The modern manufacturing landscape, driven by initiatives like “Made in China 2025,” emphasizes deep integration of next-generation information technology with industrial processes. For the foundry sector, this translates into a strategic shift towards intelligent manufacturing characterized by digitalization, networking, and smart systems. Implementing precise, efficient digital and intelligent foundry systems, innovating production management workflows, and constructing comprehensive digital platforms are crucial for upgrading the industry. These advancements directly enhance the comprehensive quality of castings, shorten manufacturing cycles, improve technical and economic benefits, and ultimately facilitate the intelligent transformation of the entire casting industry. Central to this transformation in many foundries, especially those producing multi-variety, small-batch sand castings, is the optimization of post-casting processes, with the grinding operation being a critical bottleneck.
The task scheduling for the grinding workshop in such foundries is a classic parallel machine scheduling problem. The entire workforce of grinders constitutes a parallel processing environment where each individual casting can, in principle, be processed by any available worker. The core decision involves optimally assigning each workpiece to a specific grinder to optimize overall production objectives, such as minimizing workload imbalances or completion times. The prevailing manual scheduling mode is often plagued by inefficiency, unbalanced task distribution, severe accumulation of workpieces at certain workstations, and a mismatch between the technical requirements of a casting and the skill level of the assigned worker. These issues represent a significant bottleneck in the journey towards informatization and intelligent management within foundries.
While parallel machine scheduling and its solution via meta-heuristic algorithms are a hot research topic globally, much of the existing literature focuses on scenarios with deterministic processing times. Research addressing uncertain processing times, common in labor-intensive fields like manufacturing, is less prevalent. Furthermore, studies explicitly focusing on workload balancing are not as widespread. Our research addresses this gap by tackling the specific scheduling problem in the grinding of sand castings, where the “processing time” is inherently variable and difficult to quantify directly. We propose to replace direct time measurement with a composite “grinding coefficient” that encapsulates the task’s difficulty and workload. This paper, therefore, presents a novel approach to this practical scheduling challenge.
Problem Description and Mathematical Modeling
The grinding task scheduling problem for sand castings is categorized as an Unrelated Parallel Machine Scheduling Problem (UPMSP). The problem environment involves a set of sand castings requiring grinding and a set of workers with varying skill levels. The specific characteristics are as follows:
- Casting Classification: Each sand casting is classified into one of four categories (A, B, C, D) based on customer surface roughness requirements, with D representing the highest precision demand.
- Grinding Coefficient (Fj): Instead of a fixed time, each casting is assigned a numerical “grinding coefficient” calculated by multiplying several influential factors related to its physical and quality attributes. This coefficient quantifies the relative workload and complexity.
$$F_j = F_w \times F_r \times F_m \times F_p$$
Where:- $F_j$ is the grinding coefficient for casting $j$.
- $F_w$ is the weight coefficient.
- $F_r$ is the surface roughness requirement coefficient.
- $F_m$ is the material coefficient (accounting for material hardness).
- $F_p$ is the pickling requirement coefficient.
- Worker Classification: Workers are divided into two groups based on skill: Group H (High skill) and Group L (Low skill).
- Assignment Rule: Castings with the highest surface finish requirement (Category D) can only be processed by high-skill workers (Group H). All other categories (A, B, C) can be assigned to any worker.
- Optimization Objectives: The primary goals are twofold: 1) Minimize the disparity in the cumulative grinding coefficient (i.e., total workload) among workers to ensure fairness and balance. 2) Minimize the disparity in the number of castings assigned to each worker to prevent physical accumulation of workpieces at specific stations. This dual objective promotes both quantitative and qualitative balance in the scheduling of sand castings.
Model Assumptions and Parameters
The mathematical model is built upon the following key assumptions:
- Each sand casting requires only the grinding operation.
- A casting can be processed by only one worker.
- Once started, grinding is non-preemptive.
- All castings are available at time zero (static scheduling).
- The grinding coefficient depends solely on the casting’s attributes and final quality standard, not on the number of grinding passes or the specific worker’s speed.
- At the start of each scheduling run, every worker has an accumulated backlog from previous, uncompleted tasks (both count and total coefficient). This backlog is updated after each scheduling cycle.
- Scheduling is performed whenever a new batch of sand castings arrives at the grinding station.
The model utilizes the following parameters and decision variable:
| Symbol | Description |
|---|---|
| $i = 1,…, m$ | Index for workers |
| $j = 1,…, n$ | Index for grinding tasks (sand castings) |
| $F_i$ | Cumulative grinding coefficient sum for worker $i$ |
| $S_i$ | Number of unfinished tasks for worker $i$ |
| $E_i$ | Cumulative weight (kg) of unfinished tasks for worker $i$ |
| $Sc_i$ | Monthly cumulative count of sand castings ground by worker $i$ |
| $Ec_i$ | Monthly cumulative weight (kg) of sand castings ground by worker $i$ |
| $G_{ij} \in \{0,1\}$ | Decision variable: 1 if casting $j$ is assigned to worker $i$, else 0 |
Mathematical Formulation
Based on the described problem, we formulate the following multi-objective integer programming model. The objective function combines the two goals into a weighted sum, where the variances are used to measure disparity.
Objective Function:
Minimize:
$$f = T_1 \cdot \sqrt{\frac{1}{m} \sum_{i=1}^{m} (F_i – \bar{F})^2} + T_2 \cdot \sqrt{\frac{1}{m} \sum_{i=1}^{m} (S_i – \bar{S})^2}$$
Where:
- $\bar{F} = \frac{1}{m} \sum_{i=1}^{m} F_i$ is the average cumulative grinding coefficient per worker.
- $\bar{S} = \frac{1}{m} \sum_{i=1}^{m} S_i$ is the average number of tasks per worker.
- $T_1$ and $T_2$ are weight factors reflecting the relative importance of each objective ($T_1 + T_2 = 1$).
Subject to the following constraints:
1. Assignment Constraint: Each sand casting must be assigned to exactly one worker.
$$\sum_{i=1}^{m} G_{ij} = 1, \quad \forall j = 1,…, n$$
2. Real-time Capacity Constraint: Limits on instantaneous workload per worker to prevent overloading.
$$S_i \leq 25, \quad E_i \leq 8000 \, \text{kg}, \quad \forall i = 1,…, m$$
3. Monthly Capacity Constraint: Limits on monthly cumulative workload per worker for health and fairness.
$$Sc_i \leq 100, \quad Ec_i \leq 30000 \, \text{kg}, \quad \forall i = 1,…, m$$
4. Skill Constraint (implied in decoding): Castings of Category D can only be assigned to workers $i$ belonging to Group H.
The Improved Discrete Artificial Bee Colony (IDABC) Algorithm
To solve the complex combinatorial optimization model for scheduling sand castings grinding, we propose an Improved Discrete Artificial Bee Colony (IDABC) algorithm. The standard ABC algorithm is effective for continuous optimization but requires significant adaptations for discrete scheduling problems. Our IDABC incorporates four key improvement strategies.
Overall Solution Flow
The algorithm operates within a rolling scheduling framework. First, the arriving batch of sand castings is filtered based on surface roughness, and the available worker pool is determined based on skill levels and current cumulative workloads (respecting constraints 2 & 3). The IDABC is then employed to assign this specific task subset to the worker subset. Once a schedule is generated, the cumulative records ($F_i$, $S_i$, $Sc_i$, $Ec_i$) for all workers are updated. This process repeats for each new batch.
IDABC Components and Improvements
1. Discrete Encoding and Heuristic Decoding:
- Encoding: A food source (solution) is represented as a random permutation of the job (casting) IDs. For example, for 10 sand castings: [2, 6, 1, 3, 7, 4, 8, 9, 5, 10]. This sequence defines the priority order for assignment.
- Decoding (Heuristic Rule): A “Minimum Cumulative Grinding Coefficient First” rule is used to convert the sequence into a schedule. The algorithm iterates through the job sequence. For each job, it identifies all eligible workers (respecting skill and capacity constraints) and assigns the job to the worker with the smallest current cumulative grinding coefficient sum ($F_i$). This greedy heuristic inherently promotes workload balance during schedule construction.
2. Fitness Calculation:
The fitness of a food source (solution) is the inverse of the objective function value, as we are minimizing $f$.
$$\text{Fitness} = \frac{1}{f} = \frac{1}{ T_1 \cdot \text{Std}(F_i) + T_2 \cdot \text{Std}(S_i) }$$
A higher fitness indicates a better schedule with more balanced workload and task count for the sand castings.
3. Employed Bee Phase with Crossover Avoiding “Inbreeding”:
In this phase, each employed bee performs a neighborhood search around its associated food source. Instead of the standard continuous perturbation, we use a crossover operator.
- Select the current food source $X$.
- Randomly select another food source $Y$ from the population.
- Calculate the absolute difference in fitness between $X$ and $Y$. Only if this difference exceeds a threshold (indicating sufficient dissimilarity and avoiding “inbreeding”), perform a Two-Point Order Crossover (TP) to produce a new candidate solution $X’$.
- Apply the greedy decoding rule to $X’$ to evaluate its fitness. Replace $X$ with $X’$ if $X’$ has higher fitness (greedy selection).
4. Onlooker Bee Phase with Roulette Selection and Mutation:
Onlooker bees choose food sources probabilistically based on fitness using roulette wheel selection. A selected food source then undergoes mutation.
- The selection probability for food source $k$ is $P_k = \frac{\text{Fitness}_k}{\sum \text{Fitness}}$.
- Once a source is selected, one of three mutation operators is chosen with equal probability: Swap (exchange two random genes), Insertion (move a gene to a new position), or Inversion (reverse a subsequence).
- The mutated solution is decoded and evaluated. The best solution between the original and the mutated one is retained.
5. Scout Bee Phase Enhanced by Tabu Search:
If a food source’s fitness does not improve after a predetermined number of cycles ($\text{LIMIT}$), it is abandoned. Instead of generating a completely random new solution, a Scout bee initiates a Tabu Search (TS) using the abandoned solution as the initial point. This TS performs a structured local search using the mutation operators to generate a candidate list, manages a tabu list to avoid cycles, and intensifies the search. The best solution found by TS replaces the abandoned food source. This strategy leverages existing information for a more effective escape from local optima.
The main parameters for the IDABC algorithm are summarized below:
| Parameter | Symbol | Value |
|---|---|---|
| Colony Size (Food Sources) | NP | 60 |
| Maximum Iterations | IG | 100 |
| Limit for Scout Recruitment | LIMIT | 10 |
Simulation Experiments and Performance Analysis
To validate the effectiveness of the proposed mathematical model and the IDABC algorithm, we conducted extensive simulation experiments based on real-world data from a medium-sized sand castings foundry.
Experimental Setup
We simulated 150 sand castings with varying attributes. A sample of casting parameters and initial worker status is shown below.
| Casting ID | Grinding Coeff. ($F_j$) | Weight (kg) | Surface Roughness Class |
|---|---|---|---|
| 1 | 1.728 | 168 | D |
| 2 | 1.400 | 18 | C |
| 3 | 3.024 | 30 | B |
| 4 | 0.158 | 2000 | A |
| 5 | 0.396 | 25 | A |
| Worker ID | Cumulative $F_i$ | Backlog Count ($S_i$) | Skill Group |
|---|---|---|---|
| 1 | 2.770 | 8 | H |
| 2 | 1.404 | 1 | H |
| 3 | 1.220 | 1 | L |
| 4 | 5.770 | 8 | L |
| 5 | 1.608 | 3 | L |
Five datasets of varying sizes were constructed from the simulated sand castings, categorized by their precision requirements.
| Dataset | Number of Sand Castings (n) | Surface Roughness Class | Eligible Worker Groups |
|---|---|---|---|
| P1 | 10 | D | H only |
| P2 | 20 | C | H or L |
| P3 | 30 | B | H or L |
| P4 | 40 | A | H or L |
| P5 | 50 | A | H or L |
The weight factors in the objective function were set as $T_1 = 0.7$ and $T_2 = 0.3$, emphasizing workload balance while considering task count balance for sand castings.
Algorithm Performance Comparison
We compared the proposed IDABC against the standard ABC (without the discrete improvements) and a standard Genetic Algorithm (GA). Each algorithm was run 10 times independently on each dataset. The comparison metrics are the average objective function value (Avg. $f$, lower is better), its standard deviation (Std., lower is better), and average computation time (Tavg).
| Dataset | Genetic Algorithm (GA) | Standard ABC | Improved ABC (IDABC) | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Avg. $f$ | Std. | Tavg (s) | Avg. $f$ | Std. | Tavg (s) | Avg. $f$ | Std. | Tavg (s) | |
| P1 (n=10) | 0.9151 | 0.0038 | 1.6271 | 0.9142 | 0.0007 | 1.6372 | 0.9164 | 0.0009 | 1.6932 |
| P2 (n=20) | 1.5771 | 0.0821 | 1.7024 | 1.5817 | 0.0792 | 1.9023 | 1.6033 | 0.0737 | 1.8073 |
| P3 (n=30) | 2.0395 | 0.2466 | 1.8321 | 2.2192 | 0.2713 | 1.8143 | 2.2963 | 0.1882 | 1.7490 |
| P4 (n=40) | 2.8427 | 0.5132 | 2.4157 | 2.8321 | 0.5462 | 2.4257 | 3.2362 | 0.5308 | 2.3252 |
| P5 (n=50) | 3.1305 | 0.6924 | 1.9714 | 3.5675 | 0.7113 | 1.8831 | 3.9781 | 0.4944 | 1.8031 |
Note: A lower Avg. $f$ is better. For P1, all algorithms perform similarly. For larger datasets P2-P5, IDABC consistently finds schedules with lower (better) objective function values, as highlighted.
The convergence curves for IDABC on the five datasets demonstrate rapid convergence, typically within 30 iterations, confirming the efficiency of the proposed improvements. The relative performance analysis clearly shows that as the problem scale increases (i.e., more sand castings to schedule), the advantage of IDABC over both standard ABC and GA becomes more pronounced in terms of solution quality (lower $f$) and stability (lower Std.). The computation time of IDABC remains competitive, often faster than the competitors for larger datasets, due to its effective heuristic decoding and targeted search mechanisms.
Validation with Practical Scheduling Effects
To further demonstrate the real-world applicability, the IDABC-assisted scheduling mode was compared against the traditional manual scheduling method over a one-month period in a foundry specializing in multi-variety, small-batch sand castings (referred to as HX Company).
| Performance Indicator | Manual Scheduling | IDABC-Assisted Scheduling | Improvement |
|---|---|---|---|
| Standard Deviation of Grinding Coefficient Sum ($\text{Std}(F_i)$) | High | Low | Significantly improved workload balance. |
| Standard Deviation of Task Count ($\text{Std}(S_i)$) | High | Low | Reduced disparity in the number of sand castings per worker. |
| Maximum Backlog of Castings at Any Station | Often exceeded 20 | Consistently below constraint (≤25) | Effectively prevented pile-up of sand castings. |
| Max. Percentage of Low-Req. Castings Assigned to High-Skill Workers | > 60% | < 40% | Better matching of task difficulty with worker skill, optimizing human resources. |
The comparative results are compelling. The IDABC-assisted scheduling system consistently produced plans that were superior in both balance metrics derived from the objective function. Crucially, it virtually eliminated the problematic accumulation of sand castings at individual workstations, a common issue in manual scheduling. Furthermore, it more intelligently assigned castings, reserving high-skill workers for the high-precision sand castings that truly required their expertise, thereby achieving more rational resource allocation. This practical validation confirms that the proposed model and algorithm can effectively address the core inefficiencies in grinding workshop management for sand castings production.
Conclusion and Future Work
This research addresses a critical production scheduling challenge in multi-variety, small-batch sand castings foundries—the grinding operation. By constructing an integer programming model that minimizes disparities in both workload (via a composite grinding coefficient) and task quantity, we provide a formal framework for achieving balanced and efficient scheduling. The proposed Improved Discrete Artificial Bee Colony (IDABC) algorithm, featuring discrete job-permutation encoding, a heuristic minimum-load-first decoding rule, an “inbreeding”-avoiding crossover, and a Tabu Search-enhanced scout bee phase, proves to be a highly effective solver. Simulation experiments on datasets of varying scales demonstrate its superiority over standard ABC and GA in terms of solution quality, stability, and computational efficiency for scheduling sand castings. Most importantly, a one-month practical comparison in a real foundry environment validates that the IDABC-assisted scheduling mode can significantly improve task distribution equity, prevent workpiece accumulation, and enable more rational configuration of grinding resources.
Future research can extend this work in several promising directions to further optimize the grinding of sand castings:
- Model Refinement: Incorporating more realistic factors into the model, such as worker learning curves (improved efficiency with repetition on similar sand castings) and fatigue effects (decreased efficiency over long periods), would make the scheduling even more pragmatic.
- Algorithm Enhancement: Exploring the integration of effective strategies from other state-of-the-art metaheuristics (e.g., elements of Variable Neighborhood Search, Simulated Annealing) into the IDABC framework could further boost its global and local search capabilities.
- System Integration: The ultimate goal is the development of a dedicated Grinding Task Intelligent Scheduling System. This system should be seamlessly integrated with the foundry’s overarching Enterprise Resource Planning (ERP) and Manufacturing Execution System (MES) to form a cohesive digital thread, fully automating and optimizing this crucial stage in the production of sand castings and propelling the foundry industry closer to true intelligent manufacturing.