Adaptive Genetic Algorithm Integrated with Ant Colony Optimization for Multi-Task Agricultural Machinery Scheduling

Abstract

Efficient scheduling of agricultural machinery is critical for optimizing resource utilization and reducing operational costs in modern farming operations. This study proposes an Adaptive Genetic Algorithm integrated with Ant Colony Optimization (AGA-ACO) to solve the multi-task machinery scheduling problem. The problem is formulated as a Vehicle Routing Problem with Time Windows (VRPTW), considering time constraints, machinery heterogeneity, and task dependencies. The AGA-ACO algorithm employs a two-phase optimization strategy: genetic algorithms for global exploration and ant colony optimization for local refinement through pheromone-guided search. Experimental evaluation using real-world agricultural data from Hangzhou demonstrates that AGA-ACO achieves cost reductions of 5.92–10.87% compared to genetic algorithms, 5.47–7.75% compared to ant colony optimization, and 6.23–9.51% compared to particle swarm optimization, while converging with fewer iterations. The algorithm maintains stable convergence and high robustness across different farmland scales, reducing computational time while preserving solution quality. A scheduling management system integrating IoT sensors, MQTT protocols, and GIS technologies validates the practical applicability of the proposed approach. This research provides a replicable framework for agricultural machinery optimization, contributing to the advancement of sustainable and precision agriculture.

1. Introduction

Modern agriculture is rapidly transforming toward mechanization and intelligent management to enhance productivity, reduce labor costs, and ensure food security for a growing global population [1]. Agricultural mechanization has become particularly crucial in addressing the challenges of labor shortages, aging rural populations, and the increasing demand for agricultural products [2]. In this context, the efficient scheduling of agricultural machinery emerges as a critical factor in optimizing resource utilization and minimizing operational costs [3,4].

At the field level, agricultural operations are highly time-sensitive, requiring precise coordination of multiple machines across farmlands within strict time windows. Tasks such as plowing, seeding, fertilizing, and harvesting must be completed at optimal times to ensure crop quality and maximize yields [5]. This is particularly evident in rice cultivation, where agronomic requirements impose strict temporal constraints, and during peak farming seasons, various operational stages are tightly interconnected with compressed scheduling windows. The complexity of scheduling is further amplified by factors including machinery heterogeneity, varying field sizes, different task requirements, and geographical dispersion of farmlands [6]. Traditional experience-based manual scheduling approaches fail to address these multifaceted challenges, often resulting in resource underutilization, operational delays, and increased costs.

China, as the world’s largest agricultural producer, exemplifies these challenges at scale. The country has achieved remarkable progress in agricultural mechanization, with the comprehensive mechanization rate of crop cultivation and harvest in China reaching 71% in 2020 and expected to rise to 75% by 2025 [7]. In 2020, agricultural machinery cooperatives in China serviced 56.509 million hectares, with cross-regional operations covering 19.9 million hectares [8]. Despite this achievement, efficient machinery scheduling remains a significant bottleneck, particularly in regions with fragmented land holdings and complex cropping patterns. The development of scientific scheduling methodologies has thus become essential for advancing agricultural modernization, optimizing the substantial investment in machinery resources, and ensuring sustainable food production [4].

Agricultural machinery scheduling optimization has been extensively studied through intelligent algorithms. Existing research has established numerous models and algorithms, particularly through intelligent optimization techniques such as Particle Swarm Optimization (PSO) [9,10], Simulated Annealing (SA) [11], Tabu Search (TS) [6,12], Neighborhood Search [13,14], Ant Colony Optimization (ACO) [15,16], Genetic Algorithms (GA) [17], and hybrid optimization methods [18,19]. Particularly, adaptive genetic algorithms have demonstrated superior performance over traditional fixed-parameter GAs. Srinivas and Patnaik pioneered adaptive probabilities of crossover and mutation based on fitness values, establishing the theoretical foundation for parameter self-adaptation [20]. This seminal work has inspired diverse applications: from optical metasurface design, where AGA handles high-dimensional optimization with fabrication constraints [21], to tourism flow forecasting, where seasonal patterns are captured through adaptive parameter tuning [22], and magnetic hysteresis modeling utilizing self-adaptive mechanisms [23]. These studies collectively demonstrate that adaptive parameter control significantly enhances convergence efficiency while maintaining population diversity, eliminating the need for manual parameter tuning.

These algorithms demonstrate significant potential in agricultural scheduling. For instance, Wang et al. developed a time window-constrained machinery scheduling model using GA [24], while Guan et al. (2009) proposed a two-stage metaheuristic framework integrating SA and GA for dynamic resource allocation in sugarcane production [25]. Orfanou et al. (2013) introduced a biomass harvesting task sequencing method to optimize machine schedules and total operational costs [26]. Chen et al. (2023) designed a three-stage scheduling framework that reduces multi-field collaborative operation cycles through dynamic task prioritization and resource constraint modeling [27]. Sethanan and Neungmatcha (2020) enhanced PSO with collision avoidance mechanisms, achieving dual optimization of harvest distance and yield in sugarcane operations [3]. For dynamic environments, Cao et al. (2021) proposed a multi-machine collaborative task allocation model that reduces delays by responding to machinery failures and emergent tasks [28]. Wang and Huang (2022) further integrated CPLEX with time window constraints in a shared machinery framework, reducing cross-regional scheduling costs by 28% [29]. He et al. (2018) minimized wheat harvesting periods in fragmented fields using a hybrid TS approach under temporal consistency constraints [30].

The agricultural machinery scheduling problem fundamentally extends the classical Vehicle Routing Problem (VRP), first proposed by Dantzig and Ramser (1959) for fuel truck fleet optimization [31]. In agriculture, VRP-based models abstract farmlands as nodes to optimize route costs, providing theoretical foundations for scheduling. Zhang et al. established a time-optimal machinery scheduling model using improved GA [32], while Lin et al. (2019) developed a harvest planning model to maximize farmer profits [33]. For complex constraints, Basnet et al. integrated time windows, operational costs, and machinery-driver assignments to minimize total scheduling time [34]. Vázquez et al. (2021) integrated multi-crop rotation constraints within a multi-machine scheduling framework [35]. Chen et al. (2021) analyzed pandemic-induced scheduling costs and formulated a total cost minimization model [36].

The integration of metaheuristic algorithms has significantly advanced this field. Seyyedhasani et al. (2018) modeled scheduling as a multi-vehicle collaborative VRP and solved it using an improved tabu-search algorithm [37]. Cerdeira et al. (2017) investigated a variant of the Traveling Salesman Problem (TSP) incorporating cluster constraints, time windows, and city processing time constraints [38]. They proposed a hybrid algorithm integrating Tabu search and simulated annealing to optimize operational routes for combined harvesters in an agricultural machinery cooperative [38]. Wang et al. (2020) proposed a hybrid algorithm, MOSA-ACO, which integrates Simulated Annealing with Ant Colony Optimization to address the periodic vehicle routing problem [39]. The algorithm demonstrated strong performance in solving multi-objective optimization problems [39]. Notably, the multi-task, multi-machine scheduling problem exhibits the characteristics of a Multi-Depot VRP with Time Windows (MDVRPTW)—an NP-hard extension [40] that incorporates multiple depots, time windows, and heterogeneous fleets, thus greatly increasing complexity [41]. Most existing studies focus on single-task or fixed-sequence scheduling, leaving a gap in the context of multi-task, multi-machine operations in complex agricultural environments.

Despite these advances, significant challenges remain for the highly time-sensitive, multi-machine, multi-task coordination required in complex agricultural environments. Diverse terrain conditions, varying field characteristics, and inter-machine task conflicts render traditional models inadequate. To address these limitations, this study proposes an Adaptive Genetic Algorithm-Ant Colony Optimization (AGA-ACO) framework that synergistically combines GA’s global exploration capabilities with ACO’s local exploitation precision.

The main contributions of this research are:

• Dual-Objective Scheduling Model: A VRP-based model incorporating time window constraints and machinery heterogeneity is established, targeting both minimal total scheduling cost and shortest operational duration. This model addresses the multi-machine, multi-task scheduling problem in modern agriculture and ensures operational efficiency.
• Hybrid Optimization Algorithm: An adaptive crossover probability adjustment strategy and dynamic pheromone update mechanism are designed to synergize GA’s global search capabilities with ACO’s local path optimization. Comparative experiments demonstrate AGA-ACO’s superior convergence speed, solution accuracy, and resistance to local optima over standalone GA and ACO.

This study provides theoretical support and methodological innovation for intelligent agricultural machinery scheduling in agricultural regions and charts a new technological path toward unmanned, high-efficiency precision agriculture. The proposed framework enables real-time coordination of autonomous machinery fleets in farmlands, significantly reducing operational delays and resource conflicts—key steps toward scalable unmanned farming in agricultural regions.

2. Materials and Methods

2.1. Problem Description

In modern agricultural operations, the efficient scheduling of agricultural machinery is crucial for optimizing operational costs and ensuring the timely completion of field tasks. This study addresses a multi-task, multi-machine scheduling problem in which agricultural machinery from a central depot must be allocated to execute various operational tasks across multiple farmlands.

The scheduling problem involves multiple farmlands, each requiring specific agricultural operations to be performed within designated time windows. Each farmland may require multiple types of agricultural operations, such as plowing, seeding, fertilizing, and harvesting, which must be executed by appropriate machinery types. The agricultural machinery depot maintains various types of machines, each designed to perform specific operational tasks. The scheduling system must dispatch appropriate machinery from the depot to designated farmlands for task execution, ensuring that each required operation is completed efficiently while minimizing overall operational costs.

Let the set of farmlands be defined as $F=\{F_1,F_2,\dots ,F_m\}$, where $F_i$ represents the $i$-th farmland. The attributes of $F_i$ are described as $\left\{\mathrm{L}\mathrm{o}\mathrm{c}{F}_i,S_i,T_i\right\}$, where $\mathrm{L}\mathrm{o}\mathrm{c}{F}_i$ represents the entrance location of farmland $F_i$, $S_i$ denotes the area of farmland $F_i$ and $T_i$ represents the task list required for farmland $F_i$.

Similarly, let the set of agricultural machines be denoted as $M=\left\{M_1,M_2,\dots ,M_n\right\}$, where $M_j$ represents the j-th machine. The attributes of $M_j$ are characterized by $M_j=\left\{{\mathrm{L}\mathrm{o}\mathrm{c}\mathrm{M}}_j,{ \mathrm{transV}}_j,{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{V}}_j,{\mathrm{T}\mathrm{y}\mathrm{p}\mathrm{e}}_j\right\}$, where ${\mathrm{L}\mathrm{o}\mathrm{c}\mathrm{M}}_j$ indicates the current location of machinery $M_j$,${ \mathrm{transV}}_j$ represents the average travel speed during inter-farmland transitions, ${\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{V}}_j$ denotes the operational efficiency, and ${\mathrm{T}\mathrm{y}\mathrm{p}\mathrm{e}}_j$ indicates the task type that machine $M_j$ is designed to perform.

The primary objective is to develop an optimal scheduling strategy that minimizes the total operational cost while ensuring efficient task completion. The total cost encompasses machinery transfer costs, labor costs, operational costs, waiting costs, and penalty costs associated with time window violations.

2.2. Model Building

2.2.1. Model Assumptions

Based on the characteristics of the multi-task, multi-machine scheduling problem, the hypothetical preconditions for this scheduling problem are first declared:

• All farmland parameters (area, location, required tasks, time windows) and machinery specifications (type, quantity, efficiency, speed) are known and constant.
• Each machine can only execute tasks matching its type and operates on one task at a time, with total deployment not exceeding available inventory.
• For computational tractability, farmland geometries are approximated as regular polygons to enable efficient area calculation and distance estimation in the scheduling model.

4.

Tasks within each farmland must be executed sequentially according to the order specified in the task list $T_i$. For example, if $T_i= [$1,2], task type 1 must be completed before task type 2 can begin.

5.

Dynamic environmental factors (e.g., weather variations, soil moisture fluctuations) are excluded from operational efficiency calculations.

6.

A soft time window constraint is implemented for each farmland, where all required tasks must be completed within the specified time window. Deviations incur quantified penalty costs.

2.2.2. Objective Function

The multi-task, multi-agricultural machinery scheduling model aims to minimize the total cost, which includes machinery transfer cost, labor cost, operational cost, waiting cost, and penalty cost. The cost objective functions are defined as follows:

1. Agricultural Machinery Transfer Cost:

$$C_{trans}=\sum _{i=0}^{m+1} \sum _{p=0}^{m+1} \sum _{j=1}^n C_{tj}·tran{sT}_{ipj}·y_{ipj}$$

(1)

2. Agricultural Machinery Labor Cost:

$$C_{labor}=\sum _{i=1}^m\sum _{j=1}^n \sum _{t{\in T}_i}{C}_{gj}·work{T}_{ijt}·x_{ijt}$$

(2)

3. Agricultural Machinery Operational Cost:

$$C_{op}=\sum _{i=1}^m\sum _{j=1}^n \sum _{t{\in T}_i}{C}_{wj}·work{T}_{ijt}·x_{ijt}$$

(3)

4. Agricultural Machinery Waiting Cost:

$$C_{\mathrm{w}\mathrm{a}\mathrm{i}\mathrm{t}}=\sum _{i=1}^m\sum _{j=1}^n \sum _{k{\in T}_i}{C}_{dj}·wait{T}_{ijt}·z_{ijt}$$

(4)

5. Time Window Violation Penalty:

$$C_{\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{y}}=\sum _{i=1}^m{P}_i$$

(5)

Thus, the total scheduling cost is given by:

$$min C_{total}=C_{trans}+C_{labor}+C_{op}+C_{wait}+C_{penalty}$$

(6)

2.2.3. Constraints

According to the analysis of the multi-task, multi-agricultural machinery scheduling process of agricultural operations, the constraints are determined as follows:

$${\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{T}}_{ipj}={\displaystyle \frac{d_{ip}}{{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{V}}_j}};(i,p\in \{\mathrm{0,1},\dots ,m+1\},j\in \{\mathrm{1,2},\dots ,n\})$$

(7)

$$\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}{T}_{ijt}=\left\{\begin{array}{l}{\displaystyle \frac{S_i}{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}{V}_j}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}, if {Type}_j=t\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}, otherwise. \end{array};\left(\forall i,j,t\right)\right.$$

(8)

$$\sum _{j:Typ{e}_j=t} x_{ijt}=1;\left(\forall i\in \{1,\dots ,m\},\forall t\in T_i\right)$$

(9)

$$x_{ijt}=0; if Typ{e}_j\ne t$$

(10)

$$f_{i,T_i\left[p\right]}\le s{t}_{i,T_i[p+1]};\forall p\in \left\{\mathrm{1,2},\dots ,\left|T_i\right|-1\right\}$$

(11)

$$s{t}_{i,t}=\sum _{j:{ Type}_j=t} s{t}_{ijt}\cdot x_{ijt};\left(\forall i\in \{1,\dots ,m\},\forall t\in T_i\right)$$

(12)

$$f_{i,t}=s{t}_{i,t}+\sum _{j:Typ{e}_j=t} { \mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k} }_{ijt}\cdot x_{ijt};\left(\forall i\in \{1,\dots ,m\},\forall t\in T_i\right)$$

(13)

$$E{T}_i\le s{t}_{i,T_i\left[1\right]} \mathrm{and} f_{i,T_i\left[\left|T_i\right|\right]}\le L{T}_i;(\forall i\in \{1,\dots ,m\left\}\right)$$

(14)

$$P_i=\left\{\begin{array}{l}\left(E{T}_i-s{t}_{i,T_i\left[1\right]}\right)\cdot C_{ei}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em} \mathrm{i}\mathrm{f} s{t}_{i,T_i\left[1\right]}<E{T}_i\\ 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hspace{1em}\hspace{1em} \mathrm{i}\mathrm{f} E{T}_i\le s{t}_{i,T_i\left[1\right]} \mathrm{a}\mathrm{n}\mathrm{d} f_{i,T_i\left[\left|T_i\right|\right]}\le L{T}_i\\ \left(f_{i,T_i\left[\left|T_i\right|\right]}-L{T}_i\right)\cdot C_{li}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em} \mathrm{i}\mathrm{f} f_{i,T_i\left[\left|T_i\right|\right]}>L{T}_i\end{array}\right.$$

(15)

$$a{t}_{ijt}=\sum _{p=1}^{m+1} \left(\underset{t^{\prime }\in T_p}{max} \left(f_{p,t^{\prime }}\cdot x_{pj{t}^{\prime }}\right)+\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}{T}_{pij}\right)\cdot y_{pij};(\forall i,j,t)$$

(16)

$$wait{T}_{ijt}=max\left(0,s{t}_{ijt}-a{t}_{ijt}\right);(\forall i,j,t)$$

(17)

$$\sum _{p=0}^{m+1} y_{pij}=\sum _{p=0}^{m+1} y_{ipj};(\forall i\in \{\mathrm{0,1},\dots ,m+1\},j\in \{1,\dots ,n\})$$

(18)

$$\sum _{i=1}^m y_{0ij}\le 1;(j\in \{\mathrm{1,2},\dots ,n\})$$

(19)

$$\sum _{i=1}^m y_{i,m+1,j}\le 1;(j\in \{\mathrm{1,2},\dots ,n\})$$

(20)

$$x_{ijt}=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em} , \mathrm{i}\mathrm{f} \mathrm{m}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{e} M_j \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{s} \mathrm{t}\mathrm{a}\mathrm{s}\mathrm{k} t \mathrm{a}\mathrm{t} F_i\\ 0 \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}, \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e} \end{array}\right.$$

(21)

$$y_{ipj}=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em} , \mathrm{i}\mathrm{f} \mathrm{m}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{e} M_j \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{s} \mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m} \mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} i \mathrm{t}\mathrm{o} \mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} p\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em} , \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e} \end{array}\right.$$

(22)

$$z_{ijt}=\left\{\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em} , \mathrm{i}\mathrm{f} \mathrm{m}\mathrm{a}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{e} M_j \mathrm{w}\mathrm{a}\mathrm{i}\mathrm{t}\mathrm{s} \mathrm{b}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e} \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{t}\mathrm{a}\mathrm{s}\mathrm{k} t \mathrm{a}\mathrm{t} F_i\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em} , \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e} \end{array}\right.$$

(23)

Constraint (7) calculates transfer time between locations based on machinery speed. Constraint (8) determines operational time considering machinery efficiency and task compatibility. Constraint (9) enforces the task assignment constraint, ensuring that each required task at every farmland is assigned to exactly one machine of the appropriate type. Constraint (10) establishes the machine-task compatibility constraint, preventing machines from being assigned to tasks that do not match their operational capabilities.

Constraint (11) imposes the task sequence constraint, ensuring that tasks within each farmland are executed in the predetermined order specified in the task list. Constraints (12) and (13) define the temporal relationships between task executions, where Constraint (12) establishes the actual start time of each task based on the assigned machinery, and Constraint (13) calculates the completion time as the sum of start time and operational duration. Constraint (14) ensures all operations comply with farmland time windows $[E{T}_i,L{T}_i]$. Constraint (15) quantifies the time window violation penalties, imposing costs for premature task initiation or delayed task completion relative to the specified time windows.

Constraint (16) calculates the arrival time of machinery at farmlands based on previous task completions and transfer times from either the depot or other farmland locations. Constraint (17) determines the waiting time incurred when machinery arrives at farmlands before the scheduled task commencement time. Constraint (18) maintains the flow balance constraint for machinery movement, ensuring that the number of arrivals equals the number of departures at each location for every machine. Constraints (19) and (20) specify that each machine can depart from and return to the depot at most once during the scheduling horizon, ensuring realistic operational constraints. Constraints (21–23) define the binary decision variables that govern the scheduling system.

2.2.4. Variable Description

The parameter configuration used in the proposed AGA-ACO algorithm is summarized in

Table 1. Symbol description.

Symbol	Description
$F=\{F_1,F_2,\dots ,F_m\}$	Set of farmlands
$M=\left\{M_1,M_2,\dots ,M_n\right\}$	Set of agricultural machinery
$F_i$	The i-th farmland
${LocF}_{\mathrm{i}}$	$\mathrm{Entry} \mathrm{location} \mathrm{of} \mathrm{farmland} F_i$
$S_i$	$\mathrm{Area} \mathrm{of} \mathrm{farmland} F_i$, hm2
$T_i$	$\mathrm{Task} \mathrm{list} \mathrm{for} \mathrm{farmland} F_i$
$T_i\left[p\right]$	$\mathrm{The} p\text{-th} \mathrm{task} \mathrm{in} \mathrm{the} \mathrm{task} \mathrm{list} T_i$
$M_j$	The j-th agricultural machinery
${\mathrm{T}\mathrm{y}\mathrm{p}\mathrm{e}}_j$	$\mathrm{Task} \mathrm{type} \mathrm{that} \mathrm{machine} M_j$ can perform
$Q_j$	$\mathrm{Maximum} \mathrm{operational} \mathrm{capacity} \mathrm{of} M_j$, hm2
${LocM}_j$	$\mathrm{Current} \mathrm{location} \mathrm{of} \mathrm{machinery} M_j$
${\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{V}}_j$	$\mathrm{Average} \mathrm{travel} \mathrm{speed} \mathrm{of} M_j$ during transfer, km/h
${\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{V}}_j$	$\mathrm{Average} \mathrm{operational} \mathrm{efficiency} \mathrm{of} M_j$, hm2/h
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}{T}_{ipj}$	$\mathrm{Transfer} \mathrm{time} \mathrm{of} M_j$ from location i to p, h
$work{T}_{ijt}$	$\mathrm{Working} \mathrm{time} \mathrm{of} M_j$$\mathrm{for} \mathrm{task} t \mathrm{at} \mathrm{farmland} F_i$, h
$wait{T}_{ijk}$	$\mathrm{Waiting} \mathrm{time} \mathrm{of} M_j$$\mathrm{before} \mathrm{task} t \mathrm{at} \mathrm{farmland} F_i$, h
${st}_{ijt}$	$\mathrm{Start} \mathrm{time} \mathrm{of} \mathrm{machine} M_j$$\mathrm{for} \mathrm{task} t \mathrm{at} \mathrm{farmland} F_i$, h
${st}_{i,t}$	$\mathrm{Start} \mathrm{time} \mathrm{of} \mathrm{task} t \mathrm{at} \mathrm{farmland} F_i$, h
$f_{i,t}$	$\mathrm{Completion} \mathrm{time} \mathrm{of} \mathrm{task} t \mathrm{at} \mathrm{farmland} F_i$, h
${at}_{ijt}$	$\mathrm{Arrival} \mathrm{time} \mathrm{of} \mathrm{machine} M_j$$\mathrm{for} \mathrm{task} t \mathrm{at} \mathrm{farmland} F_i$, h
$C_{tj}$$, C_{gj}$$, C_{wj}$$, C_{dj}$	$\mathrm{Unit} \mathrm{transfer}, \mathrm{labor}, \mathrm{operation} \cos \mathrm{t}, \mathrm{waiting} \cos \mathrm{t} \mathrm{of} M_j$(CNY/h)
$C_{ei},C_{li}$	$\mathrm{Unit} \mathrm{penalty} \cos \mathrm{t} \mathrm{for} \mathrm{early} \mathrm{arrival}/\mathrm{late} \mathrm{completion} \mathrm{at} \mathrm{farmland} F_i$(CNY/h)
$x_{ijt}$	$\mathrm{Binary}: 1 \mathrm{if} \mathrm{machine}{ M}_j$$\mathrm{performs} \mathrm{task} t \mathrm{at} \mathrm{farmland} F_i$, 0 otherwise
$y_{ipj}$	$\mathrm{Binary}: 1 \mathrm{if} \mathrm{machine} M_j$ travels from location i to p, 0 otherwise
${\mathrm{z}}_{\mathrm{i}\mathrm{j}\mathrm{t}}$	$\mathrm{Binary}: 1 \mathrm{if} \mathrm{machine} M_j$$\mathrm{waits} \mathrm{before} \mathrm{task} t \mathrm{at} \mathrm{farmland} F_i$, 0 otherwise
$d_{ip}$	$\mathrm{Distance} \mathrm{from} \mathrm{location} i$ to p, km
$\left[E{T}_{\mathrm{i}},L{T}_{\mathrm{i}}\right]$	$\mathrm{Time} \mathrm{window} \mathrm{for} \mathrm{completing} \mathrm{all} \mathrm{tasks} \mathrm{at} \mathrm{farmland} F_i$, h
$P_i$	$\mathrm{Total} \mathrm{penalty} \cos \mathrm{t} \mathrm{for} \mathrm{farmland} F_i$, CNY

.

2.3. AGA-ACO Algorithm

Agricultural machinery scheduling problems present a complex optimization challenge involving multiple constraints such as time windows, spatial relationships, and resource allocation. While traditional genetic algorithms (GA) have demonstrated broad applicability to combinatorial problems, they often suffer from premature convergence to local optima and rely on empirical parameter tuning. To address these limitations, we propose AGA-ACO (Adaptive Genetic Algorithm with Ant Colony Optimization), a hybrid metaheuristic framework enhanced through three innovations:

(1)

Autonomous adaptation mechanism: Dynamically adjusts crossover and mutation probabilities based on real-time population fitness, eliminating manual parameter setting.

(2)

Elite preservation with diversity maintenance: A fitness-proportional (roulette-wheel) selection scheme guarantees the retention of high-quality solutions, while explicitly enforcing diversity thresholds to prevent genetic stagnation.

(3)

Pheromone-guided local intensification: Borrowing from ant colony optimization (ACO), pheromone trails focus local search on promising task–machine assignments under tight time-window constraints, thus refining solution quality without sacrificing convergence speed.

This hybrid approach leverages GA’s global exploration capabilities and ACO’s local exploitation precision to achieve accelerated convergence while maintaining solution diversity, particularly crucial for the spatiotemporal optimization required in agricultural machinery coordination, where both routing efficiency and temporal compliance must be jointly optimized.

2.3.1. Algorithm Framework

On the basis of the mathematical model presented, an Adaptive Genetic Algorithm with Ant Colony Optimization (AGA-ACO) is proposed to solve the multi-task scheduling problem of agricultural machinery. The hybrid algorithm framework iterates through genetic operations (selection, crossover, mutation) and ACO-guided local search. Elite solutions trigger pheromone updates that guide subsequent ACO local search operations, creating a synergistic optimization process where GA provides global exploration and ACO enhances local solution quality. The overall algorithm flow is shown in Figure 1.

2.3.2. Algorithm Steps

(1)

Parameter Initialization. The algorithm begins by loading all necessary input parameters, which include the operational parameters of agricultural machinery, geospatial attributes of farmlands, and task specifications. Subsequently, core algorithmic parameters are configured to initialize the optimization process.

(2)

Encoding and Decoding Scheme. The algorithm employs a priority-based integer encoding strategy for solution representation. Each chromosome consists of n genes corresponding to n farmlands, where the gene value g(i) represents the operational priority of farmland i. Lower values of g(i) indicate higher priority for task execution. The initial population is generated through a randomized priority assignment process. During decoding, farmlands are sorted according to their priority values, and agricultural machinery is then assigned to execute tasks following this prioritized sequence. The assignment considers both spatial proximity between machinery and farmlands, as well as temporal constraints.

As shown in Figure 2, where the number of farmlands is 9, $g\left(i\right)$ denotes the priority of operation point $i$, where the smaller number of $g\left(i\right)$ means the higher priority of the operation point. The order of operation of agricultural machinery obtained after decoding is 7-2-8-1-5-4-9-6-3.

(3)

Fitness. The fitness function evaluates individual quality during evolution. To minimize total scheduling costs, the fitness is defined as the reciprocal of the cost function:

$$f={\displaystyle \frac{1}{1+C_{\mathrm{normalized} }}}$$

(24)

where $C_{\mathrm{normalized} }$ is the cost scaled between the minimum and maximum values within the population using min-max normalization:

$$C_{\mathrm{normalized} }={\displaystyle \frac{C-C_{\min }}{C_{\max }-C_{\min }}}$$

(25)

This formulation ensures that individuals with lower costs receive higher fitness values, with fitness values ranging from 0.5 (worst individual) to 1.0 (best individual).

(4)

Selection. High-quality individuals are selected without duplication to ensure population diversity and improve convergence efficiency and algorithm accuracy for effectively preserving high-quality solutions. In the selection phase, both roulette wheel selection and elitist selection are used to ensure that individuals with higher fitness have a greater probability of survival, while outstanding individuals are preserved and passed on to the next generation [42]. The selection probability for individual i is:

$$P_i={\displaystyle \frac{f_i}{\sum _{i=1}^n f_i}}$$

(26)

where n is the population size, and f_i is the fitness of individual i. This strategy biases selection toward high-fitness solutions while preserving diversity.

(5)

Adaptive crossover. In traditional genetic algorithms, the crossover and mutation probabilities remain fixed throughout the evolutionary process, which often leads to premature convergence or inefficient exploration. Although these parameters are critical factors affecting algorithm performance, determining their optimal values remains challenging. Extensive research has demonstrated that adaptive control of these parameters can significantly improve algorithm efficiency [20,43].

Building upon previous studies on dynamic parameter control strategies [39], this paper proposes an enhanced adaptive crossover mechanism that dynamically adjusts crossover probability based on individual fitness levels and population diversity.

The crossover probability $P_c$ is dynamically adjusted based on individual fitness to balance exploitation and exploration, defined as:

$$P_c=\left\{\begin{array}{c}{P}_{c1}\ast {\displaystyle \frac{1}{\left(P_{c1}-P_{c2}\right)+e^{\frac{f^{\prime }-f_{\mathrm{avg} }}{f_{\max }-f_{\mathrm{avg} }}}}},f^{\prime }\ge f_{\mathrm{avg} }\\ {{ k}_1\ast P}_{c1},{ f}^{\prime }<f_{\mathrm{avg} }\end{array}\right.$$

(27)

where $P_{c1}$ and $P_{c2}$ denote the maximum and minimum crossover probabilities, respectively; $f^{\prime }$ represents the higher fitness value between the two individuals selected for crossover; $f_{\mathrm{avg} }$ and $f_{\max }$ are the average and maximum fitness values in the current population; $k_1\in$[0.5, 1] is a scaling factor.

This adaptive mechanism protects elite genetic material by assigning exponentially decreasing crossover probabilities to high-fitness individuals above the population average, while simultaneously promoting exploration through higher crossover rates for suboptimal solutions scaled by $k_1$. The exponential function ensures smooth probability transitions across different fitness levels, preventing abrupt changes that could destabilize the search process.

(6)

Adaptive mutation. While crossover exploits existing genetic information, mutation serves as a critical diversification operator that introduces novel genetic variations to escape local optima. To complement the adaptive crossover strategy and further enhance the algorithm’s search capability, we propose a self-adaptive mutation mechanism with similar fitness-based adjustment principles. The mutation probability $P_m$ is dynamically adjusted based on individual fitness:

$$P_m=\left\{\begin{array}{c}{P}_{m1}\ast {\displaystyle \frac{1}{\left(P_{m1}-P_{m2}\right)+e^{\frac{f-f_{\mathrm{avg} }}{f_{\max }-f_{\mathrm{avg} }}}}},f\ge f_{\mathrm{avg} }\\ k_2\ast P_{m1}, f<f_{\mathrm{avg} } \end{array}\right.$$

(28)

where $P_{m1}$ and $P_{m2}$ represent the maximum and minimum mutation probabilities, respectively; $f$ is the individual’s fitness value; $k_2\in$ [0.5, 1] regulates mutation intensity. The parameters for adaptive crossover and mutation were determined through a series of preliminary experiments. The values were chosen from a range of candidate values and the selected ones provided a good balance between exploration and exploitation.

This mutation mechanism preserves elite genetic structures through minimal mutation rates for high-fitness individuals, while applying stronger mutation to suboptimal individuals to escape local optima. As the population converges, mutation intensity automatically adjusts—decreasing for elite individuals while maintaining sufficient diversity. Combined with adaptive crossover, this dual-adaptive strategy forms a self-regulating system that balances convergence pressure and population diversity throughout the evolutionary process.

(7)

ACO-Enhanced Optimization Mechanism. The hybrid algorithm integrates an ant colony optimization (ACO) mechanism, a probabilistic algorithm designed for optimal path finding [44], to refine task-machine assignments by leveraging pheromone-guided local search and heuristic-driven exploration. The pheromone matrix ${\tau }_{(i,j,m)}$, initialized as ${\tau }_{(i,j,m)}\left(0\right)=1.0$, quantifies the desirability of assigning machine m to task j at farmland i. The heuristic factor ${\eta }_{(i,j,m)}$ balances temporal feasibility and cost efficiency, defined as:

$${\eta }_{(i,j,m)}=\left[1-{\displaystyle \frac{max\left(0,t_{jm}-l_i\right)}{l_i}}\right]\times {\displaystyle \frac{1}{1+C_{\mathrm{estimated} }}}$$

(29)

where $t_{jm}$ is the execution time of machine m for task j, $l_i$ denotes the latest service time of farmland i, and $C_{\mathrm{estimated} }$ aggregates transfer, operational, and labor costs. The selection probability $P_{(i,j,m)}^k$ for ant k to choose machine m follows a pheromone-heuristic balance rule:

$$P_{(i,j,m)}^k={\displaystyle \frac{{\left[{\tau }_{(i,j,m)}\right]}^{\alpha }{\left[{\eta }_{(i,j,m)}\right]}^{\beta }}{\sum _{m^{\prime }\in M_{ij}} {\left[{\tau }_{\left(i,j,m^{\prime }\right)}\right]}^{\alpha }{\left[{\eta }_{\left(i,j,m^{\prime }\right)}\right]}^{\beta }}}$$

(30)

The parameters $\alpha =1$ and $\beta =2$ balance pheromone influence and heuristic information based on preliminary experiments.

Pheromone trails are updated using an evaporation-reinforcement strategy:

$${\tau }_{(i,j,m)}(t+1)=(1-\rho ){\tau }_{(i,j,m)}\left(t\right)+{\mathsf{\Delta }\tau }_{(i,j,m)}$$

(31)

where the evaporation rate $\rho \in \left(\mathrm{0,1}\right)$ ensures adequate pheromone persistence while preventing excessive accumulation, and the pheromone increment ${\mathsf{\Delta }\tau }_{(i,j,m)}$ is calculated as:

$${\mathsf{\Delta }\tau }_{(i,j,m)}={\displaystyle \frac{\mathrm{Q}}{1+C_{\mathrm{total}}}}$$

(32)

where Q provides sufficient reinforcement strength and $C_{\mathrm{total} }$ represents the total cost of the current solution. Pheromone updates occur only when improved solutions are discovered, ensuring that the pheromone matrix learns exclusively from high-quality assignments.

To prevent premature convergence and maintain exploration diversity, pheromone values are bounded within the range [${\tau }_{min}$, ${\tau }_{max}$] = [0.5, 5.0] after each update:

$${\tau }_{\left(i,j,m\right)}(t+1)=max(0.5,\mathrm{m}\mathrm{i}\mathrm{n}(5.0,{\tau }_{\left(i,j,m\right)}(t+1))$$

(33)

This bounding mechanism maintains exploration diversity by ensuring minimum selection probability (${\tau }_{min}=0.5$) for all feasible assignments while preventing excessive pheromone accumulation (${\tau }_{max}=5.0$) that causes premature convergence on suboptimal solutions.

Post-genetic operations, the ACO-driven local search selectively refines high-fitness solutions by stochastically reassigning tasks to machines with higher pheromone-heuristic scores, prioritizing allocations that minimize time window violations and resource conflicts. The local search mechanism randomly selects tasks from current solutions, identifies compatible machines, calculates selection probabilities using the pheromone-heuristic balance rule, and updates assignments only when solution quality improves.

The optimized solutions replace their original counterparts in the genetic algorithm population, facilitating a synergistic exchange between global search (GA) and local refinement (ACO). This integration ensures computational stability while addressing the spatiotemporal and economic constraints inherent in agricultural machinery scheduling.

(8)

Conflict Detection and Resolution. To ensure operational feasibility and prevent scheduling conflicts during solution construction, the algorithm implements a farmland occupation tracking mechanism. This mechanism maintains a dynamic registry of scheduled operations for each farmland and validates each new task assignment to prevent temporal overlaps.

During the chromosome decoding process (Step 2) and local search phase (Step 7), the algorithm maintains an occupation registry for each farmland. This registry stores the time intervals of all scheduled operations, recording when each operation starts and finishes. When assigning a new task, the algorithm performs the following conflict detection procedure:

• Calculate completion time: Determine when the proposed operation would finish based on its start time and duration.
• Check for conflicts: Compare the proposed operation time interval with all existing operations at the same farmland. Two operations overlap if one starts before the other finishes.
• Resolve conflicts: If overlap is detected, adjust the start time to begin after all conflicting operations complete.
• Validate time window: Verify that the adjusted assignment satisfies the farmland time window constraint. If the completion time exceeds the deadline, reject the assignment.
• Update registry: Upon successful validation, add the new operation interval to the registry for future conflict checks.

The detection mechanism addresses three types of scheduling conflicts commonly encountered in agricultural machinery coordination: spatial conflicts from limited field capacity, resource conflicts over shared infrastructure access and agronomic conflicts requiring safety intervals between operations.

This runtime detection system complements Constraint (11)’s task sequencing by preventing temporal overlaps when different machines operate at the same location. With minimal computational overhead, it ensures all generated schedules are operationally feasible and directly deployable without manual intervention.

(9)

Preservation of Elite Solutions. A crucial aspect of the algorithm involves safeguarding the best-performing individuals to mitigate population degeneration. This is achieved through a strict elitism mechanism where the optimal solution in each generation is preserved unchanged. One copy of this elite individual is exempt from crossover and mutation operations and is automatically carried forward to the next generation, thereby ensuring that the solution quality does not deteriorate throughout the evolutionary process.

(10)

Iterative Procedure. The optimization process is conducted iteratively. The algorithm proceeds cycle-by-cycle through the steps of selection, crossover, mutation, and local refinement. It halts when the stopping criterion is met, which is typically either a sufficiently high solution quality (fitness) has been attained or a computational budget has been exhausted. The final solution is then retrieved by decoding the best individual in the population.

3. Results

3.1. Experimental Setup

To validate the effectiveness of the proposed AGA-ACO algorithm, comprehensive experiments were conducted using real-world agricultural data from the Hangzhou region, which represents typical intensive farming conditions in eastern China. A discrete-event simulation framework in Python 3.9 incorporated actual agronomic requirements and machinery specifications from local cooperatives. The computational experiments were executed on a Windows 10 platform equipped with an Intel^® Core™ i7-8750H processor (6 cores @ 2.20 GHz; Intel Corporation, Santa Clara, CA, USA) and 16 GB DDR4 RAM (Samsung Electronics Co., Ltd., Suwon, South Korea).

3.1.1. Dataset Description

Table 2. Basic Information of Farmland Operations.

Farmland ID	Operation Area (hm2)	Coordinate X (km)	Coordinate Y (km)	Operation Task	Time Window (h)
1	33.2	31	78	2, 3, 4	[3, 20]
2	28.7	11	79	2, 3	[6, 18]
3	25.8	79	44	3, 4	[5, 21]
4	29.2	32	65	2, 4	[15, 35]
5	43.3	33	41	2, 3, 4	[13, 35]
6	16.7	52	56	2, 3, 4	[3, 18]
7	31.5	43	44	3, 4	[10, 30]
8	19.2	66	35	2, 3, 4	[8, 30]
9	24.0	27	31	2, 3, 4	[6, 45]
10	28.2	45	62	2, 3, 4	[12, 40]

presents the farmland operation information, primarily covering the farmland operation area, farmland coordinates, operation types, and time windows for farmland access. Specifically, operation task numbers 1 to 5 correspond to rotary tillage, rice transplanting, weeding, pesticide spraying, and transportation operations, respectively.

The experimental dataset uses a coordinate system where distances are measured in kilometers. All farmlands and the machinery depot are positioned within an 80 km × 80 km operational area. All agricultural machines are initially stationed at the central depot located at coordinates (30, 60), representing a typical centralized machinery management system adopted by agricultural cooperatives. This coordinate system allows for realistic distance calculations while maintaining the generalizability of the optimization results.

Table 3. Available Agricultural Machinery Information.

ID	Type	Efficiency (hm2/h)	Speed (km/h)	Relocation Cost (¥/h)	Operational Cost (¥/h)
1	Rotary Tiller	6.1	13	65	105
2	Rotary Tiller	6.5	15	71	112
3	Rice Transplanter	9.3	16	59	119
4	Rice Transplanter	10.2	17	66	125
5	Weeder	6.5	14	70	108
6	Weeder	7.3	16	73	112
7	Sprayer	5.1	12	66	169
8	Sprayer	5.8	13	73	160
9	Transport Vehicle	8.9	18	55	102
10	Transport Vehicle	8.1	19	64	109

presents the available agricultural machinery information, including machinery specifications and cost parameters.

The experimental design follows established practices in agricultural machinery scheduling research, with parameters validated against industry standards and local cooperative operational data to ensure both realism and computational tractability. The dataset structure facilitates reproducibility while maintaining the generalizability of optimization results for similar agricultural regions.

3.1.2. Algorithm Parameter Configuration

To ensure fair comparison and reproducibility, the AGA-ACO algorithm was initialized with carefully calibrated parameters based on preliminary sensitivity analysis. The population size was set to 300 individuals to balance solution diversity with computational efficiency. The maximum number of evolutionary iterations was limited to 100, with convergence criteria defined as no improvement in the best solution for 20 consecutive iterations. For the genetic component, the crossover probability was set to 0.9 and the mutation probability to 0.05, following established best practices in evolutionary computation for combinatorial optimization problems. The ACO component employed 50 ants with a pheromone importance factor (α) of 1.0 and a heuristic information factor (β) of 2.0, parameters empirically determined to balance exploration and exploitation.

For comparative analysis, the conventional GA, baseline ACO, and PSO algorithms were configured with identical population sizes and iteration limits. The PSO algorithm parameters were set as follows: inertia weight w = 0.9, cognitive and social acceleration coefficients c1 = c2 = 2.0. All parameter values were validated through preliminary experiments to ensure unbiased comparison.

3.2. Simulation Results and Analysis

3.2.1. Performance Evaluation

The initial experimental evaluation focused on a 10-farmland scenario comprising 26 distinct operational tasks. Figure 3 presents the comprehensive scheduling results obtained from the best solutions after five independent runs of AGA-ACO, including Gantt charts and convergence curves. The Gantt charts utilize color-coded blocks to represent individual farmland operations, with block dimensions indicating operation duration and machinery allocation. In the Gantt charts, the notation $F(i,j)$ indicates that the machinery unit is executing task j on farmland i. White blocks represent machinery idle time, while gray blocks denote machinery transfer time between farmlands, providing a comprehensive visualization of both productive and non-operational time allocation.

For comparison, Figure 4 displays the scheduling results and convergence plots of the conventional GA, ACO, and PSO algorithms under the same 10-farmland scenario.

The AGA-ACO algorithm demonstrated superior performance across multiple evaluation metrics.

Table 4. Scheduling results of the AGA-ACO algorithm.

Machinery ID	Scheduling Time (h)	Scheduling Cost (¥)	Operation Route 1
3	15.88	2326.64	2→8→10→4
4	16.46	2302.05	1→9→7→5
5	18.91	2068.89	3→2→7
6	33.30	3948.08	1→6→9→8→10→5
7	35.23	5138.18	1→9→7→10
8	32.72	5674.00	6→3→8→5→4
Total	152.49	21,458.83

¹ Operation route indicates the sequence of farmland plots visited by each machinery.

presents the detailed scheduling solution, revealing an optimized machinery routing strategy that minimizes both scheduling cost and idle time.

3.2.2. Scalability Analysis

To evaluate algorithm scalability and robustness, experiments were conducted on increasingly complex scenarios with 15, 20, and 30 farmland plots.

Table 5. Performance Comparison Across Different Problem Scales.

Algorithm	Farmland Plots	Tasks	Time (h)	Cost (¥)	Iterations
AGA-ACO	15	38	216.24	35,766.41 ± 1326	47 ± 1.86
	20	50	284.47	52,214.63 ± 1845	53 ± 2.35
	30	68	412.18	68,694.24 ± 2146	61 ± 2.88
GA	15	38	235.41	40,125.67 ± 2318	66 ± 2.56
	20	50	288.49	56,654.32 ± 2672	70 ± 3.56
	30	68	432.25	73,024.63 ± 2634	65 ± 3.77
ACO	15	38	223.59	38,764.37 ± 1885	67 ± 2.47
	20	50	296.47	55,234.73 ± 2418	66 ± 3.12
	30	68	421.24	74,325.61 ± 2872	68 ± 3.21
PSO	15	38	226.36	39,524.26 ± 1617	41 ± 3.18
	20	50	297.34	57,256.32 ± 2623	52 ± 3.58
	30	68	433.56	73,265.36 ± 3012	64 ± 3.49

summarizes the comparative performance across different problem scales, with each result averaged over five independent runs to ensure statistical validity.

The results demonstrate consistent performance advantages of AGA-ACO across all problem scales. Cost reductions range from 5.92% to 10.87% compared to GA, 5.47% to 7.75% compared to ACO, and 6.23% to 9.51% compared to PSO, where the adaptive genetic operators and pheromone-guided local search synergistically refine solutions. Convergence efficiency remains remarkably stable, with AGA-ACO achieving competitive iteration counts while maintaining superior solution quality. The relatively small standard deviations in both cost and iterations further validate the robustness and reliability of the proposed approach across multiple independent runs.

3.2.3. Convergence Behavior Analysis

Figure 5 presents detailed convergence trajectories for different problem scales, revealing distinct algorithmic behaviors and validating the effectiveness of the hybrid approach.

The convergence analysis reveals two key insights:

(1)

Superior Optimization Trajectory: AGA-ACO demonstrates consistently better performance throughout the entire optimization process. It generates superior initial solutions with costs 15–20% lower than standard algorithms due to adaptive initialization, then maintains effective exploration-exploitation balance through adaptive operators and pheromone-guided diversification. This dual mechanism prevents the premature convergence observed in standalone GA and ACO, where both algorithms progressively lose optimization momentum and stagnate at suboptimal solutions, particularly evident in Figure 5c,d.

(2)

Convergence Efficiency: The hybrid approach achieves optimal solutions with significantly fewer iterations across all problem scales. Most notably in the 30-farmland scenario (Figure 5d), AGA-ACO achieves convergence significantly faster than GA, ACO, and PSO while maintaining superior solution quality. This efficiency gain stems from the synergistic two-phase optimization strategy—genetic algorithm for rapid global exploration followed by ant colony optimization for targeted local refinement—enabling the algorithm to identify and exploit high-quality solution regions more effectively than either component algorithm alone.

Notably, AGA-ACO’s performance advantage becomes more pronounced as problem complexity increases, demonstrating its superior scalability in handling larger agricultural scheduling instances. While PSO exhibits faster convergence in smaller problems (15 farmlands), AGA-ACO maintains a better balance between convergence speed and solution quality across all scales, particularly excelling in the more challenging 20 and 30 farmland scenarios where operational constraints and resource coordination become increasingly complex.

3.2.4. Algorithm-Component Comparison

To quantify the marginal contribution of each algorithmic component, four configurations were benchmarked under identical parameter settings:

GA-Base: fixed crossover and mutation probabilities ($P_c$ = 0.9, $P_m$ = 0.05);

GA-ACO: base GA + ACO local search (no adaptation);

AGA: adaptive Pc and Pm only;

AGA-ACO: full proposed method (adaptive + local search);

Experiments were conducted on 15, 20, and 30 farmland scenarios, each averaged over five independent runs.

Table 6. Progressive component impact across problem scales.

Algorithm	15 Farmlands			20 Farmlands			30 Farmlands
	Cost	Iter	Gap (%) 1	Cost	Iter	Gap (%)	Cost	Iter	Gap (%)
GA-Base	40,126 ± 2318	66	+12.2	56,654 ± 2672	70	+8.5	73,025 ± 2634	65	+6.3
AGA	36,890 ± 1520	52	+3.1	53,850 ± 1923	59	+3.1	71,187 ± 2566	58	+3.6
GA-ACO	37,315 ± 1950	58	+4.3	54,180 ± 2341	62	+3.8	70,220 ± 2879	67	+2.2
AGA-ACO	35,766 ± 1326	47	0.0	52,215 ± 1845	53	0.0	68,694 ± 2146	61	0.0

¹ Gap denotes the mean cost increase relative to AGA-ACO; positive values indicate additional expense.

summarizes the average total cost, the number of iterations required to reach convergence, and the relative performance gap compared to the full AGA-ACO configuration.

Three trends are visible:

• The AGA configuration removes roughly three-quarters of the excess cost observed in GA-Base and shortens convergence time by 10–15%. Dynamically self-tuned crossover and mutation probabilities help maintain population diversity and protect high-fitness routes from premature loss. This continuous adaptation supplies a steady stream of promising task–machine combinations, yielding near-optimal solutions without additional computational overhead from local search.
• Introducing ACO refinement into a fixed-parameter GA (GA-ACO) reduces the total cost by another 2–3%, demonstrating the effectiveness of pheromone-guided task reassignment in enhancing local solution quality. However, because crossover and mutation remain static, the algorithm still risks premature convergence once the population becomes overly homogeneous—limiting the ACO’s ability to further explore new high-quality routes. As a result, GA-ACO’s performance remains slightly inferior to that of the adaptive GA.
• The full hybrid model achieves the best balance between exploration and exploitation. Its marginal improvement over AGA decreases from 3.1% to 3.6% and finally to 2.2% as the number of farmlands increases, indicating that with larger problem sizes, opportunities for route sharing diminish and constraint saturation increases. Nonetheless, the ACO refinement continues to provide meaningful improvements in both convergence smoothness and final solution quality.

In summary, Table 6 quantitatively demonstrates that the adaptive mechanism primarily stabilizes the convergence dynamics of the genetic process, while the ACO-based local search contributes precision refinements during the final optimization stage. Their synergistic integration ensures the most effective balance between computational effort and scheduling cost across all tested problem scales.

3.3. System Development

To validate the practical applicability of the proposed AGA-ACO algorithm, a comprehensive agricultural machinery scheduling management system was developed. The system architecture integrates multiple advanced technologies, including intelligent sensors, the Internet of Things (IoT), positioning systems, remote sensing, and Geographic Information Systems (GIS), providing precise scheduling and management support for agricultural operations. The system follows a hierarchical design pattern based on the “Perception-Transmission-Service-Application” closed-loop framework, enabling data-driven decision-making and intelligent management integration.

The system employs the MQTT protocol for its communication infrastructure, enabling real-time sensor data acquisition from agricultural machinery and bidirectional command transmission. This lightweight messaging protocol ensures efficient data exchange even in areas with limited network connectivity, which is common in rural agricultural environments.

The system’s core functionality encompasses agricultural machinery management, scheduling optimization through the AGA-ACO algorithm, real-time operation monitoring, historical data analysis, and an integrated agricultural expert system for operational guidance. Figure 6 illustrates the agricultural machinery management interface, which displays the current list of available machinery along with their operational status. The interface provides comprehensive access to machinery attributes, current operational states, and historical operation trajectories. Users can query specific machinery information through intuitive search and filter functions, facilitating efficient fleet management.

The real-time scheduling visualization, presented in Figure 7, demonstrates the practical implementation of the AGA-ACO algorithm. The interface displays optimized routing paths for each machinery unit departing from the central depot to assigned farmland plots based on task requirements. The visualization is implemented using the AMap (Gaode) API. Some non-English labels appearing on the base map correspond to real geographic names automatically generated by the mapping service; these have no scientific implication and do not affect the interpretation of the results. The visualization employs an interactive map overlay where machinery icons represent current positions, and polylines indicate planned routes. When users hover over machinery icons, the system displays real-time operational information, including current task progress, estimated completion time, and operational parameters.

4. Discussion

The proposed AGA-ACO algorithm demonstrates a significant advancement in solving the multi-task agricultural machinery scheduling problem, yielding notable improvements in cost efficiency, convergence speed, and scalability across various operational scales. However, the study’s findings should be interpreted within the context of its model assumptions, which present certain limitations and avenues for future research. A primary limitation lies in the model’s reliance on static and predetermined parameters, which may not fully encapsulate the dynamic and uncertain realities of agricultural operations. Factors such as abrupt weather changes, sudden disasters, or real-time fluctuations in soil conditions and task priorities are not accounted for, potentially affecting the model’s applicability in highly volatile environments.

While the experimental validation utilized farmland data from the Hangzhou region, the proposed AGA-ACO framework demonstrates strong potential for broader geographical adaptation. The algorithm’s core components—including the adaptive parameter control mechanism and conflict resolution system—provide inherent flexibility to accommodate diverse climatic and topographic conditions. Future application to different agricultural regions would primarily require parameter adjustments in machinery efficiency coefficients, time window constraints, and transfer costs to reflect local operational characteristics, rather than fundamental algorithmic modifications.

Future research should prioritize enhancing the model’s resilience and adaptability. Integrating real-time data assimilation from IoT and remote sensing technologies could enable dynamic rescheduling in response to unforeseen events like extreme weather or disasters. Additionally, expanding the optimization objectives beyond economic efficiency to include environmental considerations—such as fuel consumption, carbon emissions, and soil compaction—would align the scheduling system with sustainable agriculture principles. The exploration of deep reinforcement learning could enable the system to learn from operational history and continuously improve its scheduling policies.

In a broader perspective, this research contributes a robust computational framework to the digital transformation of agriculture. By addressing complex spatiotemporal constraints in machinery coordination, the AGA-ACO algorithm lays the foundation for intelligent agricultural management systems. As precision agriculture continues to evolve toward autonomous operations, scheduling frameworks that balance computational efficiency with real-world adaptability will become increasingly critical for achieving resilient and sustainable food production systems.

5. Conclusions

This study addressed the multi-task agricultural machinery scheduling problem by proposing an adaptive genetic algorithm integrated with ant colony optimization (AGA-ACO). The hybrid framework successfully overcomes the limitations of traditional scheduling approaches in handling time window constraints, machinery heterogeneity, and task dependencies in complex agricultural environments.

The AGA-ACO algorithm achieved consistent performance advantages across all problem scales. Compared with the benchmark algorithms, it reduced total scheduling cost by 5.92–10.87% relative to GA, 5.47–7.75% relative to ACO, and 6.23–9.51% relative to PSO, while maintaining competitive convergence speed. The adaptive genetic operators and pheromone-guided local search work synergistically to refine solutions, yielding higher-quality schedules with fewer iterations. The relatively small standard deviations of both cost and convergence iterations further confirm the robustness and reliability of the proposed method across multiple independent runs.

The scheduling model effectively captures agricultural operational complexities by formulating the problem as a VRPTW variant. The developed management system, integrating IoT sensors, MQTT protocols, and GIS visualization, demonstrates the practical feasibility of the proposed approach for real-time agricultural operations. Moreover, the algorithm exhibits strong scalability, with its performance advantage becoming more pronounced as the problem size and scheduling complexity increase, indicating its adaptability to large-scale agricultural applications.

This research contributes a novel metaheuristic integration framework for NP-hard scheduling problems and provides a replicable methodology for optimizing agricultural machinery operations. The AGA-ACO algorithm and its system implementation can be extended to diverse agricultural contexts requiring multi-machine coordination under time and resource constraints. Future work will extend the model to incorporate dynamic environmental factors, enhancing its applicability across diverse agricultural ecosystems. As agriculture evolves toward increased mechanization, intelligent scheduling systems become critical for ensuring operational efficiency and sustainable development.