Effective control of the heat-treatment operation is essential to reducing manufacturing cost in mould manufacturing. The heat-treatment shop floor is a flow shop with parallel batch processors and incompatible jobs. The jobs differ from each other in product types, sizes, release times, and due-dates. The scheduling objective is to minimize manufacturing cost, including energy cost, subcontracting cost, and jobs’ tardiness penalties cost. A hierarchical production planning structure is proposed, which contains three decision-making levels: (1) balancing capacity and demand, (2) machining at the quenching stage, and (3) machining at the tempering stage. At the first level, the periodic rolling scheduling heuristic is proposed for the purpose of balancing the capacity and the demand in the coming period in the heat-treatment shop floor. At the second and third levels, two new look-ahead batching heuristics are proposed. An extensive computational experiment is conducted to verify the effectiveness of the proposed heuristics.
Product quality, delivery, and price are three elements for which customers are concerned in the mould market. Most jobs of mould production need a heat-treatment operation between cutting machining and finish machining. The heat-treatment process is very important to ensure the precision, intension, and the life-span of mould production (cf. [
Heat-treatment shop floor is a flow shop which contains the quenching process and the tempering process. The flow chart of heat-treatment shop floor is shown in Figure
The flow chart of heat-treatment shop floor.
Heat-treatment is usually not the first operation in the mould manufacturing system. Jobs arrive at the heat-treatment shop floor in a dynamic way from upstream operations. Since the mould is an order oriented production, the job processing time is not always the same as expectation for a new mould. If the production planning of the heat-treatment shop floor is made by a fixed period, the actual batches and machining times will be usually different than the plan, since the arrival times of the jobs at the heat-treatment shop floor are always different from expectation. How to establish an effective scheduling mechanism to improve the robustness of the production plans, this issue has plagued the workshop manager. The heat-treatment furnace is the bottleneck resource in the mould manufacturing system. On the production season of demand greater capacity, in order to reduce the amount of lateness, manager often subcontracts some mould jobs to other companies. However, subcontracting needs longer preparation times and higher cost. Therefore, how to choose the suitable subcontracting jobs is another decision issue in this study.
This paper focuses on the dynamic control of parallel batch machines in the mould heat-treatment shop floor, which is subject to nonidentical jobs with different product types, sizes, release times, and due-dates. A hierarchical production planning structure is proposed to minimize manufacturing cost, including energy cost, subcontracting cost, and jobs’ tardiness penalties cost. A periodic rolling scheduling strategy is proposed to select the subcontracting jobs. Based on a look-ahead batch control strategy, two machining control strategies at the quenching stage and tempering stage are proposed, respectively.
The remainder of the paper is organized as follows. In Section
Production control of a batch process has been widely studied in the literature. There are two decision approaches to assign resources to jobs: scheduling and real-time control.
Scheduling is the task of allocating available production resources (machine, labor, material, etc.) to jobs over time with full knowledge of future job arrivals and system status (cf. [
Real-time control strategies of batch processing machine (BPM) can be categorized into two policies, threshold policy and look-ahead policy. The threshold policy is applied when there is no future arrival information available. Neuts [
The look-ahead policy is applied when limited near-future arrival information is available to the decision-maker. Glassey and Weng [
Based on LAB strategy, three control strategies are proposed in this paper for the three decision-making stages in a mould heat-treatment shop floor with the objective of minimizing manufacturing cost, including energy cost, subcontracting cost, and jobs’ tardiness penalties cost. The scheduling environment we study is in a flow shop with parallel batch machines and nonidentical jobs. To the best of our knowledge, this is the first application of LAB strategy in this regard.
There are
The objective is to minimize manufacturing cost C, which includes jobs’ tardiness penalties cost
The following notations are used throughout this paper: I: Set of product types W: Capacity of a machine
The objective function is described as follows:
The near-future necessary information can be obtained from the computer integrated manufacturing environment. However, due to various stochastic and uncertain sources in mould manufacturing, the arrival times of jobs are always inaccurate. Generally speaking, only the arrival times of jobs in the most upstream operation are relatively accurate (called predicted future), while the arrival times of jobs in other upstream operations are difficult to predict accurately, which are called the roughly predicted future or random future. The division of the jobs in the upstream operation is shown in Figure
The division of the jobs in the upstream operation.
The mould jobs arrive at the heat-treatment shop floor in a dynamic way from upstream shop floors. The workshop manager must make the production planning for the jobs and choose some jobs to be subcontracting when the demand is greater than the capacity. Traditionally, the production planning is made by a fixed period. However, if the fixed period is too long, the actual batches and machining times of the batches will be different from the production planning, since the release times of the jobs in the roughly predicted future are inaccurate; if the fixed period is too short, there is not enough time for subcontracting, since subcontracting needs a longer preparation time. Therefore, the batch plan, machining plan, and subcontracting plan are hard to be made simultaneously. Hierarchical production planning is proposed here to solve the issue. As shown in Figure
The flow chart of hierarchical production planning.
A periodic rolling scheduling strategy (PRS) is used to balance capacity and demand in the coming period, which does not need to assign the mould lot to a specific machine. The decision point is rolling by a fixed period. The subcontracting strategy is often used in the heat-treatment shop floor. Since subcontracting requires a longer preparation time, the control decision needs to be better at looking forward. More upstream operations should be considered in the predictive horizon. The upcoming jobs in the predicted future and roughly predicted future will be considered at the decision point. The jobs needed to be processed in this workshop will be placed in the buffer, which is assumed to have an unlimited storage capacity.
The quenching stage machining strategy (QSM) is used to choose the suitable jobs and put them in a batch and determine the starting time of the batch. The decision point is defined as one of the quenching furnaces becomes available or a new job arrives at an idle quenching furnace. Robustness of the control decision is important at this stage. Hence, only the jobs in the buffer and the upcoming jobs in the predicted future are considered at the decision point.
Assume that all the furnaces for quenching and tempering have the same capacity. Every batch will not be split between the quenching stage and tempering stage. When there is more than one batch waiting for processing, the controller must choose which batch to be processed first. Traditionally, the first come first served (FCFS) rule is applied to solve this issue. However, the jobs with higher priority usually cannot be processed first by the FCFS rule, which usually causes a lot of tardiness penalties cost. The tempering stage machining (TSM) strategy is used to choose the suitable batch to be processed first. The decision point is defined as one of the quenching furnaces becomes available or a new batch arrives at an idle tempering furnace.
There are 10 jobs to be processed. All the jobs are in the buffer or in the predicted future. The release times (
The parameters of the jobs.
J1 | J2 | J3 | J4 | J5 | J6 | J7 | J8 | J9 | J10 | |
| ||||||||||
| 0 | 0 | 0 | 0 | 1 | 3 | 4 | 0 | 0 | 2 |
| ||||||||||
| 18 | 19 | 20 | 21 | 22 | 23 | 18 | 24 | 23 | 22 |
| ||||||||||
| 60 | 50 | 30 | 35 | 35 | 30 | 30 | 20 | 40 | 40 |
| ||||||||||
| 10 | 20 | 30 | 40 | 50 | 10 | 20 | 30 | 40 | 40 |
| ||||||||||
i | A | A | A | A | A | A | A | A | B | B |
The periodic rolling scheduling heuristic is introduced in this section. The purpose of this heuristic is to balance the capacity and the demand in the coming period in the heat-treatment shop floor. Look-ahead in the decision point for near-future arrivals of the upcoming jobs within a fixed period is called the look-ahead window in this section. Since subcontracting needs preparation time, a wider look-ahead window can get better control. However, it is difficult to accurately predict the arrival times of the jobs which arrive a week later. Hence, the width of look-ahead window is set to be 6 days. Also, the adjustment of the look-ahead window is allowed.
Since the arrivals of the jobs are not in a uniform manner, the jobs may arrive intensively in some periods of the look-ahead window, while being rare in arrival in other periods. In order to control the load in each period of the look-ahead window, the look-ahead window is uniformly divided into several control cycles. The number of control cycles in the look-ahead window can be set by the controller. Since the jobs need not be assigned to a specific machine in this scheduling strategy, hence only the capacity of each control cycle needs to be calculated, without the need to analyze the capacity of each machine. The capacity of a control cycle is equal to the sum of working hours of all the machines in this control cycle. The jobs will be classified according to product types and sorted in nondecreasing order of due-dates. Then, place the jobs in batches and calculate the demands of the quenching stage and the tempering stage, respectively. The demand of the quenching stage is equal to the sum of the quenching processing times of all the batches. The demand of the tempering stage is calculated in the same way.
Once the demand is greater than the capacity at one of the two stages in a control cycle, some suitable jobs are chosen to be subcontracting or left to the next control cycle. Which are the suitable jobs? The suitable jobs are chosen by the following two principles. (1) The total size of the subcontracting jobs is as light as possible, since subcontracting needs more cost. (2) Assuming the jobs which left to the next control cycle will be started at the earliest time of the next control cycle, if the amount of tardiness by subcontracting is less than that when they left to the next control cycle, the job will be subcontracting. The procedure of the period rolling scheduling strategy is proposed as follows.
Set the width of look-ahead window to be 20 hours. By running the PRS algorithm, the J8 in Example
Step cycle, respectively. Step and which arrive in the current control cycle. Step in the same batch.If the machine capacity is exceeded when a job is added to a batch, close the batch and place the job in a new batch. Step capacitiesat both stages respectively, put the jobs into the buffer and go to step Step time of the next control cycle. If the amount of tardiness by subcontracting is less than when left to the next control cycle, the job will be subcontracting; Else, the job will be left to next control cycle. Go to step 4. Step
The quenching stage machining heuristic is to choose the suitable jobs and put them in the batch for the current machine processing at the quenching stage. The LAB strategy has been widely used to control a single batch machine with a single product type. In this section, the LAB strategy will be refined for multiple parallel batch machines and multiple product types. The quenching stage machining heuristic follows a four-step algorithm.
In common practice, at any decision point, a batch is formed from the arrived jobs which are in the buffer. However, there are often cases of a high priority job arriving in the near future, which needs to be given priority for loading. Therefore, look-ahead at the decision point for the upcoming jobs in the predicted future. In order to ensure the robustness of scheduling, only the jobs in the buffer and the upcoming jobs in the predicted future are considered at the decision point.
For each product type, at the decision point or at any time a new job arrives, a suggested scenario is formed. Each suggested scenario forms a batch in which all the arrived jobs of the same type are considered. The suggested start processing time of the batch is the latest arrival time of all the jobs considered. The size of the batch is equivalent to either partial or full machine capacity. If the total size of the considered jobs in the suggested scenario is larger than the machine capacity, the suitable jobs are chosen by combining the earliest due-date (EDD) rule and largest size (LS) rule.
According to the data in Example
An example for constructing suggested scenarios for product type A.
The suggested scenario, where the sum of surplus capacity cost and penalty cost is the lowest, is set to be the suggested decision. The surplus capacity cost is defined as energy wastage due to the batch being not full. Suppose that there are n0 jobs of type i in the buffer and q0 jobs of type i in the predicted future. At a particular decision point
Suppose that the
The best suggested decision will be obtained by evaluating all the suggested decisions for each product type. The following principles will be used to select the best suggested decision.
After four steps of the QSM algorithm, the processing sequence of the jobs in the quenching stage will be determined. The Gantt chart for the jobs in Example
The Gantt chart for the jobs at the quenching stage.
When one of the tempering furnaces becomes available and there is only one batch waiting for processing, or when the new batch arrives at an idle tempering furnace, the batch will put on the tempering furnace immediately. However, when one of the tempering furnaces becomes available and there is more than one batch waiting for processing, the suitable batch must choose to be processed first. An algorithm based on a look-ahead policy is proposed in the following.
Set t1 as the idle time of the current machine; assuming each batch is processed at t1, calculate the tardiness penalty cost for each batch, respectively, by the following equation:
Set t2 as the idle time of the next available machine; assuming each batch is processed at t2, calculate the tardiness penalty cost for each batch, respectively, by the following equation:
Calculate
In Example
The Gantt chart for the jobs at the tempering stage.
An experiment is designed to demonstrate the potential of the proposed heuristics. The PRS heuristic is used to select the subcontracting jobs, and then the QSM heuristic and the TSM heuristic are used to put the jobs in batches and determine the order of each batch at both stages. The proposed heuristics will be compared with other rules. The setting minimum waiting jobs (SMWJ) rule and the setting maximum waiting time (SMWT) rule are widely adopted in the real-world mould heat-treatment shop floor. In SMWJ, once one of the types of the cumulative sizes of the arrived jobs reaches a minimum value, put these jobs in a batch and arrange the batch on the furnace. In SMWT, once the waiting time reaches a maximum value, the types in which the cumulative sizes of the arrived jobs are largest form a batch to be processed immediately.
At the quenching stage, consider the SMWJ rule only as a benchmark first. When one of the quenching furnaces becomes available, start the SMWJ rule. The minimum size is set to be 80% batch capacity W. After a lot of experiments, it is found that the performance of setting the minimum size to 60%, 70%, and 90% is inferior to 80%, so this paper only records the experimental results of 80%. Combine the SMWJ and SMWT rules as another benchmark for the quenching stage. When one of the quenching furnaces becomes available, start the SMWJ rule first; once the waiting time reaches the maximum value and no batch is formed yet, the types in which the cumulative sizes of the arrived jobs are largest form a batch. The maximum waiting time for SMWT is set to be (1/3) the mean quenching processing time (PT). At the tempering stage, consider the FCFS rule as a benchmark. It is a simple rule which is much applied in both practice and theory. The constructed four control strategies and their corresponding settings are listed in Table
Design of the simulation study.
Factor | Settings | |
---|---|---|
1 | Control strategy | PQT: PRS + QSM + TSM |
| ||
QT: QSM + TSM | ||
| ||
SF: SMWJ (80% | ||
| ||
SSF: SMWJ (80% | ||
| ||
2 | Number of product types | 2, 4 and 6 |
| ||
3 | Release time distribution ( | uniform |
| ||
4 | Due-date distribution ( | |
| ||
5 | Traffic intensity ( | 0.6, 0.7, 0.8, 0.9, 1, 1.1 and 1.2 |
| ||
6 | Processing time | Operation in the most upstream of quenching: uniform(1,20) hours |
| ||
Quenching ( | ||
| ||
Tempering ( | ||
| ||
7 | Subcontracting parameters | Preparation time: 10 hours |
| ||
Duration time: 1.2 | ||
| ||
Cost:10 | ||
| ||
15 | ||
| ||
8 | Tardiness penalty cost rate | 10 |
| ||
9 | Energy cost rate | 100 |
| ||
10 | Capacity of furnace | 500 L |
| ||
11 | Size of job | Uniform(20, 200) L |
| ||
12 | Number of furnaces ( | (7, 5) |
The release times of jobs are related to the traffic intensity
In this paper, since the jobs are nonidentical in size and there are multiple product types and machines, redefine the traffic intensity
Considering the production characteristics in the company, the length of the predictive horizon is set to be 144 hours. All of the jobs in the predictive horizon are generated by the settings in Table
The preceding section discussed the design of the computational experiment, the outcomes of which will be analyzed in this section. The manufacturing costs for alternative product types and traffic intensities are displayed in Figures
Manufacturing cost.
2 product types
4 product types
6 product types
Figures
Compared with the three figures in Figure
Figures
Tardiness penalties cost.
2 product types
4 product types
6 product types
This paper is motivated by mould heat-treatment processing in dynamic environments. The heat-treatment shop floor is a flow shop with parallel batch processors and nonidentical jobs. The jobs differ from each other in product types, sizes, and due-dates. The objective is to minimize the manufacturing cost, which includes energy cost, subcontracting cost, and tardiness penalties cost. The heat-treatment shop floor is in a sequential production system where jobs arrive in a dynamic way from upstream shop floors. According to the accuracy of arrival information, the arrival times of the jobs are divided into three situations: predicted future, roughly predicted future, and random future. A hierarchical production planning structure is proposed for the problem, which contains three decision-making levels:
The PRS algorithm is proposed to balance the capacity and the demand in the coming period in the heat-treatment shop floor. Some jobs are chosen to be subcontracted when the demand is greater than the capacity. Since subcontracting needs preparation time, a wider look-ahead window can obtain better control. Hence, the upcoming jobs in the predicted future and roughly predicted future are considered at this stage. Two new LAB strategies are exploited for the machining at the quenching stage and the tempering stage, called QSM algorithm and TSM algorithm, respectively. The LAB strategy is refined for multiple parallel batch machines and multiple product types in these two algorithms. Robustness of the control decision is important at the quenching stage. Hence, only the jobs in the buffer and the upcoming jobs in the predicted future are considered at the decision point.
An extensive computational experiment is conducted to compare the performance of the proposed strategies with the benchmark strategies. Results indicate that the manufacturing cost can be obviously decreased by running the PRS algorithm, QSM algorithm, and TSM algorithm, especially when the demand is greater than the capacity. However, the PRS strategy does not contribute much to reducing manufacturing cost when the traffic intensity values are low or moderate. The performance of the PRS strategy is better when the number of product types is low.
In future work, this study can be extended to address different optimization objectives, such as inventory-related performance. Since high holding cost of products is not common in practice, other interesting extensions of the model may address the limitations on buffer size, setup times, and transportation times between stations. The effectiveness of these strategies might be improved by considering these characteristics in the problem.
The data used to support the findings of this study are available from the corresponding author upon request.
The experimental data are generated by random way, and the method of generating data is explained in detail in Section
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This article is supported by the Hanshan Normal University doctoral startup project (QD201804024), the Pearl River S&T Nova Program of Guangzhou (201710010004), and the Special Plan Young Top-notch Talent of Guangdong (2016TQ03X364).