\cdot (1+R),Ub)\end{array}\right.$$

(14)

$$R=1-t/T_{\max }$$

(15)

In the equation, X* signifies the optimal position within the current population. Lb stands for the lower boundary of the search space, whereas Ub represents the upper boundary. Lb* and Ub* denote the lower and upper limits of the breeding zone, respectively. R signifies the inertia coefficient, and T_max indicates the maximum iteration count.

Upon locating a secure spot, the female dung beetle selects this spot for egg-laying in the breeding ball. In the case of the DBO, every female dung beetle is expected to yield a single egg ball in each cycle. The procedure for dynamically updating the egg ball’s position is as follows:

$$x_i(t+1)=X^{*}+b_1\cdot (x_i(t)-L{b}^{*})+b_2\cdot (x_i(t)-U{b}^{*})$$

(16)

In the equation, x_i(t) denotes the position of the i-th dung beetle at the t-th iteration, while b₁ and b₂ are two independent random vectors, each with dimensions 1 × D, where D represents the dimensionality of the optimization problem. The egg ball’s position is strictly confined to a specified range, namely the breeding area.

Upon the successful hatching of the egg ball, it transforms into a diminutive dung beetle that ventures out to scavenge. Consequently, creating an ideal feeding zone is crucial to direct the dung beetles in their quest for sustenance. Dynamic simulation of the foraging zone employs a boundary approach, described as follows:

$$\left\{\begin{array}{c}L{b}^b=\max (X^b\cdot (1-R),Lb)\\ U{b}^b=\min (X^b\cdot (1+R),Ub)\end{array}\right.$$

(17)

In the equation, X^b signifies the globally optimal position, with Lb^b and Ub^b representing the lower and upper boundaries of the optimal foraging zone, respectively. Upon determining this zone, the positional update procedure for the foraging dung beetles is outlined as follows:

$$x_i(t+1)=x_i(t)+C1(x_i(t)-L{b}^b)+C_2(x_i(t)-U{b}^b)$$

(18)

In the equation, x_i(t) denotes the position of the i-th foraging dung beetle at the t-th iteration. C₁ is a randomly generated number conforming to a normal distribution, whereas C₂ is a random number within the range of (0, 1).

In the population, some dung beetles will steal dung balls from other dung beetles. The position update formula for the stealing dung beetles is as follows:

$$x_i(t+1)=X^b+F\cdot g\cdot (\left|x_i(t)-X^{\ast }\right|+\left|x_i(t)-X^b\right|)$$

(19)

In the equation, x_i(t) represents the position of the i-th stealing dung beetle during the t-th iteration. The g is a 1 × D random vector that follows a normal distribution, and F is a constant.

3.2. Improved the Dung Beetle Optimization Algorithm

DBO, functioning as a smart optimization algorithm, fundamentally operates as a stochastic search algorithm, creating an initial population via random initialization. Nonetheless, this method may result in an irregular spread of beetle locations, diminished worldwide exploration potential, and limited population variety, rendering it susceptible to local optimums. Chaotic mapping is employed in the initial phase of the DBO population to tackle these problems, resulting in a highly varied starting population. Chaotic mapping leads to the creation of chaotic sequences, random in nature, generated by a basic deterministic system, and characterized by randomness, ergodicity, and regularity. The implementation of chaotic mapping in the initialization of populations enables individuals to maximize the utilization of solution space information, thus enhancing overall search efficiency [28]. Typical chaotic mappings encompass Logistic, Cubic, Bernoulli, Singer, and others. Our compilation includes nine varieties of chaotic mappings, with Figure 1 illustrating the histogram of these mappings, which produces 105 sequence values ranging from 0 to 1.

As shown in Figure 1, the sequences produced by the Bernoulli mapping exhibit a more uniform distribution when compared to alternative methods. This uniformity aids in broadening the search area for the dung beetle population in space, enhances the diversity of the population’s positions, and mitigates to a certain extent the algorithm’s propensity to converge prematurely on local optima. Consequently, we opt for the Bernoulli mapping to initialize the population, with its corresponding mathematical model given by:

$$x_{i+1}=\left\{\begin{array}{c}{x}_i/(1-\rho ),x_i\in (0,1-\rho )\\ (x_i-1+\rho )/\rho ,x_i\in (1-\rho ,1)\end{array}\right.$$

(20)

In the equation, x_i is the current value of the generated i-th generation chaotic sequence, and ρ is the control parameter.

In addition to refining the initialization phase, Equations (16), (18) and (19) reveal that the positional updates of breeding, foraging, and thieving beetles are closely tied to their current individual optimal values. This correlation may prompt premature convergence of the population during iterations, trapping it in local optima. To counteract this, we incorporate the Lévy flight strategy into the positional updates of these beetle types. The Lévy flight strategy possesses a long-tailed probability distribution. By incorporating step sizes derived from Lévy flights, the beetles undertake random walks in the search space, bolstering the algorithm’s exploratory capacity. Specifically, subsequent to the initial positional update, we further apply the Lévy flight strategy to refine individual positions. This ensures the beetle population maintains diversity in later stages, facilitating their escape from local optima and enhancing their overall search effectiveness. The Lévy flight strategy is defined as follows:

$$x_i(t+1)=x_i(t)\cdot \mathrm{Levy}(\lambda )$$

(21)

In the equation, Levy(λ) denotes a Lévy distribution with parameter λ. We define it as follows:

$$\mathrm{Levy}(\lambda )={\displaystyle \frac{\gamma \cdot E}{{\left|F\right|}^{0.5}}}$$

(22)

$$\gamma ={\displaystyle \frac{\Gamma (1+\lambda )\cdot \sin (\pi \lambda /2)}{\Gamma ((1+\lambda )/2)\cdot \lambda \cdot 2^{((\lambda -1)/2)}}}$$

(23)

In the equation, E and F follow a normal distribution, and Γ(x) = (x − 1)!.

The updated position update methods for the breeding beetles, foraging beetles, and thieving beetles are as follows:

$$x_i(t+1)=\mathrm{Levy}(\lambda )\cdot (X^{*}+b_1\cdot (x_i(t)-L{b}^{*})+b_2\cdot (x_i(t)-U{b}^{*}))$$

(24)

$$x_i(t+1)=\mathrm{Levy}(\lambda )\cdot (x_i(t)+C1(x_i(t)-L{b}^b)+C_2(x_i(t)-U{b}^b))$$

(25)

$$x_i(t+1)=\mathrm{Levy}(\lambda )\cdot (X^b+F\cdot g\cdot (\left|x_i(t)-X^{\ast }\right|+\left|x_i(t)-X^b\right|))$$

(26)

To explore the values of ρ in the Bernoulli mapping and λ in the Lévy flight strategy, we first conducted preliminary experiments. We set ρ = 1 and the ratio ρ:λ = {2:1, 3:1, 4:1, 5:1, 3:2, 4:3, 5:2, 5:3, 5:4}. The results indicated that the algorithm performed optimally when ρ:λ = 1:3. Subsequently, we set ρ = {0.1, 0.2, …, 0.9}. The results showed that the algorithm performed best when ρ = 0.5. Thus, the final values we determined were: ρ = 0.5 and λ = 1.5.

In summary, we refer to the algorithm optimized by the Bernoulli mapping and Lévy flight strategy as the BL-DBO (Algorithm 1). The pseudocode for the BL-DBO is as follows:

Algorithm 1 BL-DBO

Establish the initial population and set the parameters number_Dung Beetle = N Max − iter = T Min − Bounds = Lb Max − Bounds = Ub Set the initial positions for the population utilization Equation (20) Establish initial global fitness values using the fitness evaluation Equation (1) While (t < T) do For i = 1:N do If i ∈ rolling_dung_beetle then If τ < 0.9 then    # τ ∈ (0, 1)   Adjust the position of dung beetle X(i) according to Equation (11)   Update the fitness values NewFit[i] based on Equation (1) Else   Revise the position of dung beetle X(i) using Equation (13)   Recalculate the fitness values NewFit[i] employing Equation (1) End If End If If i ∈ breeding_dung_beetle then   Modify the position of dung beetle X(i) based on Equation (24)   Renew the fitness values NewFit[i] according to Equation (1) End If If i ∈ foraging_dung_beetle then   Adjust the location of dung beetle X(i) using Equation (25)   Reevaluate the fitness values NewFit[i] based on Equation (1) End If If i ∈ stealing_dung_beetle then   Alter the position of dung beetle X(i) according to Equation (26)   Recalculate the fitness values as NewFit[i] using Equation (1) End If Greedy choice If NewFit[i] < Fitness then Fit[i] = NewFit[i] End If End For t = t + 1 End While Provide the global optimal position, BestPosition, and its corresponding fitness value, BestFitness

3.3. Ablation Test

To verify that each improvement strategy has a positive impact on the optimization of the DBO, an ablation experiment was conducted. The comparison includes the original DBO, the DBO that only introduces the Bernoulli mapping (DBO-1), and the BL-DBO. The computer configuration used for the experiments was an Intel Core i7-12700H CPU with a clock frequency of 2.30 GHz, 32 GB of RAM, and the Windows 11 operating system, with programming executed in MATLAB R2022a. The standard test functions selected were Sphere, Schwefel 2.22, Ackley, and Rastrigin. The Sphere and Schwefel 2.22 functions serve as unimodal benchmarks to assess an algorithm’s developmental capabilities. On the other hand, the Ackley and Rastrigin functions, being multimodal with one global optimum and several local optima, evaluate an algorithm’s global search prowess and its ability to avoid getting trapped in local minima. Their mathematical expressions are as follows:

$$F1(x_1,x_2)={\displaystyle \sum _{i=1}^n{x}_i^2}$$

(27)

$$F2(x_1,x_2)={\displaystyle \sum _{i=1}^n\left|x_i\right|}+{\displaystyle \prod _{i=1}^n\left|x_i\right|}$$

(28)

$$F3(x_1,x_2)=-20\exp (-0.2\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{x}_i^2}})-\exp ({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\cos 2\pi x_i})+20+e$$

(29)

$$F4(x_1,x_2)={\displaystyle \sum _{i=1}^n[x_i^2-10\cos (2\pi x_i)+10]}$$

(30)

The graphs of the four test functions are shown in Figure 2, Figure 3, Figure 4 and Figure 5.

Detailed information for the standard test functions Sphere, Schwefel 2.22, Ackley, and Rastrigin is shown in

Table 1. Detailed information on standard test functions.

Function	Name	Dimension	Range	Optimal
F1	Sphere	30	[−100, 100]	0
F2	Schwefel 2.22	30	[−10, 10]	0
F3	Ackley	30	[−32, 32]	0
F4	Rastrigin	30	[−5.12, 5.12]	0

.

To ensure the reliability of the experimental results, all algorithm parameters are set as follows: the population size of the DBO is 90, the number of iterations is 500, and the ratios of rolling beetles, breeding beetles, foraging beetles, and stealing beetles are 6:6:7:11. Each algorithm runs independently 30 times for each standard test function. The experimental data are used to plot the optimization process graphs for the Sphere, Schwefel 2.22, Ackley, and Rastrigin functions, comparing the changes in fitness values of the DBO, DBO-1, and BL-DBO during the iteration process, as shown in Figure 6, Figure 7, Figure 8 and Figure 9. The x-axis denotes the number of iterations, whereas the y-axis represents the fitness values.

As shown in Figure 6, Figure 7, Figure 8 and Figure 9, the BL-DBO achieved convergence values that are closest to the optimal values of the functions F1, F2, and F3, indicating that the BL-DBO has a stronger local escape capability. The DBO-1 optimized the initial values based on the DBO, demonstrating that the Bernoulli mapping plays a beneficial role in the selection of initial values. The BL-DBO, built on the DBO-1, achieved faster convergence and better convergence values. This indicates that the revised positional update techniques for breeding, foraging, and stealing beetles have improved both the performance and efficiency of the algorithm. In summary, the proposed improvement strategies for different components have positively contributed to the optimization of the DBO, and the integration of the two strategies is effective.

3.4. Experimental Materials

To ensure the representativeness and practicality of the research results, the Shunnong Fruit Modern Agricultural Park in Shunping County, Baoding City, Hebei Province, and the Taihang Mountain Road First Station of Hebei Agricultural University, along with the Shijiazhuang Fruit Tree Research Institute, were selected as experimental bases. Both bases are located in a temperate monsoon climate and contain several small garden areas, with an altitude of approximately 33 m. The Shunnong Fruit Modern Agricultural Park is situated in Beidabei Village, Dabi Township, Shunping County (38.97° N, 114.93° E), covering a planned area of 8.13 km², with the core area spanning 2.35 km². It primarily focuses on apple cultivation, featuring varieties such as Fuji, Guoguang, Gala, and Qin’guan apples. There are eight apple orchards using dwarf rootstock and high-density cultivation methods, as shown in Figure 10.

The Shijiazhuang Fruit Tree Research Institute is located in Taoyuan Town, Chang’an District, Shijiazhuang City (22.73° N, 112.93° E), covering a planned area of over 0.9 km², primarily focused on apple cultivation. The varieties cultivated include Jinzong, Jihong, and Guohong apples. There are seven apple orchards using dwarf rootstock and high-density cultivation methods, as shown in Figure 11.

We converted the latitude and longitude of the two bases storing agricultural machinery and the entrances and exits of each small park into Cartesian coordinates.

Table 2. Basic information of the experimental base.

Base	Factory Coordinates	Small Park Number	Entrance and Exit Coordinates	Quantity of Agricultural Machinery Required per Unit Time
Shunnong Fruit Modern Agricultural Park	(35.25, 101)	f1	(95.6, 286.7)	2
		f2	(49.36, 193.77)	3
		f3	(53.77, 148.22)	2
		f4	(52.98, 54.33)	2
		f5	(99.32, 152.64)	1
		f6	(112.43, 163.27)	4
		f7	(112.69, 204.89)	3
		f8	(129.32, 221.36)	4
Shijiazhuang Fruit Tree Research Institute	(80.22, 93.62)	f9	(48.32, 296.11)	2
		f10	(48.14, 267.13)	3
		f11	(51.36, 154.78)	4
		f12	(79.65, 116.42)	1
		f13	(76.54, 153.85)	4
		f14	(81.93, 193.62)	4
		f15	(77.35, 246.21)	3

shows the basic information of the operation parks.

4. Results and Discussion

To verify the effectiveness of the scheduling model proposed in this research in reducing the required quantity and cost of agricultural machinery purchases, the experiment assumes the machinery is a lawnmower with a purchase price of CNY 50,000. The path costs from the factory to each small park and between the small parks are considered uniform. We generate results using three different scheduling methods: human experience-based scheduling, DBO algorithm scheduling, and BL-DBO scheduling, as shown in

Table 3. Usage of agricultural machinery in Shunping Shunnong Orchard.

Method	Farm Machinery Number	f1	f2	f3	f4	f5	f6	f7	f8
Human experience	A1	√ 1			√			√
	A2	√			√			√
	A3		√			√		√
	A4		√				√√		√
	A5		√				√√		√
	A6			√√					√√
DBO	A1	√		√			√		√√
	A2	√		√			√		√√
	A3		√		√		√	√
	A4		√		√		√	√
	A5		√			√		√
BL-DBO	A1	√√	√√√
	A2			√√	√√	√
	A3						√√√√		√√
	A4							√√√	√√

¹ The agricultural machinery is working within the unit time.

and

Table 4. Usage of agricultural machinery at Shijiazhuang Fruit Tree Institute.

Method	Farm Machinery Number	f1	f2	f3	f4	f5	f6	f7
Human experience	A1	√ 1		√		√	√
	A2	√		√		√	√
	A3		√	√		√		√√
	A4		√	√		√		√
	A5		√		√		√√
DBO	A1	√√			√		√√
	A2		√√√				√√
	A3			√√√√				√√
	A4					√√√√		√
BL-DBO	A1	√√	√√√
	A2			√√√√	√
	A3					√√√√		√√
	A4						√√√√	√

¹ The agricultural machinery is working within the unit time.

.

To further observe the usage of each agricultural machine, we have presented the operating sequence of each machine in Figure 12 and Figure 13.

Based on the results of Table 3 and Table 4, as well as Figure 12 and Figure 13, we have calculated the scheduling frequency of each agricultural machinery, as shown in

Table 5. Statistics on dispatch times for various agricultural machinery.

Base	Farm Machinery Number	Human Experience	DBO	BL-DBO
Shunnong Fruit Modern Agricultural Park	A1	2	3	1
	A2	2	3	2
	A3	2	3	1
	A4	2	3	1
	A5	2	3	-
	A6	1	-	-
Shijiazhuang Fruit Tree Research Institute	A1	3	2	1
	A2	3	1	1
	A3	3	1	1
	A4	3	1	1
	A5	2	-	-

.

According to Table 3, in the scheduling experiments conducted at the Shunping Shunnong Modern Agricultural Park, the method of scheduling based on human experience requires a total of six agricultural machines to complete the tasks, with a maximum time requirement of four units, resulting in an average utilization rate of 87.5%. The scheduling results using the DBO algorithm require five agricultural machines to complete the tasks, with a maximum time requirement of five units, leading to an average utilization rate of 84%. The scheduling results using the BL-DBO require four agricultural machines to complete the tasks, with a maximum time requirement of six units, achieving an average utilization rate of 87.49%. Compared to human experience, the BL-DBO can save CNY 100,000 in agricultural machine purchase costs, and compared to the DBO algorithm, it can save CNY 50,000.

According to Table 4, in the scheduling experiments conducted at the Shijiazhuang Fruit Tree Research Institute, the method of scheduling based on human experience requires a total of five agricultural machines to complete the tasks, with a maximum time requirement of five units, resulting in an average utilization rate of 84.0%. The scheduling results using the DBO algorithm require 4 agricultural machines to complete the tasks, with a maximum time requirement of 6 units, achieving an average utilization rate of 87.50%. The scheduling results using the BL-DBO also require four agricultural machines to complete the tasks, with a maximum time requirement of six units, resulting in an average utilization rate of 87.50%. Compared to human experience, the BL-DBO can save CNY 50,000 in agricultural machine purchase costs.

From Figure 12 and Figure 13, it can be observed that the scheduling results calculated by the BL-DBO concentrate on the agricultural machine operation sites, with the machines primarily working in adjacent small orchards, avoiding long-distance transfers. The operational tasks are clear and simple. For hilly and mountainous orchards, this approach reduces the challenges of scheduling agricultural machines between different small orchards to some extent.

From Table 5, it can be seen that using human experience for scheduling results in an average of 2 and 2.8 scheduling instances per agricultural machine at the Shunnong Fruit Modern Agricultural Park and the Shijiazhuang Fruit Tree Research Institute, respectively. Using the DBO method results in an average of 3 and 1.25 scheduling instances per agricultural machine at these locations. Using the BL-DBO method results in an average of 1.25 and 1 scheduling instances per agricultural machine at the same locations.

In summary, compared to scheduling methods based on human experience and the DBO algorithm, the BL-DBO not only reduces agricultural machine purchase costs to some extent but also significantly decreases the number of scheduling instances of agricultural machines across different small orchards, reducing energy loss from machines operating outside work sites and further lowering fruit production costs.

5. Conclusions

This research establishes a scheduling model based on the scheduling requirements within the agricultural machinery area of the orchard. Using the conventional DBO algorithm, we produce uniformly distributed initialization sequences via the Bernoulli map derived from chaos theory. Subsequently, we incorporate a Lévy flight strategy into the iterative process to mitigate the risk of the algorithm converging to local optima, thereby presenting the BL-DBO tailored for addressing scheduling models in the region. To verify the positive effects of each optimization method, we conducted tests using four standard benchmark functions. Finally, the superiority of the BL-DBO in solving the scheduling model in the area was validated using real data from the Shunping Shunnong modern orchard and the Shijiazhuang fruit research station. The results show that, on the one hand, the scheduling results obtained from the BL-DBO can save fruit growers between CNY 50,000 and 100,000 in machinery purchase costs. On the other hand, compared to manual scheduling experience and the scheduling results derived from the traditional DBO algorithm, the scheduling results produced by the BL-DBO can significantly reduce the turnover of agricultural machinery between different small orchards, decrease energy loss during machinery scheduling, and further lower the production costs of the fruit.

There are certain limitations to this study, as our model was built based on assumptions about various conditions. For example, we assumed that the agricultural machinery would return to the facility after completing all operational tasks. Although, in most cases, the energy of the machinery does not need to be replenished during a single operation, if the machinery runs out of energy midway through the task and needs to return for a recharge, our model would need to be reconstructed. In the future, our team will continue to optimize this operational model and incorporate more orchard information to enhance our database, aiming to promote the widespread application of this model.