Multi-Cooperative Agricultural Machinery Scheduling with Continuous Workload Allocation: A Hybrid PSO Approach with Sparsity Repair

Abstract

Scheduling agricultural machinery across multiple cooperatives is often inefficient because existing rigid, discrete assignment models fail to flexibly coordinate shared resources under tight time windows. To address this limitation, we develop a simulation-based framework for the Multi-cooperative Agricultural Machinery Scheduling Problem (MAMSP) underpinned by a Continuous Collaborative Workload Sharing (CWS) formulation. To mitigate the solution fragmentation inherent in continuous optimization, we propose a Hybrid Particle Swarm Optimization with Sparsity Repair (HPSO-SR). The algorithm integrates a stochastic initialization strategy to enhance global exploration, a mutation injection mechanism to avoid swarm stagnation, and a sparsity repair operator that prunes uneconomical fractional assignments, yielding operationally feasible sparse schedules. A real-world case study from Liyang, China, augmented by synthetic instances of varying scales (small, medium, and large), was conducted to benchmark the proposed approach against a rule-based heuristic, a Genetic Algorithm (GA-CWS), and Simulated Annealing (SA-CWS) under a unified decoding scheme. The results show that HPSO-SR consistently achieves the lowest objective values, reducing the total cost by 74.43% relative to GA-CWS and 59.20% relative to SA-CWS in the medium-scale case. By deliberately trading off minimal additional transfer cost against improved timeliness, the obtained schedules nearly eliminate delay penalties. Sensitivity analysis and mechanism ablation studies further confirm that the sparse solutions exhibit structural resilience and that the proposed repair strategy is essential for algorithmic convergence, supporting the reliability of the proposed approach for time-critical, high-stakes agricultural operations.

1. Introduction

As modern agriculture advances, production systems are shifting from traditional small-scale farming towards large-scale, intensive, and intelligent production models. Recent advancements in agricultural engineering have significantly enhanced machinery performance, ranging from agricultural robotics applications [1] and intelligent control systems [2] to mechanical efficiency optimization [3] and operator ergonomics [4]. However, as hardware capabilities expand, the complexity of managing these assets has evolved from simple oversight to complex logistical operations planning. This process necessitates the integration of production schedules with machinery dispatching across strategic, tactical, and operational levels under resource and time-window constraints [5].

The rise in the sharing economy has further catalyzed innovative service models, notably “machinery-hailing” and on-demand farming services [6]. These models are particularly important in regions characterized by fragmented land holdings, where the rapidly growing rental market has generated complex, large-scale scheduling challenges [7]. The agricultural machinery scheduling problem inherently involves coordinating multiple sequential operations under dynamic conditions. Factors such as heterogeneous farmland readiness and narrow, weather-dependent workability windows create a need for advanced decision-support tools for the time-constrained dispatch of shared machinery [8,9]. Furthermore, scheduling frameworks must possess sufficient robustness to manage dynamic uncertainties; failures to proactively mitigate disruptions during critical periods, such as the harvest season, can result in substantial logistical breakdowns [10]. As smart agriculture advances towards integrated logistics involving autonomous robot fleets, the demand for effective methodologies capable of managing these complex tasks continues to grow [11].

Substantial research has been directed towards developing scheduling models and optimization algorithms to address the multi-objective nature of agricultural machinery scheduling problems. Evolutionary algorithms, particularly the Non-Dominated Sorting Genetic Algorithm II (NSGA-II), have been widely adopted as baselines for resolving conflicting objectives [12,13]. To augment computational efficiency, researchers have devised various hybrid approaches that couple Genetic Algorithms (GA) with local search heuristics, such as Simulated Annealing (SA) or neighborhood search, thereby improving convergence in machinery routing [14,15,16]. However, classical evolutionary approaches often encounter limitations in the high-dimensional continuous search spaces required for precise resource allocation.

Recent advances indicate a shift towards Swarm Intelligence (SI) algorithms, which offer faster convergence and greater adaptability. For instance, Enhanced Particle Swarm Optimization (EPSO) variants have proven effective in complex multi-robot path planning by smoothing trajectories and evading local optima [17]. Similarly, synergistic approaches combining clustering with PSO have shown promise in optimizing task allocation for multi-robot systems [18]. Concurrently, research has focused on adapting scheduling models to spatially challenging environments. Operations in hilly or fragmented farmlands introduce topological complexities, prompting the development of algorithms such as the enhanced NSGA-III, which exhibits robust performance at larger scales [19]. The dynamic scheduling of unmanned aerial vehicles (UAVs) necessitates real-time decision-making; in this context, integrating NSGA-III with ant colony optimization (ACO) has been shown to strengthen multi-machine cooperation [20]. Moreover, integrating Lévy-distributed Simulated Annealing with hybrid genetic–ACO models facilitates effective operations in both static and dynamic scenarios, outperforming canonical SA and GA baselines [21,22].

In the context of cross-regional collaboration, frameworks such as HTSMOGA have utilized priority-based population generation to enhance economic efficiency [23]. Similarly, collaborative scheduling for harvesting and grain transportation has integrated diverse strategies to optimize multi-stage operations involving heterogeneous machinery [24,25]. Recent studies on coverage path planning indicate that precise region segmentation and path-orientation optimization are critical for improving operational efficiency in complex fields [26]. Building on these insights, hybrid PSO schemes have achieved measurable gains in routing performance for cross-regional scenarios [27], and advanced frameworks such as the Hybrid Particle Swarm Optimization–Neighborhood Strategy Search (HPSO-NS) continue to drive methodological developments in this area [28].

Recent advancements have further expanded the scope of cooperative scheduling. For instance, Pan et al. [29] proposed a deep reinforcement learning framework for cooperative scheduling under emergency constraints, highlighting the growing focus on dynamic responsiveness in multi-machine systems. Similarly, Kong and Liu [30] introduced a spatiotemporal clustering approach for multi-station machinery scheduling to optimize resource allocation across dispersed depots. These studies underscore the trend towards intelligent, data-driven adaptation in large-scale agricultural logistics.

However, a critical structural limitation persists in most existing studies (e.g., [14,15,16,23,24,25,26,27,28,29,30]): they predominantly adopt rigid discrete task-assignment schemes. Under this paradigm, each farmland is assigned to a single machine or a fixed subset, restricting the associated decision variables to binary values. This discreteness prevents algorithms from fine-tuning workload shares to fit operations tightly within time windows. Although some recent works explore multi-machine cooperation, they rarely treat workload as a fully continuous variable. Critically, when continuous optimization is attempted, standard metaheuristics often suffer from solution fragmentation—where workloads are atomized into inefficient fractions across too many machines—thereby resulting in excessive transfer costs.

To address these gaps, we develop a simulation-based scheduling framework for multi-cooperative agricultural machinery systems underpinned by Continuous Collaborative Workload Sharing (CWS). Specifically, this study pursues three main research objectives. First, we formulate the Multi-cooperative Agricultural Machinery Scheduling Problem (MAMSP) as a mixed-integer programming model that explicitly captures the continuous collaborative workload allocation for each farmland. Second, we develop a Hybrid Particle Swarm Optimization with Sparsity Repair (HPSO-SR). Unlike standard metaheuristics, this algorithm integrates a stochastic initialization strategy to enhance global exploration, a mutation injection mechanism to avoid stagnation, and a sparsity repair strategy to prune uneconomical fractional assignments and enforce a sparse, operationally feasible solution structure. Third, we validate the proposed framework via a real-world case study in Liyang, China, augmented by synthetic instances of varying scales. Benchmarked against a rule-based heuristic, GA-CWS, and SA-CWS, the proposed HPSO-SR consistently demonstrates higher cost efficiency and logistical resilience, backed by statistical significance tests and mechanism ablation studies.

2. Materials and Methods

2.1. Problem Description

This study investigates the Multi-cooperative Agricultural Machinery Scheduling Problem (MAMSP) in regions where multiple cooperatives collaboratively provide plowing services to spatially dispersed farmland. The study area consists of M agricultural machinery cooperatives and N farmland units requiring service. Each cooperative m manages a fleet of $K_m$ tractors stationed at a designated depot, resulting in a total regional fleet of $K={\mathit{\Sigma }}_{m=1}^M{K}_m$. Each farmland i is described by its geographic coordinates, required a plowing area $A_i$ and a time window $\left[{ET}_i,{LT}_i\right]$, indicating the agronomically preferred period for operation.

A defining characteristic of the MAMSP is the Continuous Collaborative Workload Sharing (CWS) mechanism. Unlike classical Vehicle Routing Problems (VRPs), where each customer is typically served by a single vehicle, the CWS model allows multiple tractors—whether from the same or different cooperatives—to jointly process a single farmland. The total workload $A_i$ can be continuously partitioned among collaborating tractors, enabling flexible resource allocation under tight time-window constraints. This flexibility increases the computational complexity of the problem as it requires synchronized scheduling across organizational boundaries.

The decision problem involves three coupled components: (1) task assignment (allocating tractors to specific farmlands); (2) routing sequence (the visiting order of each tractor); and (3) continuous workload allocation (the area $s_{ik}$ processed by tractor k at farmland i). The objective is to minimize the total scheduling cost, comprising transfer costs, operation costs, waiting costs (for early arrival), and delay penalty costs (for late completion). Since the planning horizon is fixed and each farmland must be fully plowed, the total amount of operational work $\sum A_i$ is constant across all feasible schedules. As a result, the total operation cost is invariant and is recorded only for economic accounting, while the effective optimization is driven by minimizing the sum of transfer, waiting, and delay penalty costs.

2.2. Problem Assumptions

To formulate the MAMSP as a tractable mathematical model, the following assumptions are adopted, reflecting common agricultural logistics practice:

(1)

Tractor homogeneity. All tractors across cooperatives are assumed to have identical specifications, including operation efficiency w, transfer speed v, and unit operation cost ϕ. This reflects the practical tendency of cooperatives to procure standardized equipment to simplify maintenance and cost accounting. Heterogeneity in fleet composition is left for future model extensions.

(2)

Deterministic parameters and complete service. All problem parameters (farmland areas, time windows, operation efficiency, and cost coefficients) are deterministic and known a priori. Each farmland must be fully processed within the planning horizon; partial service is not permitted. Consequently, the total operational workload is fixed across all feasible schedules. Stochastic factors such as weather disruptions or equipment breakdowns are not considered in this static planning phase.

(3)

Geographic distance calculation. Spatial distances between locations are computed using the Haversine formula based on GPS coordinates, approximating the Earth’s spherical geometry to ensure routing accuracy.

(4)

Closed-loop depot-to-depot routing. A depot-to-depot policy is enforced: each tractor must depart from its own home cooperative and return to the same location after completing its assigned tasks.

(5)

Soft time-window structure. Each farmland i is associated with a soft time window $\left[{ET}_i,{LT}_i\right]$, representing the agronomically optimal operation period. Service may start before ${ET}_i$ or finish after ${LT}_i$, incurring linear waiting costs or delay penalties, respectively. This soft constraint structure allows the optimization model to trade off timeliness against routing efficiency when necessary.

(6)

Non-preemptive sequential execution. For each tractor, operations at assigned farmlands are performed sequentially without preemption. Once a tractor begins working in a farmland, it must complete its assigned workload before transferring to the next location. While the operations of an individual tractor are sequential, multiple tractors may process the same farmland simultaneously under collaborative operation.

(7)

Implicit setup times. Equipment preparation and field setup times are not modeled explicitly as separate decision variables; instead, they are incorporated into the average operation efficiency parameter.

(8)

Single-horizon planning. The planning horizon is assumed to be short enough that tractor availability is not constrained by daily working-hour regulations or mandatory rest periods. Driver work-hour limits and multi-day scheduling are therefore not explicitly modeled and are left for future extensions of the framework.

2.3. Notations and Parameters

The main indices, sets, parameters, and decision variables used in this study are summarized in Appendix A (

Table A1. Indices and sets.

Symbol	Description	Unit/Type
i, j	Indices for locations, where i, j$\in${1, 2, …, M+N}	Index
m	Index for cooperatives (depots), m$\in${1, 2, …, M}	Index
M+1, …, M+N	Indices for farmland locations	Index set
N	Total number of farmlands requiring service	Count
M	Total number of cooperatives	Count
k	Index for tractors, k$\in${1, 2, …, K}	Index
K	Total number of tractors, where $K={\sum }_{m=1}^M{K}_m$	Count
Km	Number of tractors owned by cooperative m	Count

,

Table A2. Parameters.

Symbol	Description	Unit/Type
Ai	Required plowing area at farmland i, i$\in${M + 1, …, M + N}	hm2
ETi	Earliest acceptable start time for farmland i	h
LTi	Latest acceptable completion time for farmland i	h
dij	Distance between locations i and j (computed by Haversine formula using GPS coordinates)	km
w	Operation efficiency of each tractor	hm2/h
v	Transfer speed of tractors	km/h
ρ	Transfer cost coefficient	CNY/km
ϕ	Operation cost coefficient	CNY/hm2
α	Waiting cost rate for early arrival	CNY/h
β	Delay penalty cost rate for late completion	CNY/h
c(k)	Home cooperative index of tractor k, c(k)$\in${1, …, M}	Index

and

Table A3. Decision and auxiliary variables.

Symbol	Description	Unit/Type
xijk	Binary variable, equals 1 if tractor k travels from location i to location j, 0 otherwise	{0,1}
yik	Binary variable, equals 1 if tractor k performs plowing at farmland i, 0 otherwise	{0,1}
sik	Continuous variable; plowing area completed by tractor k at farmland i	${\mathrm{hm}}^2,s_{ik}\ge 0$
STik	Start time of tractor k at farmland i	h
FTik	Completion time of tractor k at farmland i	h
STi	Earliest start time among all tractors serving farmland i	h
FTi	Latest completion time among all tractors serving farmland i	h

) for readability.

2.4. Mathematical Formulation

The objective is to minimize the total cost of a schedule, decomposed into four components:

$$\mathrm{m}\mathrm{i}\mathrm{n}Z=C_{transfer}+C_{operation}+C_{waiting}+C_{delay}.$$

(1)

(1)

Transfer cost

$$C_{transfer}=\rho {\sum }_{k=1}^K{\sum }_{i=1}^{M+N}{\sum }_{j=1}^{M+N}{d}_{ij}\cdot x_{ijk}.$$

(2)

This term represents the total travel cost of all tractors moving between locations (cooperatives and farmlands) in the region.

(2)

Operation cost

$$C_{operation}=\varphi {\sum }_{k=1}^K{\sum }_{i=M+1}^{M+N}{s}_{ik}.$$

(3)

This term represents the cost of all plowing operations performed by the tractors.

(3)

Waiting cost (Farmland-level)

Let ${ST}_i$ denote the earliest start time among all tractors that actually plow farmland i. The waiting cost is computed at the farmland level as

$$C_{waiting}=\alpha {\sum }_{i=M+1}^{M+N}\mathrm{m}\mathrm{a}\mathrm{x}\left(0,{ET}_i-{ST}_i\right).$$

(4)

This penalizes early arrivals when operations at farmland i start before the beginning of its time window ${ET}_i$. Each farmland is penalized at most once for an early start, even if multiple tractors collaborate on that farmland.

(4)

Delay cost (Farmland-level)

Similarly, let ${FT}_i$ denote the latest completion time among all tractors that serve farmland i. The delay penalty cost is

$$C_{delay}=\beta {\sum }_{i=M+1}^{M+N}\mathrm{m}\mathrm{a}\mathrm{x}\left(0,{FT}_i-{LT}_i\right).$$

(5)

This penalizes late completion when the collaborative operations at farmland i extend beyond the end of its time window ${LT}_i$. Each farmland is penalized at most once for late completion, regardless of how many tractors jointly serve it.

Constraints

Constraint 1: Workload completion requirement

$${\sum }_{k=1}^K{s}_{ik}=A_i,\forall i\in \left\{M+1,\dots ,M+N\right\}.$$

(6)

This ensures that the required plowing area of each farmland is fully completed by the assigned tractors.

Constraint 2: Workload-assignment relationship

$$\begin{gathered} s_{ik}\le A_i\cdot y_{ik},\forall i\in \left\{M+1,\dots ,M+N\right\},\forall k; \\ {s}_{ik}\ge \epsilon \cdot y_{ik},\forall i\in \left\{M+1,\dots ,M+N\right\},\forall k. \end{gathered}$$

(7)

where $\epsilon >0$ is a small positive constant (set to 10⁻⁴ for numerical stability). These constraints link the continuous workload variable $s_{ik}$ with the binary assignment variable $y_{ik}$, ensuring that a tractor can only process farmland i if it is assigned to it, and that any assigned tractor contributes a positive workload.

Constraint 3: Flow conservation at farmland locations

$${\sum }_{j=1}^{M+N}{x}_{ijk}={\sum }_{j=1}^{M+N}{x}_{jik}=y_{ik},\forall i\in \left\{M+1,\dots ,M+N\right\},\forall k.$$

(8)

This ensures that if tractor k serves farmland i, then it must both arrive at and depart from that location exactly once.

Constraint 4: Departure from and return to the home cooperative

For each tractor k, let $c\left(k\right)\in \left\{1,\dots ,M\right\}$ denote its home cooperative. Then

$$\begin{gathered} {\sum }_{j=M+1}^{M+N}{x}_{c\left(k\right),j,k}\le 1,\forall k; \\ {\sum }_{i=M+1}^{M+N}{x}_{i,c\left(k\right),k}={\sum }_{j=M+1}^{M+N}{x}_{c\left(k\right),j,k},\forall k. \end{gathered}$$

(9)

Each tractor departs from its home cooperative at most once and, if it departs, must return to the same cooperative after completing all assigned tasks.

Constraint 5: Time consistency and implicit subtour elimination

For each tractor k and farmland i, the service completion time is defined by

$${FT}_{ik}={ST}_{ik}+{\displaystyle \frac{s_{ik}}{w}},\forall i\in \left\{M+1,\dots ,M+N\right\},\forall k.$$

(10a)

The start time at the first farmland visited by tractor k must be consistent with the travel time from its home cooperative:

$${ST}_{ik}\ge {\displaystyle \frac{d_{c\left(k\right),i}}{v}}-Q\left(1-x_{c\left(k\right),i,k}\right),\forall i\in \left\{M+1,\dots ,M+N\right\},\forall k.$$

(10b)

For any route leg from farmland i to j traveled by tractor k (i.e., $x_{ijk}=1$), the start time at j must respect the completion time at i and the travel time between i and j:

$${ST}_{jk}\ge {FT}_{ik}+{\displaystyle \frac{d_{ij}}{v}}-Q\left(1-x_{ijk}\right),\forall i,j\in \left\{M+1,\dots ,M+\mathrm{N}\right\},\forall k.$$

(10c)

where Q is a sufficiently large constant. Because travel times $d_{ij}/v$ and service times $s_{ik}/w$ are strictly positive, Equations (10a)–(10c) enforce strictly increasing times along any sequence of visited locations for a given tractor. As a result, positive-length cycles are infeasible, and time consistency implicitly eliminates disconnected subtours in the route structure.

Constraint 6: Farmland-level time aggregation

To compute waiting and delay penalties at the farmland level, the farmland-level times ${ST}_i$ and ${FT}_i$ are linked to the tractor-level times ${ST}_{ik}$ and $F{T}_{ik}$ as follows:

$${ST}_i=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{ST}_{ik}:s_{ik}>0\right\},{FT}_i=\mathrm{m}\mathrm{a}\mathrm{x}\left\{{FT}_{ik}:s_{ik}>0\right\},\forall i\in \left\{M+1,\dots ,M+N\right\}.$$

(11)

Equation (11) ensures that ${ST}_i$ equals the earliest start time and ${\mathrm{F}\mathrm{T}}_{\mathrm{i}}$ equals the latest completion time among all tractors that actually serve farmland i. The aggregated times are then used in the farmland-level cost terms Equations (4) and (5).

Constraint 7: Non-negativity and binary constraints

$$s_{ik}\ge 0,x_{ijk}\in \left\{0,1\right\},y_{ik}\in \left\{0,1\right\},\forall i,j,k.$$

(12)

Note that, under Assumption (1) and Assumption (2), every farmland i must be fully plowed within the planning horizon, and all tractors share the same unit operation cost ϕ. Together with Constraint (6), this implies that $C_{operation}=\varphi {\sum }_{k=1}^K{\sum }_{i=M+1}^{M+N}{s}_{ik}=\varphi {\sum }_{i=M}^{M+N}{A}_i$, which is constant with respect to the decision variables. Subtracting this constant term from Equation (1) does not change the set of optimal solutions. Hence, in the algorithmic implementation (Section 2.5), we minimize the sum of transfer, waiting, and delay penalty costs, while still reporting all four components in the numerical analysis to provide a complete economic cost decomposition.

Thus, the above formulation provides a rigorous mathematical description of the MAMSP logic. However, the non-linear aggregation of start and completion times in Equation (11) and the conditional penalty functions in Equations (4) and (5) make direct exact solution methods computationally intractable for large-scale instances. Therefore, rather than solving this static model directly, we employ it as the logical core for the simulation-based evaluation embedded in the solution algorithm. The proposed HPSO-SR described in Section 2.5 utilizes a decoding procedure that simulates the temporal flow of the tractors to strictly enforce these constraints and evaluate the objective function dynamically.

2.5. A Hybrid Particle Swarm Optimization with Sparsity Repair (HPSO-SR)

2.5.1. Overall Framework

The Multi-cooperative Agricultural Machinery Scheduling Problem (MAMSP), as formulated in Section 2.4, relies on the CWS model to resolve complex time-window conflicts. Although the CWS formulation in principle enables flexible workload distribution, solving it with standard continuous metaheuristics often yields dense solutions, in which workloads are fragmented (or even atomized) into very small, uneconomical fractions across many tractors, leading to excessive transfer costs. To exploit the flexibility of the CWS model while enforcing operational efficiency, this study proposes a Hybrid Particle Swarm Optimization with Sparsity Repair (HPSO-SR). As illustrated in Figure 1, the framework is constructed on the continuous CWS encoding and integrates three key strategies to enhance performance: (1) a stochastic initialization strategy that prevents early-stage bias towards local optima; (2) a sparsity repair operator that actively prunes redundant transfers and enforces sparse collaboration; and (3) a mutation injection mechanism that mitigates premature convergence during the evolutionary process.

2.5.2. Solution Encoding and Fitness Evaluation

Particle Representation

Effective solution representation is fundamental to metaheuristic performance. As the core data structure driving the workflow in Figure 1, this study adopts a real-valued workload matrix encoding that directly represents the decision variables. Each particle position corresponds to a matrix $S\in {\mathbb{R}}^{N\times K}$, where N is the number of farmland areas, and K is the total number of tractors. The element $s_{ik}$ denotes the area (hm²) assigned to tractor k for farmland i. This structure is illustrated in

Table 1. Particle representation for MAMSP (partial).

Farmland	Tractor 1	Tractor 2	$\cdots$	Tractor K	Sum
Area 1	${\mathrm{s}}_{11}$	${\mathrm{s}}_{12}$	$\cdots$	${\mathrm{s}}_{1\mathrm{K}}$	A1
Area 2	${\mathrm{s}}_{21}$	${\mathrm{s}}_{22}$	$\cdots$	${\mathrm{s}}_{2\mathrm{K}}$	A2
$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
Area N	${\mathrm{s}}_{\mathrm{N}1}$	${\mathrm{s}}_{\mathrm{N}2}$	$\cdots$	${\mathrm{s}}_{\mathrm{N}\mathrm{K}}$	AN

.

This encoding explicitly captures collaborative workload allocation and simplifies feasibility checks, because the sum of each row must equal the total area $A_i$ of the corresponding farmland, consistent with Constraint 1 in Section 2.4. The continuous entries $s_{ik}$ allow non-uniform workload sharing among collaborating tractors on the same farmland, rather than enforcing equal splits.

Although the MAMSP is formulated as a detailed mixed-integer model in Section 2.4, HPSO-SR does not manipulate all decision variables directly. Instead, each particle encodes only the continuous workload matrix $S=\left[s_{ik}\right]$. A deterministic simulation-based decoding procedure reconstructs the associated routing and timing variables so that the resulting schedule strictly satisfies the model constraints.

Fitness Evaluation

Fitness evaluation, corresponding to the evaluation blocks in Figure 1, is based on a simulation-driven decoding procedure. Each particle is treated as a set of instructions that drives a deterministic discrete-event simulation of the tractor fleet. As the simulation progresses, it reconstructs the temporal trajectory of every tractor, checks feasibility, and dynamically calculates the start and completion times $\left({ST}_i,{FT}_i\right)$. Based on these simulated timelines, the cost components—transfer, operation, waiting, and delay costs—are computed exactly as defined in the mathematical formulation in Section 2.4.

Consistent with the modeling assumptions, the total plowing workload is fixed, so the operation-cost term is constant across all feasible schedules. Consequently, HPSO-SR evaluates each solution using a reduced objective function $Z^{\prime }$ that contains only the decision-dependent components (transfer, waiting, and delay penalties), as defined in Equation (13):

$$Z^{\prime }=C_{transfer}+C_{waiting}+C_{delay}.$$

(13)

The algorithm seeks to minimize $Z^{\prime }$ through iterative velocity and position updates, guiding the swarm towards low-cost regions of the search space. The full cost decomposition is still reported in the numerical analysis to provide a complete economic interpretation.

2.5.3. Heuristic-Based Swarm Initialization

The quality of the initial swarm has a substantial effect on the convergence behavior of PSO. This study adopts a heuristic-based initialization strategy that serves a dual purpose: it provides a high-quality “hot start” for the swarm and establishes a performance benchmark. The initialization process comprises four steps.

(1)

Demand estimation. For each farmland i, the minimum number of tractors required $K_{min,i}$ is estimated based on its area $A_i$ and the duration of its time window $\left[{LT}_i-{ET}_i\right]$, as defined in Equation (14):

$$K_{min,i}=⌈{\displaystyle \frac{A_i}{w\left({LT}_i-{ET}_i\right)}}⌉.$$

(14)

This provides a lower bound on the machinery resources required to feasibly complete the task within the soft time window of farmland i.

(2)

Proximity-based selection. To reduce potential transfer costs while maintaining initial diversity, a candidate pool of size ${2\times K}_{min,i}$ is first identified from the cooperatives geographically closest to farmland i according to the distance matrix. From this extended pool, a set of $n_i$ tractors (where ${n_i=K}_{min,i}$) is randomly selected to form the collaborative team for farmland i. This stochastic inclusion strategy prevents the swarm from becoming trapped in local optima dominated by pure proximity, ensuring a broader exploration of potential collaborative combinations.

(3)

Workload allocation. Two allocation strategies are used. For the heuristic baseline particle, the required area $A_i$ is evenly divided among the $n_i$ candidate tractors, reflecting typical manual scheduling practice. For the stochastic particles, the workload $A_i$ is partitioned into continuous shares using a random Dirichlet-like allocation. This ensures that all initial solutions satisfy the workload-completion requirement in Constraint 1 while exploring diverse distribution patterns.

(4)

Swarm generation. These steps are combined to construct the initial swarm. The deterministic heuristic schedule (even split) is injected as the first particle to guide the search. The remaining particles are generated as randomized variants around this heuristic by perturbing the workload shares. This strategy preserves a meaningful expert-like structure while providing sufficient diversity to explore the joint space of tractor assignment and continuous workload allocation.

2.5.4. Evolutionary Mechanisms

The evolution of HPSO-SR is driven by standard PSO velocity updates, augmented by two domain-specific strategies—Sparsity Repair and Mutation Injection—that refine the CWS solutions.

(1)

Velocity and position update. The trajectory of each particle is governed by standard PSO equations with a dynamic inertia weight, as given in Equation (15):

$$\begin{gathered} v_{id}^{t+1}=\omega \cdot v_{id}^t+c_1{r}_1\left({pbest}_{id}-x_{id}^t\right)+c_2{r}_2\left({gbest}_d-x_{id}^t\right); \\ {x}_{id}^{t+1}=x_{id}^t+v_{id}^{t+1}. \end{gathered}$$

(15)

where the inertia weight $\omega$ decreases linearly from 0.9 to 0.4, and the acceleration coefficients $c_1$ and $c_2$ are both set to 1.5. This mechanism enables the algorithm to transition from global exploration to local exploitation.

(2)

Sparsity Repair strategy. Standard CWS solutions generated by velocity updates often suffer from fragmentation, where tractors are assigned negligible workloads (e.g., 0.01 hm²) and thereby incur uneconomical transfer costs. To address this, a Sparsity Repair operator is applied after every position update:

• Thresholding: For each farmland i, any assigned workload $s_{ik}$ falling below a sparsity threshold (set to 5% of the total area $A_i$) is forced to zero.
• Normalization: After pruning, the remaining workloads are renormalized so that the row-sum condition in Constraint (6) is satisfied.

This strategy transforms a dense CWS solution into a sparse CWS solution, effectively balancing the flexibility of continuous sharing with the logistical efficiency of discrete assignment. Physically, this step eliminates impractical scenarios where a tractor travels a long distance to perform a negligible amount of work, which would otherwise be mathematically valid but operationally wasteful.

(3)

Mutation Injection. To mitigate the tendency of PSO to become trapped in local optima, a Mutation Injection mechanism is incorporated. At each iteration, a subset of particles is selected with mutation probability $P_m$. For each selected particle, one of the following three problem-specific mutation operators is applied with equal probability:

• Joint assignment-and-workload mutation: The set of tractors assigned to farmland i is re-selected from nearby cooperatives, and the total area $A_i$ is repartitioned among the new tractors. This operator simultaneously perturbs the collaborative team and the associated workload distribution.
• Assignment-only mutation: The set of tractors serving farmland i is changed, but the existing workload profile is preserved and mapped to the new tractors. This explores alternative combinations without altering the workload distribution ratios.
• Workload-share mutation: The current set of tractors assigned to farmland i is retained, but their workload shares are perturbed via random rescaling. This operator focuses solely on optimizing the continuous allocation ratios among the currently assigned tractors.

By injecting these domain-specific operators into the PSO framework, the algorithm prevents stagnation and ensures a thorough exploration of the joint discrete-continuous solution space.

3. Results and Discussion

3.1. Experimental Setup

To validate the proposed framework and benchmark algorithmic performance, numerical experiments were conducted using a real-world case study from Liyang City, Changzhou, Jiangsu Province, China. The dataset comprises 20 farmland units served by 5 agricultural machinery cooperatives, which collectively operate 15 tractors. Plowing areas range from 86.81 hm² to 347.70 hm², and time windows are distributed across the autumn season to reflect realistic operational constraints during the autumn plowing period.

Figure 2 illustrates the geographic distribution of the cooperatives (red stars) and farmland units (yellow circles). Coordinates were extracted from Google Maps, and the layout is typical of agricultural operations in southern China, where farmlands are spatially fragmented around cooperative depots. Transfer distances were computed using the Haversine formula to approximate geodetic travel requirements.

The operational parameters for the model were set based on local agricultural standards: tractor operation efficiency (w) was 0.7 hm²/h, transfer speed (v) was 40 km/h, and the cost coefficients for transfer (ρ), operation (ϕ), waiting (α), and delay (β) were set to 10 CNY/km, 200 CNY/hm², 70 CNY/h, and 35 CNY/h, respectively.

To systematically evaluate the proposed scheduling framework, three optimization algorithms were implemented and benchmarked against a rule-based heuristic (simulating traditional proximity-based manual scheduling without collaborative logic):

(1)

Genetic Algorithm with Collaborative Workload Sharing (GA-CWS): An evolutionary approach utilizing the proposed continuous encoding and crossover operators. It serves as the baseline intelligent algorithm to verify the basic effectiveness of the continuous collaborative model.

(2)

Simulated Annealing with Collaborative Workload Sharing (SA-CWS): A trajectory-based method adapted for the continuous solution space. It serves as a convergence benchmark to test the potential depth of the solution space without population diversity.

(3)

Hybrid Particle Swarm Optimization with Sparsity Repair (HPSO-SR): The proposed method. While it shares the underlying CWS model with GA-CWS and SA-CWS, it additionally integrates a Sparsity Repair (SR) strategy to counter the fragmentation tendency of continuous swarms and enforce operationally efficient sparse solutions.

The algorithms were implemented in MATLAB R2022a (MathWorks, Natick, MA, USA) and executed on a personal computer equipped with an Intel Core i7-10510U CPU (Intel Corporation, Santa Clara, CA, USA), 16 GB of RAM, and the Windows 11 operating system (Microsoft Corporation, Redmond, WA, USA). To ensure a rigorous and fair comparison, the following experimental controls were imposed:

(1)

Unified Decoding: All algorithms utilized the same simulation-based decoding function to evaluate objective costs.

(2)

Equivalent Computational Budget: The maximum number of function evaluations was fixed at 25,000 for all metaheuristics (population size 50 × 500 generations for GA/PSO, and 25,000 iterations for SA).

(3)

Statistical Validation: Each algorithm was executed 20 independent times with different random seeds to account for stochasticity and evaluate algorithmic stability.

3.2. Statistical Performance and Convergence

To evaluate stability and search performance, GA-CWS, SA-CWS, and HPSO-SR were each executed for 20 independent runs. The statistics of the objective costs (sum of transfer, waiting, and delay costs) are summarized in

Table 2. Statistical performance of scheduling algorithms over 20 independent runs.

Algorithm	Best Objective Cost (CNY)	Worst Objective Cost (CNY)	Mean Objective Cost (CNY)	Std. Deviation (CNY)
GA-CWS	8211.49	14,015.32	11,107.77	1773.49
SA-CWS	5145.89	8479.52	7370.99	1102.01
HPSO-SR	2099.30	6556.76	4077.74	1132.43

.

The statistical analysis reveals a clear performance ranking: GA-CWS < SA-CWS < HPSO-SR (where “<” denotes inferior performance in terms of cost). The proposed HPSO-SR achieves the lowest mean objective cost of 4077.74 CNY, representing a reduction of 44.68% compared with SA-CWS (7370.99 CNY) and 63.29% compared with GA-CWS (11,107.77 CNY). The worst-case performance of HPSO-SR (6556.76 CNY) also remains below the best-case performance of GA-CWS (8211.49 CNY), indicating that the proposed method consistently attains higher-quality schedules in this case study. Although SA-CWS has the smallest standard deviation (1102.01 CNY), the variability of HPSO-SR occurs within a substantially lower cost range (approximately 2100–6600 CNY). From a practical perspective, this variance reflects exploration among multiple high-quality local optima rather than instability at high cost levels.

To further examine the dynamic search behavior, the convergence curves of the best-performing run for each algorithm are plotted in Figure 3.

The convergence profiles reveal distinct search characteristics:

Rapid but premature exploitation (GA-CWS, Blue Curve): GA-CWS exhibits the steepest initial descent, achieving the lowest objective cost during the early phase (generations 0–180). This suggests that the genetic operators are highly effective at aggressive exploitation, quickly identifying schedules with relatively short transfer distances but imperfect time coordination. However, the curve then plateaus around 8200 CNY, showing clear signs of premature convergence. The loss of population diversity leaves the algorithm trapped in local optima.

Steady trajectory refinement (SA-CWS, Green Curve): SA-CWS displays a slower but more gradual improvement. By accepting non-improving moves with a decreasing probability, it maintains exploration and eventually surpasses GA-CWS around generation 300. This indicates that the continuous solution space has a structure that can be effectively exploited by trajectory-based search, albeit with limited convergence speed due to its single-point nature.

Enhanced convergence capability (HPSO-SR, Red Curve): The proposed HPSO-SR exhibits a robust convergence trajectory benefiting from the stochastic initialization strategy. While the sparsity repair operator introduces a slight overhead in the very early iterations (preventing premature convergence to dense, infeasible solutions), the algorithm rapidly identifies high-quality solution basins. Unlike the plateauing behavior of GA-CWS, HPSO-SR sustains a steep descent trajectory and quickly overtakes the baselines. It eventually converges to a substantially lower objective value of approximately 2100 CNY, significantly outperforming both GA-CWS and SA-CWS. This performance confirms that the enhanced initialization and mutation mechanisms effectively prevent stagnation, enabling the swarm to locate highly efficient sparse schedules that were previously inaccessible.

3.3. Mechanism Analysis: Cost Breakdown

To clarify the mechanisms underlying the performance differences,

Table 3. Cost comparison of the best scheduling schemes.

Cost Component (CNY)	Heuristic	GA-CWS	SA-CWS	HPSO-SR
Transfer Cost	1666.98	1972.91	1916.99	2041.51
Operation Cost	872,462.00	872,462.00	872,462.00	872,462.00
Waiting Cost	0.00	0.00	0.00	0.00
Delay Cost	108,259.99	6238.58	3228.90	57.79
Objective Cost	109,926.97	8211.49	5145.89	2099.30
Improvement vs. Heuristic	—	92.53%	95.32%	98.09%
Improvement vs. GA-CWS	—	—	37.33%	74.43%
Improvement vs. SA-CWS	—	—	—	59.20%

decomposes the cost structures of the best schedules obtained by each method. The comparison separates transfer, operation, and penalty components.

The breakdown highlights a clear trade-off structure:

Heuristic baseline: The rule-based heuristic achieves the lowest transfer cost (1666.98 CNY) by assigning each tractor to its nearest farmland, but incurs very high delay costs (108,259.99 CNY). This shows that minimizing distance alone is insufficient for time-critical agricultural logistics.

Continuous collaborative advantage (GA-CWS and SA-CWS): Both GA-CWS and SA-CWS leverage the continuous CWS model to substantially reduce delays. Relative to the heuristic, GA-CWS lowers delay costs by 94.2% (to 6238.58 CNY), and SA-CWS by 97.0% (to 3228.90 CNY). This confirms that continuous workload sharing can alleviate time-window conflicts that are difficult to resolve under rigid, discrete task assignment.

“Sparsity-for-Timeliness” Trade-Off (HPSO-SR): HPSO-SR incurs a transfer cost of 2041.51 CNY, which is marginally higher (approximately 6.5%) than that of SA-CWS (1916.99 CNY) and comparable to GA-CWS (1972.91 CNY). This indicates that tractors are dispatched strategically to form collaborative teams, yet the enhanced initialization logic effectively curbs excessive travel compared to the initial exploration phase. In return, HPSO-SR maintains negligible delay penalties (57.79 CNY), whereas SA-CWS and GA-CWS incur penalties in the thousands. In effect, the sparsity repair strategy acts as a structural filter: it discourages locally attractive but time-inefficient schedules and favors sparser collaboration patterns that accept minimal additional transfer costs to ensure timely completion. By accepting a marginal increase in transfer cost relative to SA-CWS, HPSO-SR reduces delay costs by over 3000 CNY and achieves total improvements of 59.20% over SA-CWS and 74.43% over GA-CWS.

3.4. Spatiotemporal Visualization

To further validate the optimization results, the temporal schedule and spatial trajectories generated by HPSO-SR are visualized in Figure 4 and Figure 5, respectively.

Temporal Coordination (Figure 4): The Gantt chart displays the operational timelines of all 15 tractors. The optimized schedule is remarkably compact and highly parallelized, demonstrating efficient collaborative multitasking. Despite the tight agronomic constraints, there are only isolated instances of negligible delay relative to the specified time windows (consistent with the minimal delay cost of 57.79 CNY). The visualization confirms that the stochastic-enhanced HPSO-SR successfully balances workload distribution, avoiding both resource idleness and significant delivery lateness. Furthermore, the absence of fragmented task segments validates that the sparsity repair mechanism effectively enforces operationally continuous work blocks.

Spatial Logistics (Figure 5): The route map illustrates the spatial trajectories of the fleet. Compared with the localized patterns of proximity-based heuristics, the HPSO-SR solution exhibits a highly optimized topology involving strategic cross-regional routes. Tractors are dynamically dispatched between farmland clusters (e.g., collaborative support from Cooperative C2 and C5 to Area A10) only when necessary to meet time windows. Notably, the revised routing pattern is leaner and more efficient, reflecting the reduced transfer cost (2041.51 CNY). This visually confirms that the algorithm achieves a synergistic optimization where timely service is delivered with minimal excess travel.

3.5. Mechanism Ablation Study

To rigorously quantify the contributions of the proposed mechanisms—specifically Sparsity Repair (SR) and Mutation Injection—an ablation study was conducted. We compared the full HPSO-SR against two variants: (1) HPSO-NoMut, where the mutation operator was disabled to test global search capability; and (2) HPSO-NoSR, where the sparsity repair operator was disabled to assess the impact of the sparsity constraint. All variants utilized the stochastic initialization strategy to ensure a fair baseline.

The statistical results over 20 runs are presented in

Table 4. Ablation study results comparing algorithmic variants.

Variant	Mean Cost (CNY)	Std. Dev. (CNY)	Impact Analysis
HPSO-NoMut	9820.00	4299.38	Performance degrades significantly (+140.8% cost vs. Full), indicating severe premature convergence without mutation.
HPSO-NoSR	3107.18	764.20	Theoretical cost decreases (−23.8% vs. Full), but solutions suffer from extreme fragmentation (high cardinality), rendering them operationally impractical.
HPSO-SR (Full)	4077.74	1132.43	Selected Balance. Accepts a moderate cost increase to enforce sparse, feasible coordination.

. The results reveal distinct roles for each mechanism:

Role of Mutation: The HPSO-NoMut variant yields the worst performance (9820.00 CNY) with the highest variability. This confirms that standard PSO velocity updates alone are insufficient to navigate the complex multimodal landscape of MAMSP. The mutation injection mechanism is critical for maintaining population diversity and escaping local optima.

Role of Sparsity Repair (The “Cost of Feasibility”): Interestingly, the HPSO-NoSR variant achieves a lower objective cost (3107.18 CNY) than the full HPSO-SR. This counterintuitive result arises because the unconstrained continuous CWS formulation minimizes mathematical cost by atomizing workloads into negligible fractions across many tractors to reduce transfer distances. However, such “mathematically optimal” schedules are logistically infeasible due to excessive fragmentation. HPSO-SR effectively acts as a regularizer: it accepts a 31.2% increase in theoretical cost (from 3107 to 4077 CNY) to prune these fragmented assignments. This trade-off is essential to convert a theoretical lower bound into an operationally executable sparse schedule.

3.6. Sensitivity and Robustness Analysis

To assess performance under different levels of urgency, a sensitivity analysis was conducted on the delay penalty coefficient β, which was varied from 0 (risk-free) to 70 CNY/h (severe urgency).

Table 5. Sensitivity of average objective cost to delay penalty coefficient β.

Delay Penalty (β)	GA-CWS Avg. Cost (CNY)	SA-CWS Avg. Cost (CNY)	HPSO-SR Avg. Cost (CNY)
0 (Risk-free)	1479.22	1420.50	826.38
20 (Low)	7357.11	4965.64	2916.55
35 (Base Case)	11,107.77	7370.99	4077.74
50 (High)	15,666.10	9793.85	4592.28
70 (Severe)	20,496.11	11,791.58	5357.87

reports the average objective costs for each algorithm under different values of β.

The analysis highlights the superior algorithmic resilience of HPSO-SR across two distinct phases:

Low urgency (β = 0): Even when delays are not penalized, HPSO-SR attains the lowest average cost (826.38 CNY). This indicates that the sparsity repair mechanism is adaptive: in relaxed scenarios, it primarily prunes redundant transfers and avoids unnecessary movements, outperforming both GA-CWS and SA-CWS.

Increasing urgency (β ≥ 20): As the delay penalty coefficient β increases, the average costs for GA-CWS and SA-CWS grow rapidly. From β = 35 to β = 70, their average objective costs increase by approximately 9388 CNY and 4421 CNY, respectively, reflecting limited ability to fully resolve tight time-window conflicts. In contrast, the increase for HPSO-SR over the same range is only about 1280 CNY. This remarkably flat growth confirms that HPSO-SR’s structural sparsity provides exceptional resilience against tightening time constraints, effectively absorbing urgency shocks with minimal cost escalation, and is therefore the most robust of the three algorithms in this time-critical context.

3.7. Generalizability and Scalability Analysis

To address the concern that results derived from a single case study may not be generalizable, we extended the evaluation to synthetic datasets of varying scales. Three test instances were generated based on the statistical distribution of the Liyang data: Small (10 farmlands, 5 tractors), Medium (20 farmlands, 15 tractors, corresponding to the real-world case), and Large (40 farmlands, 30 tractors). Each algorithm was executed for 20 independent runs per scale.

The statistical results (Mean Objective Cost and Wilcoxon rank-sum test p-values) are summarized in

Table 6. Statistical comparison of algorithms across different problem scales.

Scale	Algorithm	Mean Cost (CNY)	Std. Dev. (CNY)	p-Value (vs. HPSO-SR)
Small	GA-CWS	24,716.53	2581.59	3.40 × 10−8 (***)
	SA-CWS	18,418.33	935.46	3.46 × 10−7 (***)
	HPSO-SR	12,647.63	2890.95	—
Medium	GA-CWS	11,107.77	1773.49	3.40 × 10−8 (***)
	SA-CWS	7370.99	1102.01	1.11 × 10−7 (***)
	HPSO-SR	4077.74	1132.43	$-$
Large	GA-CWS	94,402.01	4410.58	3.40 × 10−8 (***)
	SA-CWS	78,022.59	2207.23	3.40 × 10−8 (***)
	HPSO-SR	40,152.08	8608.02	$-$

Note: (***) indicates statistical significance at the 0.001 level (p < 0.001).

.

The results demonstrate that HPSO-SR maintains a dominant performance advantage across all scales. In the Small scale, HPSO-SR reduces the mean cost by 31.3% compared to SA-CWS (12,647 vs. 18,418 CNY). In the Medium scale, the advantage widens to 44.7% (4077 vs. 7371 CNY). Crucially, in the Large instance, HPSO-SR achieves a massive cost reduction of 48.5% relative to SA-CWS (40,152 vs. 78,022 CNY). The statistical significance tests (p < 0.001) confirm that HPSO-SR achieves a statistically significant improvement in cost reduction compared to the baselines. This trend indicates that as the problem size and complexity increase, the “sparsity repair + stochastic initialization” strategy becomes increasingly critical for avoiding the severe solution fragmentation that plagues standard metaheuristics.

4. Conclusions

This study presented a simulation-based scheduling framework for the Multi-cooperative Agricultural Machinery Scheduling Problem (MAMSP), enabling flexible cross-cooperative resource sharing under farmland-level soft time windows. Based on a mixed-integer mathematical formulation, the work addresses situations in agricultural logistics where standard discrete assignment models have limited ability to exploit collaborative scheduling across cooperatives. By combining model formulation, algorithm design, and empirical benchmarking, three main contributions are made.

First, a Continuous Collaborative Workload Sharing (CWS) model was formulated that explicitly represents cross-cooperative resource allocation. Unlike traditional models that rely on rigid binary task assignments, this formulation allows multiple tractors to jointly serve a single farmland via continuous workload partitioning. By defining farmland-level penalties on the aggregated timeline of all participating tractors, the model provides a unified mathematical structure to represent trade-offs between cooperative resource sharing and agronomic timeliness requirements.

Second, a Hybrid Particle Swarm Optimization with Sparsity Repair (HPSO-SR) was designed to solve the high-dimensional continuous problem. Recognizing that standard continuous metaheuristics can produce solution fragmentation—where workloads are atomized into uneconomical small fractions across many tractors—a Sparsity Repair strategy was introduced. This mechanism prunes low-workload fractional assignments and guides the swarm towards sparse, operationally efficient solution structures. Together with a Mutation Injection operator to maintain diversity, the algorithm balances the flexibility of continuous workload sharing with the logistical efficiency of tractor routing.

Third, the proposed approach was empirically evaluated using both a real-world case study from Liyang, China, and synthetic instances of varying scales. Numerical experiments show that HPSO-SR reduces the objective cost by 74.43% relative to the genetic baseline and by 59.20% relative to the simulated annealing benchmark in the standard case. Scalability tests further confirm that in large-scale scenarios, the algorithm maintains a dominant advantage (over 48% cost reduction vs. SA-CWS), demonstrating robust scalability. Cost decomposition indicates that these gains are achieved mainly by accepting moderately higher transfer costs while nearly eliminating delay penalties. Mechanism ablation studies validate that the mutation operator is essential for global convergence, while the sparsity repair strategy effectively converts theoretical mathematical bounds into operationally feasible schedules. Additionally, the incorporated stochastic initialization mechanism proves critical for preventing early-stage bias, significantly boosting performance in large-scale scenarios. Furthermore, sensitivity analysis with respect to the delay penalty coefficient shows that HPSO-SR maintains comparatively low total costs as urgency increases, demonstrating superior robustness compared to competing methods.

From a managerial perspective, these findings offer critical insights for agricultural cooperative operators. Our analysis reveals a strategic trade-off: pursuing the absolute minimum transfer distance (as performed in traditional proximity-based scheduling) often leads to severe delays. Instead, the results suggest that managers should accept a moderate increase in transfer costs (approximately 6.5% higher than the theoretical minimum) to enable sparse, cross-regional collaboration. This strategy significantly mitigates the risk of delay penalties, ensuring that agronomic time windows are respected without incurring the logistical chaos of fragmented assignments.

Despite these contributions, this study has limitations that suggest directions for future research. First, the current model assumes a homogeneous fleet with deterministic parameters. In practice, agricultural logistics often involve heterogeneous tractors with different capacities and uncertainties arising from weather or equipment failures. Future work should extend the model to handle heterogeneous fleets and incorporate dynamic rescheduling mechanisms. Second, the use of farmland-level penalties (applied once per field regardless of the number of tractors) was chosen to reflect agronomic completion requirements, but this may theoretically bias the solver towards using more tractors than strictly necessary. Future research could explore alternative penalty structures, such as per-tractor operational costs, to further refine resource efficiency. Finally, while the present single-objective formulation focuses on minimizing economic costs, future studies could investigate multi-objective variants to simultaneously consider carbon emissions and workload equity among cooperatives.