A Multi-Objective Optimization Method for a Tractor Driveline Based on the Diversity Preservation Strategy of Gradient Crowding

Abstract

This study presents a multi-objective optimization method for a tractor driveline based on the diversity maintenance strategy of gradient crowding. The objective was to address the trade-off between high power and low fuel consumption rates in a tractor driveline by optimizing the distribution of driveline ratios, aiming to enhance overall driving performance and reduce fuel consumption. This method introduces a strategy for evaluating gradient crowding to reduce non-inferior solution sets during selection to ensure the uniform and wide distribution of solutions while maintaining population diversity. The transmission ratio of a tractor is optimized by varying the input of the transmission ratios in each gear, constraining the theoretical tractor driving rate, common transmission ratio, and drive adhesion limit, and introducing the diversity maintenance strategy of gradient crowding. The goal is to reduce the loss rate of driving power and specific fuel consumption as much as possible. The analysis results demonstrate that the GC_NSGA-II algorithm, incorporating the evaluation strategy of gradient crowding, achieves greater diversity and a more uniform distribution in the front end. After verifying the algorithm, the optimized tractor showed a reduction of 41.62 (±S.D. 0.44)% in the theoretical loss rate of driving power and 62.8 (±S.D. 0.56)% in the loss rate of specific fuel consumption, indicating that the tractor’s drive performance significantly improved, accompanied by a substantial reduction in the fuel consumption rate. These findings affirm the feasibility of the proposed optimization method and provide valuable research insights for enhancing the overall performance of tractors.

1. Introduction

Traditional gear ratio allocation methods for tractor transmission systems typically involve equal ratio step allocation or rely on empirical, suboptimal allocation. However, these methods do not align with the required transmission speed ratio for tractors in their working environments and fail to achieve the simultaneous objectives of high driving performance and low fuel consumption [1,2,3]. Therefore, there is an urgent need to enhance the allocation scheme for tractor-driveline transmission ratios to improve the machine’s overall operational performance.

The automotive industry has undertaken extensive research on driveline parameters and the evaluation metrics of driveline optimization to find the proper driveline ratios to match the engine by using simulation programs that calculate vehicle drive performance and fuel efficiency [4,5,6,7]. ADVISOR, an Austrian company, has developed software based on MATLAB and Simulink that enables a high degree of simulation of the vehicle driving environment for the fast and accurate analysis of the vehicle dynamics and emissions [8,9,10]; it also developed the AVL CRUISE software to freely model the vehicle in various configurations with high-precision algorithmic programs to simultaneously ensure the speed and accuracy of the simulation [11,12,13]. Domestic research on the optimization of transmission ratios in automobile drivelines began relatively late and has accomplished a series of significant research findings in this area since the 1980s [14,15,16]. Liu and Guo et al. [17] employed a linear weighted combination method to construct a mathematical model for optimizing the automobile driveline, achieving a reduction of 8.21% in fuel consumption and a 3.73% improvement in in situ starting acceleration time compared to the original vehicle. Peng and Xu et al. [18] focused their research on a pure electric vehicle, wherein they reselected the motor and implemented a three-speed driveline. They conducted simulation analysis using ADCISOR software and observed that integrating three-speed transmission resulted in a 9.2% increase in the electric vehicle range and enhanced the power and overall efficiency of the vehicle. Zheng and Chen et al. [19,20,21] employed a fast multi-objective ranking genetic algorithm for a nondominated system to perform multi-objective optimization of the power system speed ratio, considering the power and transmission speed ratio of a pure electric vehicle as the constraints. The optimization process resulted in a 3.91% enhancement in fuel efficiency and a 17.55% increase in maximum speed while achieving the desired target performance compared to the original configuration. Nevertheless, the transmission speed ratios derived from existing studies on automobiles are not entirely applicable to tractors [22]. Moreover, there is a scarcity of in-depth research on tractor transmission systems.

Multi-objective evolutionary algorithms have become one of the main methods to solve multi-objective optimization problems in tractor transmission systems. Commonly used multi-objective evolutionary algorithms include the strength Pareto evolutionary algorithm (SPEA) [23] and the nondominated sorting genetic algorithm (NSGA) [24]. The NSGA-II algorithm proposed by Deb et al. is a good example of a multi-objective evolutionary algorithm with its Pareto-dominated selection model and density estimation of the selection winner mechanism [25]. However, the algorithm’s mechanism of calculating the aggregation distance to maintain the population distributivity has certain shortcomings, which may lead to the algorithm falling into local optimality [26]. The diversity maintenance strategy based on gradient crowding represents an optimization approach wherein a suboptimal individual from a population is chosen for crowding, and then it is re-evaluated and subjected to further crowding, thereby creating a cyclical process that leads to the identification of a population that optimally satisfies the fitness condition [27,28]. This method demonstrates excellent performance in evaluating the global crowding of individuals, preserving population diversity, and obtaining compromise solutions for multi-objective optimization [29,30].

Therefore, this study is based on the premise of effectively handling the dynamics and fuel economy requirements of the whole machine, with plow operating conditions as the working environment, while retaining 10% to 20% reserve tractive effort to cope with different working conditions and resistance variations caused by farm implements. A diversity maintenance strategy based on gradient crowding [31] is proposed to optimize the driveline ratio allocation by improving the traditional multi-objective optimization algorithm to effectively reduce the set of non-inferior solutions and ensure a uniform and wide distribution of solutions, thereby maintaining the diversity of the group. By rationalizing the distribution of tractor ratios, the overall power of the machine can be effectively increased while minimizing fuel consumption, improving the economy of the machine, and ultimately improving the overall working performance of the tractor.

2. Optimization Model of Matching a Tractor Transmission System

2.1. Optimization of the Objective Function

The evaluation of the dynamics and fuel economy of the entire tractor relies on the loss rate of driving power and specific fuel consumption, with the objective function of optimizing the drive train being established [32]. The loss rate of driving power signifies the deviation between the characteristics of the tractor’s driveline and the ideal driveline. A lower loss rate for driving power indicates superior alignment regarding the power performance between the engine and driveline. For j-step, the driving power of the tractor can be defined as

$$F_j=\frac{T_e{i}_j{\eta }_c}{r}$$

(1)

where $F_j$ is the driving force of the tractor in j-step ($\mathrm{kN}$), $T_e$ is the engine torque ($\mathrm{N}\cdot \mathrm{m}$), $i_j$ is the transmission ratio in j-step, ${\eta }_c$ is the total transmission efficiency (%), and r is the driving wheel power radius ($\mathrm{m}$). The theoretical speed of the tractor in each gear is

$$v_j=\frac{0.377n_{e}r}{i_j}$$

(2)

where $n_e$ is the engine speed ($\mathrm{r}\cdot {\mathrm{m}}^{-1}$). The loss rate of driving power [33,34] can be defined as

$$\left\{\begin{array}{c}{\eta }_{Fj}=\frac{S_j-A_j}{S_j}\times 100\%\\ S_j={\displaystyle {\int }_{v_j}^{v_{j+1}}{F}_q\left(v\right)dv=0.377{\eta }_c{T}_p{n}_p\ln \left(\frac{v_{j+1}}{v_j}\right)}\\ A_j={\displaystyle {\int }_{v_j}^{v_{j+1}}{F}_{qj}\left(v\right)dv}\end{array}\right.$$

(3)

where ${\eta }_{Fj}$ is the loss rate of driving power (%), $S_j$ and $A_j$ represent the area enclosed by the ideal drive characteristic curve and the j-step drive characteristic curve with the horizontal co-ordinate (${\mathrm{m}}^2$), $T_p$ is the output torque at maximum power ($\mathrm{N}\cdot \mathrm{m}$), $n_p$ is the engine speed at maximum power ($\mathrm{r}\cdot {\mathrm{m}}^{-1}$), and $F_q$ is the driving force ($\mathrm{N}$) at the ideal characteristic.

Due to the complex working environment and demanding operating conditions faced by tractors, it is essential to evaluate their fuel economy by considering multiple operating conditions concurrently to ensure the reliability of the conclusions [35,36]. Consequently, this study integrated the definitions of the fuel consumption rate of tractor traction and the power output shaft to establish the loss rate of specific fuel consumption as the evaluation index for tractor fuel economy. A lower loss rate of specific fuel consumption indicates a more efficient economic matching between tractor powertrains. It is defined as

$${\xi }_{gj}=\frac{A_{gj}-S_{gj}}{S_{gj}}=\frac{{\displaystyle {\int }_{v_j}^{v_{j+1}}{g}_{Tj}\left(v\right)}dv-{\displaystyle {\int }_{v_j}^{v_{j+1}}{g}_{TL}\left(v\right)}dv}{{\displaystyle {\int }_{v_j}^{v_{j+1}}{g}_{TL}\left(v\right)}dv}\times 100\%$$

(4)

where ${\xi }_{gj}$ is the loss rate of the specific fuel consumption of the tractor at j-step, $g_{TL}$ is the specific fuel consumption of the tractor with ideal drive characteristics, $S_{gj}$ and $A_{gj}$ represent the area enclosed by the specific fuel consumption curve of the tractor under ideal drive characteristics and j-step, respectively, and $g_{Tj}$ is the specific fuel consumption of the tractor at j-step. Therefore, the objective function of optimizing the drive train is

$$\left\{\begin{array}{c}min f_1\left(X\right)={\displaystyle \sum _j{\eta }_{Fj}\left(x_j\right)}\\ min f_2\left(X\right)={\displaystyle \sum _j{\xi }_{gj}\left(x_j\right)}\end{array}\right.$$

(5)

2.2. Constraint of Transmission Ratio Optimization

Tractor transmission ratio distribution needs to meet the four constraints of theoretical speed: the attachment limit of the driving force, maximum tractive force, and common transmission ratio. If the design speed of the first gear is $V_a$ at the rated engine speed $n_{re}$, the design starting speed is $V_d$ at the lowest stable speed, and the maximum speed is $V_c$. According to the demand of the operating speed, the speed $v_b$ of the basic operation gear should be greater than 5 $\mathrm{km}\cdot {\mathrm{h}}^{-1}$. Then, the theoretical speed of the tractor should meet the following requirements:

$$\left\{\begin{array}{l}{v}_I=\frac{0.377n_{re}r}{i_I}\ge V_a\hfill \\ v_I^{min}=\frac{0.377n_{min}r}{i_I}\le V_d\hfill \\ v_b=\frac{0.377n_{re}r}{i_b}\ge 5\hfill \\ v^{max}=\frac{0.377n_{re}r}{i_n}\ge V_c\hfill \end{array}\right.$$

(6)

In this context, let $v_I^{min}$ represent the theoretical starting speed of the first gear ($\mathrm{km}\cdot {\mathrm{h}}^{-1}$),$v_I$ denote the theoretical speed at the rated speed of the first gear ($\mathrm{km}\cdot {\mathrm{h}}^{-1}$), $v^{max}$ represent the maximum theoretical speed ($\mathrm{km}\cdot {\mathrm{h}}^{-1}$), while $i_b$ and $i_n$ correspond to the top gear and the maximum total transmission ratio of the operating gear, respectively. Considering the adhesion conditions of the ground, the maximum driving force should be determined as follows:

$$F_q^{max}=\frac{T_e{i}_b{\eta }_c}{r}\le \phi Q_q$$

(7)

where $Q_q$ is the vertical load ($\mathrm{N}$) on the drive wheel, and $\phi$ is the adhesion coefficient (approximately 0.6–0.8).

The maximum traction force of a tractor is calculated based on the first gear of the primary operation during plowing working conditions. Additionally, a reserve traction force of 10–20% is maintained to accommodate diverse working conditions and variations in resistance due to farm implements [35]. That is

$$F_q^{max}=\frac{T_e{i}_b{\eta }_c}{r}\ge 1.2zbhk+f{G}_{a}g$$

(8)

where z is the number of plowshares, b is the width of a single plowshare, h is the depth of plowshare, k is the specific soil resistance, f is the rolling resistance coefficient of working ground, Ga is the tractor mass, and g is the gravitational acceleration. When operating under highway transportation conditions and reaching the maximum speed, the driving force of the tractor must satisfy the following requirements [33,34]:

$$F_q=\frac{T_{re}{i}_n{\eta }_c}{r}\ge f_g{G}_{a}g$$

(9)

where $T_{re}$ is the rated torque of the tractor ($\mathrm{N}\cdot \mathrm{m}$), and $f_g$ is the rolling resistance coefficient of the highway. The common ratio of adjacent transmission ratios must satisfy the following requirements: the common ratio of the main transmission ratio for power shift should be approximately 1.20–1.29, while the common ratio of adjacent gears in the shift section should be 1.10–2.39 [36,37].

3. Multi-Objective Optimization Algorithm Based on Gradient Crowding

The core ideas of the NSGA-II [23] and SPEA-II [24] algorithms are shown in Figure 1a,b. In order to address their shortcomings in the spatial search capability of nondominated solutions and in maintaining the homogeneity of the population, a multi-objective optimization method based on a diversity maintenance strategy with gradient crowding is proposed in this study. Diversity is achieved by using randomly generated populations to ensure a more uniform distribution across the population during the initialization phase. During selection, the gradient crowding strategy is employed as part of the crowding operation operator to determine which individuals should be retained or discarded so that the solution is more evenly dispersed in the evolution process.

3.1. Gradient Crowding

The distance between individuals plays a significant role in determining the degree of crowding among individuals. For a vector set A representing a population, if $a_i\in A$, $i=1,2,\dots ,\left|A\right|$, $a_i^{(k)}$ is the kth individual away from $a_i$, and $a_i^{(k)}\in A$, $a_i^{(k)}\ne a_i$, the distance between $a_i$ and the kth individual is $d_i^{(k)}$, then

$$d_i^{(k)}=\left|a_i^{(k)}-a_i\right|,k=1,2,\dots ,\left|A\right|-1$$

(10)

For the vector set A, representing the population, the kth individual away from $a_i$ is $a_i^{(k)}$, which also corresponds to some $a_j$ in the original set A. Therefore, the individual $a_i^{(k)}$, i.e., the crowding of the individual $a_j$ to $a_i$ is defined as

$$c_{ij}=c_i^{(k)}=g\left(k\right)d_i^{(k)},k=1,2,\dots ,\left|A\right|-1$$

(11)

where $c_{ij}$ denotes the value of the influence of any other individual $a_i$ on an individual $a_j$ in set A, $a_j$ has a distance $d_i^{(k)}$ from $a_i$ and is the kth of all other individuals ordered by distance from $a_i$. $g\left(k\right)$ is the gradient function. The crowding evaluation for $a_i$ at the full range is then

$$C_i={\displaystyle \sum _{k=1}^{N-1}g\left(k\right)d_i^{(k)}}$$

(12)

For all N individuals in the population, $C_i$ is the cumulation of the influence of all individuals except $a_i$ on it. According to the perspective of ranking the distance from other individuals to it, that is, the accumulation of the distance from other N − 1 individuals in the set to $a_i$ and the corresponding gradient function value of k.

Generally, the influence of all individuals on the subject should be correlated with the distance between themselves [37]. For an individual $a_i$, the influence value of the individual $a_j$ closer to it must be greater than that of the individual $a_k$ farther away. That is

$$\left\{\begin{array}{l}{d}_{ij}<d_{ik}\hfill \\ E\left(c_{ij}\right)>E\left(c_{ik}\right)\hfill \end{array}\right.$$

(13)

where $E\left(c_{ij}\right)$ is the influence value of the individual $a_j$ on the individual $a_i$, $E\left(c_{ik}\right)$ is the influence value of $a_k$ on the $a_i$. Similarly,

$$\left\{\begin{array}{l}{d}_{ij}<d_{ik}\hfill \\ E\left(c_{ij}\right)>E\left({\displaystyle {\sum }_{l=k}^N{c}_{il}}\right)\hfill \end{array}\right.$$

(14)

Hence, it can be concluded that the following results exist: $E\left(c_i^{(k)}\right)>E\left(c_i^{(k+1)}\right)$ and $g\left(k\right)>g\left(k+1\right)$, indicating that the gradient function, is a monotone decreasing function; that is, $g\left(k\right)=\frac{1}{k!}$ can simultaneously satisfy the above two constraints, and its value is positive. Assuming that the calculation accuracy of $C_i$ is $\epsilon$, it is bound to be that K makes $\epsilon \ge g\left(K\right)d_{max}$. That is, when the calculation precision of $C_i$ is $\epsilon$, the number of algorithm iterations is reduced from the original N − 1 to K, and the original algorithm is optimized to find the first K shortest distances rather than fully sorting the set of distances between all individuals and target individuals.

3.2. Segmented Elite Strategy

In multi-objective optimization problems, elite retention strategies commonly employ external sets to preserve non-inferior individuals that emerge during evolution [29]. Given that the non-inferior individuals in a contemporary population may not necessarily maintain their non-inferiority in subsequent generations, it becomes necessary to employ a strategy that ensures the final solution obtained remains non-inferior to other solutions generated by the algorithm. In this study, a segmented elite strategy was designed to implement genetic manipulation during evolution.

During the evolutionary process, elites emerge in the population assessed based on certain evaluations. In order to maintain the presence of elites, a comprehensive segmented elite strategy is formulated, in which the elite evaluation function is judged to conclude whether they are elites or to assign elite ranks. Segmentation is conducted based on the evaluation outcomes, and the number of individuals in each segment and the total number are constrained to prevent the excessive expansion of the individual sets within each segment and the overall set [38,39]. The constraint on the number of individuals is implemented through a set reduction strategy, commonly employing a diversity preservation strategy. In order to decrease the number of individuals within a segment, it is important to recognize that individuals in other segments also contribute to the diversity within the segment. If only the set of individuals in that segment is employed as the input for the reduction step, the global significance of diversity will be lost [40]. Hence, the entire set of individuals is considered during the evaluation, while only the set of individuals in the specified segment is used to exclude individuals during selection.

In general, elites can be ranked by some fitness assignment or simply by a non-inferior solution as the only judgment. The steps of the segmented elite strategy are shown in

Table 1. Steps of segmented elite strategy algorithm (Algorithm: Segmented elite strategy).

Input: P, L Output: P	// Segment P by elite rating
1: function Divide (P) 2:    Divide P into Pi by Elite Level. 3:    returen Pi 4: end function 5: for i = 1, 2, …, N do 6:    Pi = Diversity-Prune(Pi, Li, P) 7: end for 8: P = Diversity-Prune(P, L) 9: end Procedure	// Reduce P by the number of limits per segment // function to exclude individuals when special treatment, only for the set of individuals of the specified segment // Reduce P by the total number of restrictions

. The population P is segmented based on the elite evaluation. The ith elite segment is set as $P_i$, and the number of its individuals is limited to $L_i$, and the total number is limited to $L$.

3.3. Multi-Objective Optimization Algorithm for Gradient Crowding and Evaluation Metrics

The new algorithm is abbreviated as GC_NSGA-II and GC_SPEA-II, respectively, based on the classical NSGA-II algorithm and SPEA-II, which uniformly use the various initial strategy, segmented elite strategy, and binary genetic operators and replace the congestion evaluation function as the gradient crowding algorithm. The multi-objective evolutionary algorithm flowchart is shown in Figure 2.

Different to the general single-objective optimization problem, the multi-objective optimization problem has two objectives to be achieved: the degree of convergence from the solution point to the Pareto optimal solution set and a diversity of solutions [41]. This study employs convergence and diversity as indices to assess the effectiveness of the proposed improved algorithm and evaluate the advantages and disadvantages of convergence and diversity in multi-objective problems. Convergence and diversity are defined as follows: $\gamma$ is defined as a metric to quantify the solution point’s convergence level towards a known Pareto optimal set. Initially, $N_{\gamma }$ equidistant points are selected from the true Pareto optimal frontier of the target space. The distance to the nearest point among these $N_{\gamma }$ points is calculated for each solution point obtained through the algorithm. $\gamma$ represents the average distance among these distances (Figure 3).

A second performance criterion, $△$, was defined to evaluate the solution’s diversity and quantifies the distribution of solution points obtained by the algorithm. In other words, criterion $△$ serves as a measure of diversity. The algorithm generated n solution points, as depicted in Figure 4. The distances between the adjacent points were calculated and denoted as $d_1,d_2,\dots ,d_{n-1}$. Subsequently, the average value $\overline{d}$ of these n − 1 distances was computed [42].

$$\overline{d}={\displaystyle \sum _{i=1}^{n-1}\frac{d_i}{n-1}}$$

(15)

A large value for $\overline{d}$ suggests a certain degree of even distribution regarding the solution points on the surface. However, it does not exclude the possibility of extremely large distances between some pairs of points, which can inflate the average. Therefore, variable $\delta$ was introduced to represent the variance of the n − 1 distances.

$$\delta ={\displaystyle \sum _{i=1}^{n-1}\frac{d_i-\overline{d}}{n-1}}$$

(16)

A higher value for $\overline{d}$ and a lower value for $\delta$ indicate a more uniform distribution of solution points and better diversity regarding the solutions. Thus, the dispersion degree, $△$, of the solution set is defined as

$$△=\frac{\delta }{\overline{d}}$$

(17)

The diversity of the solutions set is quantified as the dispersion degree, denoted as $△$, where a smaller value indicates higher diversity.

4. Results and Discussion

4.1. Verification of Algorithm Optimization

The same parameters are employed for the algorithms NSGA-II, SPEA-II, GC_NSGA-II, and GC_SPEA-II. The parameter settings can be found in

Table 2. Parameter settings.

Algorithm	NSGA-I	GC_NSGA-II	SPEA-II	GC_SPEA-II
Population size	200	200	200	200
Evolutionary algebra	200	200	200	200
Crossover probability	1	1	1	1
Mutation probability	0.05	0.05	0.05	0.05
Number of uniformly selected points at the front end of convergence	500	500	500	500
The pre-generation ratio of diversity random		4		4
Segmented strategy		0.5, 0.4, 0.1		0.5, 0.4, 0.1

. The solution set plots produced by each of the four algorithms after the multi-objective evolution process of the five standard test functions, ZDT1, ZDT2, ZDT3, ZDT4, and ZDT6, are shown in Figure 5.

The ZDT1 test function, with a convex Pareto optimal front, tests how well the algorithm performs regarding the underlying convex front. The ZDT2 test function, with a nonconvex Pareto optimal front, tests the algorithm’s performance regarding the underlying nonconvex front. The ZDT3 test function tests the performance of the algorithm for several noncontiguous convex surfaces; the Pareto optimal front is composed of several noncontiguous convex surfaces. The ZDT4 test function, with 2¹⁹ local Pareto optimal sets, can test the algorithm’s ability to handle multi-peak function capability. Since the objective space is nonuniformly distributed, the ZDT6 test function has a global Pareto along the optimal front end, and the solutions are non-uniformly distributed; the solutions become sparser the closer to the Pareto optimal front end, meaning the closer the solution is to the Pareto-optimal front, and the denser the solution is, the further away it is. The distribution of all individuals produced during the evolutionary process is a record of the evolutionary capacity, and the graphs given by the distribution of individuals during the experiments show that the traditional NSGA-II and SPEA-II algorithms, improved by using gradient congestion, clearly produce individuals relatively efficiently during the evolutionary process, without wasting too many individuals in spaces far from the optimal front. The high density of individuals near the optimal front end can cause a good convergence trend. The convergence of individuals is efficient relative to the global evolutionary picture of the population. Furthermore, as can be seen in Figure 4, all four algorithms are able to adhere to the non-inferior front end. However, GC_NSGA-II and GC_SPEA-II, which use the Gradient function, have better diversity and are more evenly distributed across the front end. GC_NSGA-II has the best diversity performance, relatively speaking.

In order to evaluate these algorithms more objectively and systematically, a 50-evolution process was performed on each of the five standard test functions ZDT1, ZDT2, ZDT3, ZDT4, and ZDT6. $\gamma$, $\overline{d}$, $\delta$, and $△$ were calculated for each resulting solution set (

Table 3. Statistics for the convergence and diversity of the four algorithms.

Standard Test Functions	Algorithm	$\mathit{\gamma }$	$\overline{\mathit{d}}$	$\mathit{\delta }$	$△$
ZDT1	NSGA-II	0.022912	0.007967	0.007611	0.955302
	GC_NSGA-II	0.023032	0.008259	0.006376	0.771947
	SPEA-II	0.024342	0.007643	0.012948	1.694197
	GC_SPEA-II	0.019805	0.008127	0.008148	1.002563
ZDT2	NSGA-II	0.012369	0.007644	0.006560	0.858250
	GC_NSGA-II	0.011480	0.007720	0.005724	0.741448
	SPEA-II	0.010460	0.007489	0.016042	2.142011
	GC_SPEA-II	0.005046	0.007541	0.007542	1.000210
ZDT3	NSGA-II	0.002024	0.011720	0.020567	1.754830
	GC_NSGA-II	0.002127	0.011905	0.021290	1.788401
	SPEA-II	0.003303	0.011691	0.028947	2.475952
	GC_SPEA-II	0.002726	0.012225	0.022220	1.817520
ZDT4	NSGA-II	0.000473	0.007389	0.00680	0.920091
	GC_NSGA-II	0.000185	0.007634	0.005503	0.720860
	SPEA-II	0.000238	0.007284	0.010673	1.465305
	GC_SPEA-II	0.000181	0.007400	0.008518	1.151056
ZDT6	NSGA-II	0.000473	0.007389	0.006799	0.920091
	GC_NSGA-II	0.000185	0.007634	0.005503	0.720860
	SPEA-II	0.000238	0.007284	0.010673	1.465305
	GC_SPEA-II	0.000181	0.007400	0.008518	1.151056

). The improved algorithm based on the gradient function introduces a greater overhead in terms of the time spent on the testing process. The data are approximately equal to twice that of the unimproved algorithm.

All the general test results verify that the multi-objective evolutionary algorithm based on the improved gradient congestion provides significant improvement not only regarding the ability to maintain diversity but also in the ability of the algorithm to converge better. This further validates that the ability to maintain diversity leads to an improvement in convergence. According to the systematic test statistics table, it can be seen that GC_NSGA-II has the best diversity, with better convergence than the other three algorithms in most standard tests.

4.2. Optimization Analysis of Matching Transmission Ratio

A tractor model presented in

Table 4. Basic parameters of tractor and engine.

	Indicators
Engine	Model	WP6G180E330
	Calibrated power (kW)	132.5
	$\mathrm{Calibration} \mathrm{speed} (\mathrm{r}\cdot {\min }^{-1}$)	2200
	$\mathrm{Rated} \mathrm{fuel} \mathrm{consumption} (\mathrm{g}\cdot {(\mathrm{kW}\cdot \mathrm{h})}^{-1}$)	<235
Driveline	Central drive ratio	6.125
	Final Drive Ratio	6.40
Speed	Number of gears	32(F) + 32(R)
	$\mathrm{Forward} \mathrm{speed} (\mathrm{km}\cdot {\mathrm{h}}^{-1}$)	2.28–39.8
Machine parameter	Rated tractive force (kN)	57
	Machine quality (kg)	8144
	Service mass (kg)	8319

was employed as an example to assess the rationality and effectiveness of the optimized algorithm. The GC_NSGA-II algorithm, known for its superior diversity performance and convergence, was utilized to optimize the transmission ratio, and the optimization results for the tractor’s transmission ratio compared to the traditional algorithm are presented in

Table 5. Comparison of transmission ratio optimization results based on gradient crowding.

Gear Partition	Gear	Gear Name	Before Optimization	NSGA-II	SPEA-II	GC_NSGA-II
Low gear	1	Forward L-I	298.93	290.539	238.377	287.742
	2	Forward L-II	248.11	241.145	198.647	229.116
	3	Forward L-III	209.25	203.271	165.494	190.772
	4	Forward L-IV	173.68	171.640	137.820	159.082
	5	Forward M-I	191.47	189.216	151.937	175.377
	6	Forward M-II	158.92	157.049	126.108	149.280
	7	Forward M-III	134.03	132.452	106.357	129.168
	8	Forward M-IV	111.25	109.940	88.280	109.952
	9	Forward H-I	132.71	131.147	105.309	127.864
	10	Forward H-II	110.15	108.853	87.407	108.838
	11	Forward H-III	82.89	81.914	65.776	83.794
	12	Forward H-IV	77.102	76.194	61.183	80.180
	13	Forward S-I	97.45	96.303	77.329	101.341
	14	Forward S-II	80.88	79.928	64.181	84.109
	15	Forward S-III	68.21	67.407	54.127	70.933
	16	Forward S-IV	56.62	55.953	44.930	58.881
high grade	17	Forward L-I	90.155	89.094	71.541	93.755
	18	Forward L-II	74.828	73.947	59.378	77.816
	19	Forward L-III	63.108	62.365	50.078	65.628
	20	Forward L-IV	52.379	51.762	41.564	54.470
	21	Forward M-I	57.746	46.951	45.823	60.052
	22	Forward M-II	47.929	38.970	38.033	49.843
	23	Forward M-III	40.422	32.866	32.076	42.036
	24	Forward M-IV	33.550	27.278	26.623	34.890
	25	Forward H-I	40.023	32.541	31.759	41.621
	26	Forward H-II	33.218	27.008	26.359	34.544
	27	Forward H-III	28.016	22.779	22.232	29.135
	28	Forward H-IV	23.253	18.906	18.452	24.181
	29	Forward S-I	29.39	23.896	23.322	30.563
	30	Forward S-II	24.39	19.831	19.354	25.364
	31	Forward S-III	20.57	16.725	16.323	21.391
	32	Forward S-IV	17.07	13.879	13.546	17.752

.

The analysis of Table 5 reveals significant changes in the distribution of the tractor transmission ratio resulting from the optimization performed by the three algorithms. The reduction in the transmission ratio for gears 1–4 is more pronounced, while the basic operating gear remains unchanged before and after optimization. Notably, there are substantial differences in the optimization results for the transmission ratio of the transport gear. Following improvements to the traditional algorithm, the transport gear ratio obtained by GC_NSGA-II slightly exceeds the results of the other algorithms, indicating sparser gearing and a wider range of corresponding speed variations.

This study conducts a comparative analysis of the theoretical tractor speeds derived from different algorithms to further examine the variations in optimization results (Figure 6). The optimized tractor speed was significantly improved in all gears compared to the pre-optimization setup, resulting in a more reasonable speed distribution. The optimization results of the NSGA-II and SPEA-II algorithms are consistent, and the speed in the low gear increased significantly, especially the theoretical speed in the first gear, which increased by 16.82 ± 0.32% and 24.11 ± 0.44%, respectively, and the operating speed in other gear increased by 1.38–18.63%. Following the enhancements made to the GC_NSGA-II algorithm in this study, the speed of the first gear was optimized and increased by 20.15 ± 0.27%. Additionally, the speed of the highest gear decreased to 30.20 ± 0.03 km/h, while the operating speeds of the intermediate gears increased by a range of 0.48% to 5.46%. The fourth gear achieved a speed of 4.99 ± 0.04 km/h, expanding the coverage of the operating gear range. Furthermore, apart from the direct gear, the operating speeds of the transport gears were reduced by 1.79 ± 0.22%, 3.70 ± 0.31%, and 4.82 ± 0.34%, respectively. This adjustment resulted in more concentrated transmission gearing, catering to diverse operational requirements effectively.

As observed in Figure 7, the optimized gear drive experiences a slight decrease, while the driving power utilization and fuel economy show significant improvements. The gear distribution is more intensive within the main operating speed range of 5–20 km/h. The SPEA-II algorithm prioritizes tractor power performance, resulting in a more optimal power optimization for the entire machine. The loss rate of driving power optimized by the SPEA-II algorithm is reduced by 20.19 ± 0.52%, approaching the ideal drive curve, which is more significant than the 8.62 ± 0.21% reduction in fuel consumption loss. Following the optimization of the GC_NSGA-II algorithm, the tractor experiences a reduction in both the loss rate of driving power and the loss rate of specific fuel consumption. These reductions amount to 4.17 ± 0.25% and 32.75 ± 0.38%, respectively, compared to the NSGA-II algorithm. Moreover, when compared to pre-optimization, they represent reductions of 41.62 ± 044% and 62.80 ± 0.56%, respectively. These findings indicate that the optimization effect of the algorithm improved by gradient crowding exhibits better performance.

Table 6. Comparison of tractor performance indicators based on optimization algorithms.

Indicators	Before Optimization	NSGA-II (Mean ± S.D.)	SPEA-II (Mean ± S.D.)	GC_NSGA-II (Mean ± S.D.)
The loss rate of driving power (%)	36.21	22.06 ± 0.31	20.19 ± 0.52	21.14 ± 0.44
The loss rate of specific fuel consumption (%)	11.37	6.29 ± 0.33	8.62 ± 0.21	4.23 ± 0.36
$\mathrm{Maximum} \mathrm{speed} (\mathrm{km}\cdot {\mathrm{h}}^{-1}$)	31.73	32.13 ± 0.45	34.33 ± 0.23	30.20 ± 0.51
$\mathrm{First} \mathrm{gear} \mathrm{speed} (\mathrm{km}\cdot {\mathrm{h}}^{-1}$)	2.30	2.71 ± 0.60	2.87 ± 0.43	2.76 ± 0.14
The climbing gradient of the first gear of the transport gear (%)	44.03	39.57 ± 0.11	38.07 ± 0.26	45.07 ± 0.39

presents a detailed comparison of tractor performance indexes before and after optimization. It is evident that, after the improvements in this study, the GC_NSGA-II algorithm is better optimized than the NSGA-II and SPEA-II algorithms before the improvements, and the speed range of the tractor is more in line with the actual operational needs, and the speed distribution is more reasonable. Additionally, the climbing gradient of 45.07 ± 0.39% in the first gear of the transport gear exceeds the values of 39.57 ± 0.11% and 38.07 ± 0.26% achieved by the NSGA-II and SPEA-II algorithms, representing an increase of 2.35% compared to the pre-optimization period, marking a significant improvement in the overall performance of the machine.

5. Conclusions

This study introduces an optimization method for matching tractor drivelines based on a diversity maintenance strategy known as gradient crowding aimed at obtaining the optimal solution for tractor driveline ratio matching, thereby enhancing the drive performance of the entire machine, reducing fuel consumption, and improving tractor economy. The proposed method employs the evaluation strategy of gradient crowding during selection to reduce the non-inferior solution sets, ensuring a uniform and wide distribution of solutions, thus maintaining population diversity effectively.

The improved performance of the algorithm was evaluated through testing, using four sets of standard test functions, demonstrating its effectiveness. For the problem of optimizing tractor transmission ratio matching, the loss rate of tractor driving power and specific fuel consumption serve as evaluation indexes for power and fuel economy, and the optimal transmission ratio allocation was calculated based on the improved GC_NSGA-II algorithm.

The comparison of the optimization results from various algorithms demonstrates that the method proposed in this study can achieve a 41.62 ± 0.44% and 62.80 ± 0.56% reduction in the loss rate for driving power and the loss rate for specific fuel consumption, respectively, and a 2.35% increase in the climbing gradient of the first gear of the transport gear, with more reasonable gear distribution, which can adapt to more complex working conditions. The effectiveness of this paper in solving the multi-objective optimization problem is, thus, further verified.