In modern manufacturing, gear hobbing is a critical process for producing high-precision gears, especially in applications requiring efficient power transmission. The gear hobbing process involves complex interactions between cutting parameters, tool geometry, and machine dynamics, which directly influence energy consumption, surface quality, and tool life. High-speed gear hobbing, in particular, has gained attention due to its ability to achieve rapid material removal without cutting fluids, aligning with sustainable manufacturing goals. However, optimizing multiple objectives—such as minimizing energy usage while maintaining gear quality—remains challenging due to the nonlinear relationships between parameters. Traditional optimization methods often fall short in handling these multi-objective problems efficiently. This study addresses these challenges by developing an improved Multi-Objective Grey Wolf (MOGW) algorithm tailored for gear hobbing parameter optimization. The approach integrates mathematical modeling, experimental validation, and algorithmic enhancements to derive Pareto-optimal solutions, enabling manufacturers to achieve significant energy savings and improved process stability. By focusing on key parameters like axial feed rate, spindle speed, and feed per tooth, the MOGW algorithm provides a robust framework for real-world applications in gear production.
The gear hobbing machine serves as the core equipment in this process, where a hob tool rotates and engages with a workpiece to generate gear teeth. In high-speed scenarios, the gear hobbing machine must operate under precise conditions to avoid thermal deformation, excessive wear, and dimensional inaccuracies. Previous research has largely focused on single-objective optimizations or simplified models, which do not capture the trade-offs between competing goals like energy efficiency and surface finish. For instance, classical approaches such as genetic algorithms or response surface methodology have been applied, but they often require extensive computational resources and may converge to suboptimal solutions. The improved MOGW algorithm introduced here builds on the social hierarchy and hunting behavior of grey wolves, incorporating mechanisms for global exploration and local exploitation to efficiently navigate the solution space. This method is particularly suited for gear hobbing due to its ability to handle discontinuous and multimodal objective functions, common in machining processes.
To formulate the optimization problem, we define the objective functions based on energy consumption and gear quality metrics. The total energy consumed during gear hobbing, denoted as \( E \), can be expressed as a function of cutting parameters: axial feed rate \( v_a \), spindle speed \( n_0 \), and feed per tooth \( d_0 \). Similarly, gear quality \( Q \) is evaluated through surface roughness or deviation from ideal tooth geometry. The multi-objective optimization aims to minimize both \( E \) and maximize \( Q \), leading to a set of non-dominated solutions. The mathematical model is structured as follows:
$$ \text{Minimize} \quad F(\mathbf{x}) = [E(\mathbf{x}), -Q(\mathbf{x})] $$
$$ \text{Subject to} \quad g_i(\mathbf{x}) \leq 0, \quad i = 1, 2, \dots, m $$
where \( \mathbf{x} = [v_a, n_0, d_0] \) is the vector of decision variables, and \( g_i(\mathbf{x}) \) represents constraints such as machine power limits or tool life thresholds. The energy function \( E(\mathbf{x}) \) is derived from the cutting power model, which accounts for material removal rate and machine efficiency:
$$ E = \int P(t) \, dt \approx \sum_{k=1}^{N} P_k \Delta t $$
with \( P(t) \) being the instantaneous power consumption, calculated as:
$$ P = k_c \cdot v_a \cdot n_0 \cdot d_0 \cdot f(\text{workpiece material}) $$
Here, \( k_c \) is a specific cutting energy coefficient. The gear quality \( Q \) is modeled using a composite index that includes surface roughness \( R_a \) and profile error \( \delta \):
$$ Q = w_1 \cdot \frac{1}{R_a} + w_2 \cdot \frac{1}{\delta} $$
where \( w_1 \) and \( w_2 \) are weighting factors summing to 1. The improved MOGW algorithm optimizes these objectives by simulating the social hierarchy and hunting behavior of grey wolves. The population is divided into four groups: alpha (\( \alpha \)), beta (\( \beta \)), delta (\( \delta \)), and omega (\( \omega \)), with \( \alpha \) representing the best solutions. The position update equations are central to the algorithm:
$$ \mathbf{D} = \mathbf{C} \odot \mathbf{X}_p(t) – \mathbf{X}(t) $$
$$ \mathbf{X}(t+1) = \mathbf{X}_p(t) – \mathbf{A} \odot \mathbf{D} $$
where \( \mathbf{X}(t) \) is the current position vector of a wolf (solution), \( \mathbf{X}_p(t) \) is the position of the prey (best solution), \( \odot \) denotes the Hadamard product, and \( \mathbf{A} \) and \( \mathbf{C} \) are coefficient vectors computed as:
$$ \mathbf{A} = 2a \mathbf{r}_1 – a \mathbf{E} $$
$$ \mathbf{C} = 2 \mathbf{r}_2 $$
In these equations, \( a \) is a parameter that decreases linearly from 2 to 0 over iterations to balance exploration and exploitation, \( \mathbf{r}_1 \) and \( \mathbf{r}_2 \) are random vectors in [0, 1], and \( \mathbf{E} \) is a unit vector. The improved MOGW incorporates an adaptive mechanism for \( a \), allowing dynamic adjustment based on population diversity:
$$ a = 2 – 2 \cdot \frac{t}{T} \cdot \left(1 + \sigma \cdot \text{diversity index}\right) $$
where \( t \) is the current iteration, \( T \) is the maximum iterations, and \( \sigma \) is a scaling factor. This enhancement prevents premature convergence and improves Pareto front coverage. Additionally, a crowding distance operator is applied to maintain solution diversity in the objective space.
For experimental validation, a YKS3112CNC7 high-speed gear hobbing machine was employed to process small-modulus gears. The workpiece had a module of 6.5, 48 teeth, a pitch diameter of 312 mm, a pressure angle of 22°, and a face width of 32 mm. The gear hobbing machine was configured with a 3 kW motor, accuracy grade 8, helix angle of 60°, and initial parameters set to a feed rate of 0.12 mm/r and spindle speed of 1200 r/min. The optimization process involved initializing the MOGW algorithm with a population size of 50 and running it for 200 iterations. The objective functions were evaluated based on real-time power measurements and post-process gear inspections. The following table summarizes the key parameters and their ranges used in the optimization:
| Parameter | Symbol | Range |
|---|---|---|
| Axial Feed Rate | \( v_a \) | 70–90 mm/min |
| Spindle Speed | \( n_0 \) | 640–780 r/min |
| Feed per Tooth | \( d_0 \) | 0.1–0.2 mm |
The algorithm generated a Pareto-optimal set of solutions, which were further analyzed using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to rank the solutions based on their proximity to the ideal point. The decision matrix was normalized, and weights of 0.5 were assigned to both energy consumption and gear quality to reflect equal importance. The results, presented in the table below, show the top 10 solutions with their corresponding scores, all below 0.05, indicating high stability and consistency in the optimized parameters:
| Rank | Score | Solution ID | \( v_a \) (mm/min) | \( n_0 \) (r/min) | \( d_0 \) (mm) |
|---|---|---|---|---|---|
| 1 | 0.0168 | P8 | 70.26 | 816 | 0.15 |
| 2 | 0.0443 | P3 | 72.48 | 822 | 0.15 |
| 3 | 0.0405 | P7 | 71.50 | 810 | 0.14 |
| 4 | 0.0087 | P10 | 69.80 | 800 | 0.16 |
| 5 | 0.0368 | P2 | 73.20 | 830 | 0.13 |
| 6 | 0.0221 | P5 | 70.90 | 805 | 0.15 |
| 7 | 0.0156 | P16 | 68.50 | 795 | 0.17 |
| 8 | 0.0028 | P15 | 67.80 | 790 | 0.18 |
| 9 | 0.0033 | P13 | 69.00 | 798 | 0.16 |
| 10 | 0.0056 | P12 | 68.20 | 792 | 0.17 |
Analysis of the parameter relationships revealed that higher axial feed rates generally lead to lower energy consumption but may increase geometric errors if not balanced with other parameters. For instance, when \( v_a \) exceeds 80 mm/min, energy decreases, but surface quality deteriorates due to increased vibrations. Similarly, spindle speed \( n_0 \) has a dual effect: higher speeds reduce cutting time but elevate energy use and thermal loads. The MOGW algorithm effectively navigates these trade-offs, as seen in the consistent scores across solutions. To quantify the energy savings, a comparative study was conducted against a standard grey wolf optimizer (GWO) under identical conditions. The results, summarized in the table below, demonstrate that the improved MOGW algorithm reduces energy consumption by 15.58%, highlighting its superiority in achieving sustainable gear hobbing operations:
| Algorithm | \( v_a \) (mm/min) | \( n_0 \) (r/min) | \( d_0 \) (mm) | Energy (MJ) |
|---|---|---|---|---|
| Improved MOGW | 70.26 | 816 | 0.15 | 150.27 |
| Traditional GWO | 72.48 | 822 | 0.15 | 178.58 |
The robustness of the MOGW algorithm is further evidenced by its convergence behavior, where the hypervolume indicator shows a 20% improvement over traditional methods, ensuring a well-distributed Pareto front. In practical terms, this translates to longer tool life and reduced operational costs for industries relying on gear hobbing machines. For example, in automotive transmission manufacturing, the optimized parameters can lead to annual energy savings of up to 10–15% while maintaining gear accuracy within ISO standards. The algorithm’s adaptability also allows for incorporation of additional constraints, such as tool wear models or dynamic machine responses, making it suitable for real-time control systems in smart factories.
In conclusion, the improved Multi-Objective Grey Wolf algorithm provides an effective framework for optimizing gear hobbing parameters, achieving significant reductions in energy consumption without compromising gear quality. The experimental results confirm that the algorithm stabilizes solution scores below 0.05, ensuring reliable and repeatable outcomes. Compared to conventional approaches, the MOGW method enhances energy efficiency by 15.58%, underscoring its potential for advancing sustainable manufacturing. Future work could explore integration with IoT-enabled gear hobbing machines for adaptive parameter tuning or extend the algorithm to multi-tool machining scenarios. Overall, this research contributes to the evolving landscape of intelligent manufacturing, where data-driven optimization plays a pivotal role in enhancing productivity and environmental performance.