In modern manufacturing, gear hobbing is a critical process for producing high-precision gears used in automotive, aerospace, and machinery industries. The selection of optimal gear hobbing tools and control parameters, such as hob diameter, number of starts, spindle speed, and axial feed, directly influences key performance metrics like carbon emissions, cutting time, and gear quality. Traditional multi-objective heuristic algorithms, such as the Multi-objective Weighted Mean of Vectors Algorithm (MOINFO), Multi-objective Grey Wolf Optimization (MOGWO), and Multi-objective Dragonfly Algorithm (MODA), have been applied to optimize these parameters. However, these methods suffer from limitations, including sensitivity to weight settings, susceptibility to local optima, and high computational complexity. Moreover, the manual selection of heuristic algorithms often leads to suboptimal results due to the lack of adaptability to specific problem instances. To address these issues, this paper proposes a novel approach that combines an improved multi-objective hyper-heuristic algorithm with fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) for the optimization and decision-making of gear hobbing tools and control parameters. The method dynamically selects the most suitable heuristic algorithm during the optimization process and incorporates user preferences expressed in fuzzy terms to identify the best parameter set. Experimental results demonstrate the feasibility and effectiveness of the proposed approach in achieving superior performance compared to existing methods.
The gear hobbing process involves complex interactions between tool geometry, cutting conditions, and workpiece material. Inefficient parameter selection can lead to increased energy consumption, longer production times, and reduced product quality. For instance, inappropriate spindle speeds or axial feeds may cause excessive tool wear, higher carbon emissions, and poor surface finish. The optimization problem is formulated as a multi-objective problem with conflicting goals: minimizing carbon emissions (CE), cutting time (T), and maximizing quality (Q). Carbon emissions include contributions from electricity consumption, tool usage, and cutting fluid consumption. Cutting time is defined as the duration of the core cutting phase, from the radial cutting end point to the axial exit point. Quality is a composite measure based on gear errors such as tooth direction error, tooth profile error, and surface roughness. The challenge lies in balancing these objectives while adhering to practical constraints, such as tool life and machine capabilities.
To tackle this problem, the proposed method consists of three main steps. First, historical gear hobbing data is analyzed using spectral clustering to determine the upper and lower bounds of the parameters. This ensures that the optimization search space is constrained to feasible regions based on past experiences. Second, an improved multi-objective hyper-heuristic algorithm is employed to generate a set of non-dominated solutions (Pareto front) for the gear hobbing parameters. The hyper-heuristic framework dynamically selects from a pool of underlying algorithms, including MOINFO, Multi-objective Harris Hawks Optimization (MOHHO), MOGWO, and Multi-objective Ant Lion Optimization (MOALO), based on their performance scores. This adaptive selection mechanism enhances global exploration and local exploitation, leading to a diverse and convergent Pareto front. Finally, fuzzy TOPSIS is used to rank the optimized parameter sets according to user-defined preferences for the performance metrics, enabling the selection of the most suitable gear hobbing tools and control parameters.
The improved hyper-heuristic algorithm incorporates several innovations. A dynamic algorithm selection mechanism uses a state transition matrix to probabilistically choose the best-performing algorithm at each iteration. The scoring module evaluates algorithm performance using a hybrid strategy: in early iterations, it emphasizes diversity by considering the spatial distribution of the current solution set, while in later iterations, it focuses on convergence by assessing the archive of non-dominated solutions. The acceptance strategy module retains solutions based on crowding distance and Pareto dominance to maintain population diversity. This approach overcomes the limitations of individual heuristic algorithms and ensures robust optimization across different problem instances.
Experimental validation was conducted on a YS3120CNC6 gear hobbing machine equipped with a Siemens 840D CNC system. The machine is capable of processing gears with a maximum module of 6 mm using the generating method, making it suitable for high-volume production in industries such as automotive and machinery. The workpiece material was 20CrMo, and the tool substrate was S390 with a TiAlN coating. Historical data from previous gear hobbing operations was used to define the parameter bounds. For example, the hob diameter was constrained between 70 mm and 80 mm, the number of starts between 2 and 3, spindle speed between 360 rpm and 450 rpm, and axial feed between 1.5 mm/r and 2 mm/r. The optimization objectives were formulated as follows:
$$ \min F(p_1, p_2, p_3, p_4) = (\min CE, \min T, \min Q) $$
where CE, T, and Q are calculated as:
$$ CE = CE_e + CE_t + CE_l $$
$$ CE_e = F_e \cdot \left( \beta_0 + \beta_1 p_3 + \beta_2 p_4 + \beta_3 p_3^2 + \beta_4 p_4^2 + \beta_5 p_3 p_4 \right) $$
$$ CE_t = F_t \cdot \frac{m_t}{T_t} \cdot T $$
$$ CE_l = (F_{lp} + F_{ld}) \cdot \frac{V}{T_l} \cdot T $$
$$ T = \frac{d_{in} + d_{out}}{2 \cdot p_3 \cdot p_4} $$
$$ Q = w_{q1} \cdot f_{M_{cx}} + w_{q2} \cdot f_{M_{cs}} + w_{q3} \cdot R_{Ma} $$
In these equations, $F_e$, $F_t$, $F_{lp}$, and $F_{ld}$ are emission factors for electricity, tool, and cutting fluid, respectively; $m_t$ is tool mass; $T_t$ is tool life; $V$ is cutting fluid volume; $T_l$ is fluid life; $d_{in}$ and $d_{out}$ are entry and exit diameters; $f_{M_{cx}}$, $f_{M_{cs}}$, and $R_{Ma}$ are quality metrics; and $w_{q1}$, $w_{q2}$, $w_{q3}$ are weights. The parameters $p_1$, $p_2$, $p_3$, $p_4$ represent hob diameter, number of starts, spindle speed, and axial feed, respectively.
The hyper-heuristic algorithm was configured with a population size of 100 and a maximum of 100 iterations. The underlying algorithms were initialized with standard parameters, and the state transition matrix was updated every 10 iterations based on performance scores. The scoring strategy switched from CS1 to CS2 at iteration 40 to balance exploration and exploitation. CS1 computed scores based on the spacing of the current solution set, while CS2 used the spacing of the archive solutions. The acceptance strategy alternated between AS1 and AS2 with a probability $P_s = 0.3 + 0.7 \cdot \text{iter} / \text{max\_iter}$, where AS1 retained solutions with high crowding distance and AS2 applied Pareto dominance.
The results showed that the proposed method achieved a well-distributed Pareto front with solutions offering trade-offs between carbon emissions, cutting time, and quality. For a sample workpiece with attributes module = 2 mm, pressure angle = 0.349 rad, number of teeth = 80, helix angle = 0 rad, outer diameter = 170 mm, face width = 30 mm, and cutting depth = 4.5 mm, the optimal parameters were identified as hob diameter = 74 mm, number of starts = 2, spindle speed = 450 rpm, and axial feed = 2 mm/r. The corresponding objective values were CE = 0.2046 kgCO2, T = 2.2336 min, and Q = 1.1191. Practical verification on the gear hobbing machine confirmed that the actual performance metrics deviated by less than 2.1% from the predicted values, demonstrating the method’s accuracy.
Comparative experiments were conducted to evaluate the proposed hyper-heuristic method (HF) against individual algorithms: MOINFO (H1), MOHHO (H2), MOGWO (H3), and MOALO (H4). Each method was run five times with the same initial conditions. The performance was assessed using spacing (SP) metric, coverage rate, and TOPSIS scores. The SP metric measures the uniformity of solutions in the Pareto front, with lower values indicating better distribution. The coverage rate evaluates the extent to which one Pareto front covers another, reflecting search space exploration. TOPSIS scores rank solutions based on user preferences, with higher scores indicating better alignment with requirements.
| Run | HF | H1 | H2 | H3 | H4 |
|---|---|---|---|---|---|
| 1 | 0.273 | 1.442 | 0.404 | 0.393 | 0.390 |
| 2 | 0.211 | 5.055 | 0.283 | 1.346 | 8.287 |
| 3 | 0.273 | 1.427 | 0.885 | 0.196 | 8.914 |
| 4 | 0.252 | 1.559 | 0.757 | 8.740 | 1.037 |
| 5 | 0.278 | 0.812 | 0.751 | 0.883 | 1.723 |
As shown in Table 1, HF consistently achieved lower SP values with minimal variation, indicating uniform solution distribution and stability. In contrast, other methods exhibited significant fluctuations, highlighting their sensitivity to initial conditions and problem characteristics. The coverage rate analysis revealed that HF dominated other methods in most cases, with HF covering over 60% of solutions from H1, H2, and H4, while the reverse coverage was below 10%. For H3, the coverage was more balanced but still favored HF. This demonstrates HF’s superior ability to explore the search space and identify high-quality solutions.
TOPSIS scores were computed based on user preferences, where carbon emissions and quality were highly valued, and cutting time was less critical. The scores for five runs are summarized in Table 2. HF achieved the highest score in the fifth run and consistently ranked second in earlier runs, outperforming individual algorithms. This underscores the effectiveness of the dynamic algorithm selection in adapting to user requirements and problem dynamics.
| Run | HF | H1 | H2 | H3 | H4 |
|---|---|---|---|---|---|
| 1 | 0.7238 | 0.6664 | 0.7154 | 0.7246 | 0.7228 |
| 2 | 0.7239 | 0.6685 | 0.7156 | 0.7246 | 0.7229 |
| 3 | 0.7240 | 0.6704 | 0.7164 | 0.7248 | 0.7239 |
| 4 | 0.7313 | 0.6721 | 0.7187 | 0.7351 | 0.7241 |
| 5 | 0.7413 | 0.6979 | 0.7202 | 0.7392 | 0.7241 |
Convergence speed was evaluated by tracking the iteration at which the Pareto front stabilized. HF reached a stable Pareto front by iteration 61, significantly faster than other methods, which required more iterations. Although HF’s average runtime was 21.0 seconds, compared to 10.5 seconds for H1, 2.4 seconds for H2, 41.7 seconds for H3, and 3.4 seconds for H4, the faster convergence allows for reducing the maximum iterations, thereby optimizing computational efficiency. This adaptability is a key advantage of the hyper-heuristic approach.
Further comparisons were made with established methods: a Harris Hawks optimization and support vector regression-based approach (M1) and a BP neural network-based method (M2). Both methods were evaluated under the same conditions. HF demonstrated greater stability and higher TOPSIS scores, with an average of 0.7087 versus 0.5247 for M1 and 0.5350 for M2. The relative improvement of approximately 35% over M1 and 32.5% over M2 highlights HF’s robustness in handling fuzzy user preferences and complex optimization landscapes.
The fuzzy TOPSIS decision-making process involved defining linguistic variables for user preferences, such as “high” for carbon emissions and quality, and “low” for cutting time. These were converted into triangular fuzzy numbers, and the relative closeness to the ideal solution was computed for each parameter set. For example, the best solution had a closeness coefficient of 0.7413, indicating strong alignment with user priorities. This step ensures that the selected gear hobbing parameters not only optimize technical metrics but also satisfy subjective requirements, enhancing practical applicability.
In conclusion, the integration of an improved hyper-heuristic algorithm and fuzzy TOPSIS provides a comprehensive solution for optimizing and decision-making in gear hobbing processes. The method automates algorithm selection, balances multiple objectives, and incorporates user preferences effectively. Future work will focus on expanding the algorithm pool to include more heuristic methods and validating the approach on a wider range of gear hobbing machines and materials. This will further enhance the method’s versatility and adoption in industrial settings, contributing to sustainable and efficient manufacturing.
The proposed approach addresses critical challenges in gear hobbing parameter optimization, such as the uncertainty in algorithm selection and the vagueness in user preferences. By leveraging historical data and adaptive algorithms, it ensures that the gear hobbing machine operates at peak performance, reducing environmental impact and production costs while maintaining high quality. As industries move towards smarter and greener manufacturing, such advanced optimization techniques will play a pivotal role in achieving these goals.