In the rapidly expanding field of offshore wind energy, the demand for critical components such as hubs, main shafts, bearing housings, and front frames has surged. These components are typically manufactured as thick-section ductile iron castings, with single-piece weights ranging from 20 to 60 tonnes. The production of these massive ductile iron castings presents a significant challenge in the foundry melting stage. Often, the weight of a single casting exceeds the safe melting capacity of an individual induction furnace available in a foundry. This necessitates the simultaneous use of multiple furnaces to melt the required quantity of iron for a single pouring, a process known as multi-heat melting. The core scheduling problem involves efficiently allocating various casting orders of differing weights and deadlines across multiple parallel furnaces, while respecting constraints like material compatibility and furnace capacity limits. This study addresses this complex production planning issue by developing a novel multi-heat scheduling model and an efficient hybrid algorithm to solve it.
The traditional approach to furnace charge planning, or “heat planning,” assumes that each casting’s weight is less than the capacity of a single furnace. However, this assumption fails for large wind turbine components made from ductile cast iron. Our work directly tackles this gap. We consider a scenario where several furnaces operate in parallel, and a large ductile iron casting order must be split and melted concurrently across them. Smaller castings cannot be split due to gating system design. The primary objectives are to maximize furnace utilization—thereby reducing energy waste and the number of heats—and to minimize order delays by considering due-date priorities. All castings grouped into a single simultaneous heat must be of the same grade of ductile iron.
We begin by formally describing the problem. The foundry process flow includes molding, melting, and pouring. The melting stage is the bottleneck. Let there be K furnaces that can be operated simultaneously, indexed by \( k = 1, 2, …, K \). Each furnace \( k \) has a maximum safe melting capacity \( Q_k \). There are L casting orders, indexed by \( l = 1, 2, …, L \). Each order \( l \) has a gross weight \( N_l \), a due-date-related priority value \( \rho_l \), and a material type \( M_l \in \{1, 2, …, N\} \), where \( N \) is the number of ductile iron grades. The key decision variable is \( P_{lk} \), representing the proportion of order \( l \)’s weight assigned to furnace \( k \). For a standard ductile iron casting whose weight \( N_l \leq \min(Q_k) \), \( P_{lk} \) must be either 0 or 1, meaning the entire casting is assigned to one furnace or not at all. For a large ductile iron casting where \( N_l > \min(Q_k) \), \( P_{lk} \) can be fractional between 0 and 1, but the sum across all furnaces for that order must be 1 if it is scheduled.
The due-date priority for an order \( l \) is calculated based on its remaining slack time \( V_l \). A common formulation is:
$$
\rho_l = \frac{1}{1 + V_l}
$$
This gives a higher priority to orders with shorter remaining times. Our objective function combines both weight and priority to maximize the total “melting value” of a heat plan:
$$
\text{Maximize } Z = \sum_{k=1}^{K} \sum_{l=1}^{L} ( \rho_l \cdot N_l \cdot X_{lk} )
$$
where \( X_{lk} \) is an auxiliary variable indicating if order \( l \) is assigned to furnace \( k \) (\( X_{lk} = 1 \) if \( P_{lk} > 0 \), else 0). For the proportional assignment, the value contributed is \( \rho_l \cdot N_l \cdot P_{lk} \). The multiplicative form ensures we favor schedules that use furnace capacity fully with high-priority orders.
The constraints for the model are as follows:
- Complete Assignment or None: For large ductile iron castings, if scheduled, the entire weight must be allocated: $$ \sum_{k=1}^{K} P_{lk} = 1 \text{ or } 0 \quad \forall l \in \{ l: N_l > \min(Q_k) \} $$ For small ductile cast iron parts: $$ \sum_{k=1}^{K} P_{lk} \in \{0, 1\} \quad \forall l \in \{ l: N_l \leq \min(Q_k) \} $$
- Furnace Capacity: The total weight melted in any furnace must not exceed its safe capacity: $$ \sum_{l=1}^{L} N_l \cdot P_{lk} \leq Q_k \quad \forall k $$
- Material Homogeneity per Furnace: All ductile iron castings melted together in one furnace must be of the same material grade. If \( T_k \) is the material type assigned to furnace \( k \), then: $$ P_{lk} > 0 \implies M_l = T_k \quad \forall l, k $$
- Non-negativity and Bounds: $$ 0 \leq P_{lk} \leq 1 \quad \forall l, k $$
This model presents a mixed-integer nonlinear programming challenge due to the conditional constraints based on casting size and the material compatibility constraint. To solve it efficiently, we decompose the problem into three sub-problems: (1) Grouping castings that can be melted together based on material type. (2) Determining which groups of castings (batches) should be scheduled for a simultaneous heat. (3) Allocating the weights of castings within a batch across the multiple furnaces. Sub-problem (2) is essentially a 0-1 knapsack problem for each material group, where the “knapsack” capacity is the total available furnace capacity for that heat, and the “items” are casting orders with a value of \( \rho_l \cdot N_l \).
We propose a Hybrid Genetic Whale Optimization Algorithm (HGWOA) to solve the batch selection sub-problem. The standard Whale Optimization Algorithm (WOA) is inspired by the bubble-net hunting behavior of humpback whales. It has good exploration ability but can suffer from premature convergence. The Genetic Algorithm (GA) offers robust global search through selection, crossover, and mutation. Our HGWOA integrates the mechanisms of both to improve search efficiency and solution quality for scheduling ductile iron production.
The HGWOA procedure is as follows. The population is a set of binary strings (chromosomes), where each bit corresponds to a casting order (1 if selected for the current batch, 0 otherwise). The population size is NIND, and the algorithm runs for MAXGEN iterations. The fitness of an individual (scheme) is the total melting value \( \sum (\rho_l \cdot N_l) \) for the selected orders, provided the total weight does not exceed the aggregated furnace capacity and the selection satisfies an additional “size-count” constraint. This size-count constraint ensures that a batch containing large ductile cast iron parts can be feasibly distributed later; e.g., the number of castings heavier than a smaller furnace’s capacity must not exceed the number of larger furnaces. We repair infeasible chromosomes using a greedy strategy: remove orders with the lowest melting value until constraints are met, then try to add high-value orders back.
The position update in WOA is guided by a coefficient vector \( A \). In our hybrid approach, when \( |A| \geq 1 \), we perform crossover between the current individual and a random individual to encourage exploration. When \( |A| < 1 \), we perform crossover with the current best individual to favor exploitation. Mutation is applied with a certain probability to maintain diversity. The crossover and mutation operations are standard GA operators adapted for binary encoding.
After HGWOA selects an optimal batch of ductile iron castings for a heat, the third sub-problem—weight allocation across furnaces—is solved. This is a linear programming problem with the constraints defined earlier for \( P_{lk} \). We use the `intlinprog` solver in MATLAB to find the precise weight fractions \( P_{lk} \) that maximize furnace fill under the material and capacity constraints. A sample allocation result for a batch is shown in Table 1.
| Casting ID | Weight (kg) \( N_l \) | Material | Furnace 1 Allocation \( P_{l1} \) | Furnace 2 Allocation \( P_{l2} \) |
|---|---|---|---|---|
| 1 | 1,028 | QT400 | 0 | 1.0 |
| 2 | 1,240 | QT400 | 0 | 1.0 |
| 3 | 920 | QT400 | 0 | 1.0 |
| 4 (Large) | 21,800 | QT400 | 0.917 | 0.083 |
| 5 | 1,033 | QT400 | 1.0 | 0 |
| 6 | 1,100 | QT400 | 1.0 | 0 |
| Total Melted Weight per Furnace | ~20,000 kg | ~20,000 kg | ||
To validate our model and algorithm, we conducted experiments using real production data from a foundry specializing in wind turbine ductile iron castings. The setup involves two parallel medium-frequency induction furnaces, each with a safe melting capacity \( Q = 20,000 \) kg. The foundry typically runs four heats per furnace per night. The iron-to-casting yield ratio is assumed to be 1:1.1. The test dataset contains 191 casting orders categorized into three ductile iron grades: QT400, QT500, and QT600, as summarized in Table 2.
| Material ID | Ductile Iron Grade | Number of Casting Orders | Typical Applications |
|---|---|---|---|
| 1 | QT400 | 116 | Hubs, Brackets |
| 2 | QT500 | 21 | Gears, Shafts |
| 3 | QT600 | 54 | High-Stress Components |
We applied our HGWOA-based scheduling system to plan eight consecutive heats (four per furnace). A snapshot of one heat’s schedule is shown in Table 1. The large ductile iron casting (ID 4, 21,800 kg) was successfully split between the two furnaces. Furnace utilization for each heat was calculated as:
$$
\text{Furnace Utilization} = \frac{\text{Total Weight Melted in Furnace}}{Q_k} \times 100\%
$$
The average furnace utilization \( B^* \) over the eight heats was computed. We compared the performance of our intelligent scheduling against the foundry’s existing manual scheduling method for the entire dataset. The results are presented in Table 3.
| Performance Metric | Manual Scheduling | Intelligent Scheduling (Our Model) | Improvement |
|---|---|---|---|
| Average Furnace Utilization | 86.57% | 95.89% | +10.76 percentage points |
| Total Number of Heats Required | 42 | 39 | Reduction of 3 heats (-7.14%) |
The results clearly demonstrate the effectiveness of our approach. The intelligent scheduling system significantly improves furnace utilization for melting ductile cast iron, leading to better resource use and lower energy consumption per tonne of castings produced. The reduction in the total number of heats also implies savings in furnace preheating time and operational costs.
We further evaluated the efficiency of our HGWOA by comparing it with two other popular optimization algorithms: the standard Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). All algorithms were configured with a population size of 500 and a maximum of 300 generations. The algorithms were run 20 times on test datasets with varying complexity (1, 2, and 3 material types). The comparison metrics include average computation time (Time), best-found solution value (Best), and average solution value (Avg). The results are consolidated in Table 4.
| Number of Material Types | Algorithm | Avg. Time (s) | Best Solution Value (\( \times 10^4 \)) | Avg. Solution Value (\( \times 10^4 \)) |
|---|---|---|---|---|
| 1 | PSO | 2.25 | 3.1842 | 2.9935 |
| GA | 3.42 | 3.1911 | 3.0222 | |
| HGWOA (Proposed) | 2.53 | 3.1918 | 3.0230 | |
| 2 | PSO | 6.34 | 3.2178 | 3.0114 |
| GA | 10.98 | 3.2169 | 3.0928 | |
| HGWOA (Proposed) | 7.98 | 3.2113 | 3.0898 | |
| 3 | PSO | 11.12 | 3.2945 | 3.0521 |
| GA | 17.24 | 3.2961 | 3.1121 | |
| HGWOA (Proposed) | 12.18 | 3.2969 | 3.1125 |
The convergence behavior of the algorithms is also illustrative. PSO converges very quickly but tends to get trapped in local optima, resulting in lower solution quality. GA converges more slowly but generally finds better solutions. Our HGWOA strikes an excellent balance: it converges almost as fast as PSO (often within 20-30 generations) while achieving solution values comparable to or slightly better than GA. This makes HGWOA highly suitable for the dynamic scheduling environment of a ductile iron foundry, where good solutions are needed promptly.
The mathematical formulation and solution strategy provide a robust framework for multi-heat scheduling. The objective function \( Z \) effectively balances due-date adherence and capacity utilization. The constraints ensure practical feasibility. The decomposition into grouping, batching, and allocation simplifies a complex industrial problem. The HGWOA, with its tailored chromosome repair mechanism and hybrid update rules, proves to be an effective solver for the knapsack-like batching sub-problem inherent in planning for ductile iron castings.
In conclusion, we have successfully developed and validated a multi-heat scheduling model specifically designed for the production of large wind turbine components made from ductile cast iron. The model explicitly handles the splitting of oversized ductile iron castings across multiple furnaces, a scenario not addressed in previous literature. Our hybrid HGWOA solution method demonstrates superior performance in terms of solution quality and computational efficiency compared to standard algorithms. The practical application shows significant improvements in furnace utilization and a reduction in the total number of heats required, leading to tangible economic and energy-saving benefits for foundries producing heavy-section ductile iron castings. Future work could integrate constraints from upstream molding capacity and downstream pouring time windows to create an even more comprehensive production planning system for advanced ductile iron foundries.