With the rapid development of offshore wind power, the demand for large wind turbine components, such as hubs, main shafts, bearing seats, and front frames, has increased significantly. These thick-section ductile iron castings, often weighing between 20 to 60 tons, exceed the safe melting capacity of individual furnaces in many foundries. As a result, multiple furnaces must be operated simultaneously to meet the pouring requirements for these heavy ductile iron castings. This presents a complex scheduling challenge, where orders for ductile iron castings of varying weights must be allocated efficiently across available furnaces while considering constraints like delivery deadlines, furnace capacity utilization, and material compatibility. This study addresses this issue by developing a multi-heat planning model for the melting stage, optimized using a hybrid algorithm, to enhance resource allocation and production efficiency for ductile iron castings.
The production process for ductile iron castings involves several stages, including molding, core making, melting, and pouring. The melting stage is often the bottleneck, as it requires coordinating multiple furnaces to handle large ductile iron castings that cannot be melted in a single furnace due to weight limitations. For instance, a ductile iron casting weighing more than the maximum safe capacity of a furnace must be split across multiple furnaces, while smaller ductile iron castings can be grouped into batches for single-furnace processing, provided they share the same material grade. This necessitates a robust scheduling model that maximizes furnace utilization and minimizes order delays, specifically tailored for ductile iron castings in wind turbine applications.
To formalize the problem, consider a foundry with multiple melting furnaces operating in parallel. Each furnace has a maximum safe melting capacity, and each order for ductile iron castings has a specified weight, material type, and delivery deadline. The goal is to assign these ductile iron castings to furnace heats such that the total melting value—incorporating both weight and delivery priority—is maximized. The constraints include ensuring that the total weight assigned to a furnace does not exceed its capacity, that only ductile iron castings of the same material are combined in a single heat, and that large ductile iron castings are split appropriately across furnaces when necessary.
The mathematical model for this multi-heat planning problem involves several variables and parameters. Let \( K \) denote the total number of furnace heats, \( L \) the total number of tasks (each representing an order for ductile iron castings), and \( N \) the number of material types. For each task \( l \) (where \( l = 1, 2, \ldots, L \)), let \( N_l \) be the total weight of the ductile iron casting, \( V_l \) the remaining delivery time, and \( M_l \) the material type. For each furnace \( k \) (where \( k = 1, 2, \ldots, K \)), let \( Q_k \) be the maximum capacity and \( T_k \) the material type assigned to the furnace. The decision variable \( P_{lk} \) represents the proportion of task \( l \) allocated to furnace \( k \).
The priority of task \( l \) based on delivery date is defined as:
$$ \text{Priority}_l = \frac{1}{V_l} $$
The objective function aims to maximize the total melting value, which combines furnace capacity utilization and delivery priority:
$$ \text{Maximize} \quad \sum_{k=1}^{K} \sum_{l=1}^{L} P_{lk} \cdot N_l \cdot \text{Priority}_l $$
This function ensures that orders for ductile iron castings with tighter deadlines and larger weights are prioritized, thereby optimizing both resource use and customer satisfaction. The constraints are as follows:
$$ \sum_{k=1}^{K} P_{lk} = 1 \quad \text{for all } l \quad \text{(each task is fully allocated or not allocated)} $$
$$ \sum_{l=1}^{L} P_{lk} \cdot N_l \leq Q_k \quad \text{for all } k \quad \text{(furnace capacity constraint)} $$
$$ T_k = M_l \quad \text{for all } l, k \text{ where } P_{lk} > 0 \quad \text{(material compatibility constraint)} $$
$$ P_{lk} \in \{0, 1\} \quad \text{for small ductile iron castings} $$
$$ 0 \leq P_{lk} \leq 1 \quad \text{for large ductile iron castings} $$
These constraints ensure that all ductile iron castings are properly assigned, with large ductile iron castings allowing fractional allocation across furnaces and small ductile iron castings being assigned entirely to one furnace or none.
To solve this model, a heuristic approach is employed, breaking the problem into three sub-problems: grouping ductile iron castings into batches that can be melted together, scheduling these batches for pouring, and allocating the batches across multiple furnaces. The grouping sub-problem is treated as a 0-1 knapsack problem, where the goal is to select a set of ductile iron castings that maximize the total melting value without exceeding furnace capacities. This is solved using a Hybrid Genetic Whale Optimization Algorithm (HGWOA), which combines the global search capabilities of genetic algorithms with the efficient convergence of whale optimization algorithms.
The HGWOA process begins with initializing a population of solutions, where each solution is a binary string indicating whether a ductile iron casting order is included in a batch. The algorithm iteratively updates the solutions based on the best-found solutions, incorporating crossover and mutation operations from genetic algorithms and the exploration mechanisms from whale optimization. The fitness function is modified to account for both weight and priority:
$$ \text{Fitness} = \sum_{l \in \text{batch}} N_l \cdot \text{Priority}_l $$
This encourages the selection of batches that are both heavy and urgent. Constraints are enforced by repairing infeasible solutions using a greedy strategy, where orders are removed or added based on priority until capacity and material constraints are satisfied.
Once batches are formed, the allocation sub-problem assigns the ductile iron castings in each batch to specific furnaces. This is formulated as a linear programming problem with integer constraints for small ductile iron castings. Using MATLAB’s intlinprog function, the optimal proportions \( P_{lk} \) are determined to ensure that the total weight per furnace is within limits and that all ductile iron castings are fully allocated. For example, consider a batch with three ductile iron castings of weights \( N_1, N_2, N_3 \). The allocation might result in values like \( P_{11} = 1 \), \( P_{21} = 0 \), \( P_{31} = 0.8 \) for furnace 1 and \( P_{12} = 0 \), \( P_{22} = 1 \), \( P_{32} = 0.2 \) for furnace 2, indicating that casting 1 is fully assigned to furnace 1, casting 2 to furnace 2, and casting 3 is split between them.
To validate the model and algorithm, a case study was conducted using real production data from a foundry specializing in ductile iron castings for wind turbines. The foundry operates two medium-frequency furnaces, each with a maximum capacity of 20,000 kg, and typically runs four heats per furnace per night. The test dataset included 191 orders for ductile iron castings, categorized by material type: QT400, QT500, and QT600. The results showed that the model successfully scheduled all orders, with large ductile iron castings being split across furnaces and smaller ones grouped into batches.
| Material Type | Number of Ductile Iron Castings |
|---|---|
| QT400 | 116 |
| QT500 | 21 |
| QT600 | 54 |
For a sample heat, the scheduling resulted in the following allocation for ductile iron castings:
| Casting ID | Material | Weight (kg) | Furnace 1 Allocation (kg) | Furnace 2 Allocation (kg) |
|---|---|---|---|---|
| 9 | QT400 | 1,028 | 0 | 1,130.8 |
| 71 | QT400 | 1,240 | 0 | 1,364 |
| 23 | QT400 | 920 | 0 | 1,012 |
| 98 | QT400 | 21,800 | 20,000 | 3,980 |
| 15 | QT400 | 1,033 | 0 | 1,136.3 |
| 17 | QT400 | 1,100 | 0 | 1,320 |
The furnace capacity utilization was calculated for each heat using the formula:
$$ \text{Utilization} = \frac{\text{Total Weight Melted}}{\text{Furnace Capacity}} \times 100\% $$
Over eight heats, the average utilization achieved was 95.89%, compared to 86.57% with manual scheduling. This represents a 10.76% improvement, and the total number of heats required was reduced from 42 to 39, demonstrating enhanced efficiency and energy savings for processing ductile iron castings.
To evaluate the algorithm’s performance, comparisons were made with Particle Swarm Optimization (PSO) and standard Genetic Algorithm (GA). The HGWOA showed superior results in terms of solution quality and convergence speed. For instance, in tests with varying material types, HGWOA achieved higher fitness values and faster convergence, as summarized below:
| Number of Material Types | Algorithm | Average Time (s) | Best Fitness | Average Fitness |
|---|---|---|---|---|
| 1 | PSO | 2.25 | 3.1842 × 10⁴ | 2.9935 × 10⁴ |
| GA | 3.42 | 3.1911 × 10⁴ | 3.0222 × 10⁴ | |
| HGWOA | 2.53 | 3.1918 × 10⁴ | 3.0230 × 10⁴ | |
| 2 | PSO | 6.34 | 3.2178 × 10⁴ | 3.0114 × 10⁴ |
| GA | 10.98 | 3.2169 × 10⁴ | 3.0928 × 10⁴ | |
| HGWOA | 7.98 | 3.2113 × 10⁴ | 3.0898 × 10⁴ | |
| 3 | PSO | 11.12 | 3.2945 × 10⁴ | 3.0521 × 10⁴ |
| GA | 17.24 | 3.2961 × 10⁴ | 3.1121 × 10⁴ | |
| HGWOA | 12.18 | 3.2969 × 10⁴ | 3.1125 × 10⁴ |
The convergence analysis revealed that HGWOA reaches near-optimal solutions within about 20 iterations, whereas GA requires over 100 iterations, and PSO tends to get stuck in local optima. This makes HGWOA particularly suitable for dynamic production environments where rapid scheduling of ductile iron castings is essential.
In conclusion, the multi-heat planning model developed in this study effectively addresses the challenges of melting large ductile iron castings for wind turbines by optimizing furnace utilization and meeting delivery deadlines. The use of a hybrid algorithm ensures efficient and practical solutions, as demonstrated through real-world data. Future work could incorporate additional factors, such as molding capacity and pouring time constraints, to further enhance the scheduling of ductile iron castings. This approach not only improves resource allocation but also supports the sustainable growth of the wind energy sector by enabling more efficient production of critical components like ductile iron castings.