A Dual-List Feature-Driven Heuristic for the Batch Processing Machine Scheduling Problem ^†

Abstract

This paper addresses the multi-objective scheduling problem for unrelated parallel batch processing machines under different job arrival times. We propose a dual-list feature-driven (DLFD) heuristic algorithm to simultaneously minimize the completion time, total delay time, and total energy consumption. Firstly, the heuristic selects a machine based on the machine’s capacity and energy consumption characteristics. Secondly, a job is selected from two candidate job lists governed by machine capacity and batch processing time constraints, thereby reducing the search space and improving solution quality. To validate the effectiveness of the DLFD heuristic, experiments of three different scales were designed to compare its performance against classic composite dispatching rules. The results demonstrate that the proposed heuristic achieves a significantly superior Pareto front compared to the traditional rules and exhibits strong robustness in solving problems of various scales.

1. Introduction

The scheduling problem of unrelated parallel batch processing machines (UPBPM) is prevalent in various industrial applications, such as semiconductor wafer fabrication, chemical reactors, and steel heat treatment. Its core objective is to enhance key performance indicators (e.g., total completion time, energy consumption) by optimizing the job sequence on machines [1] and the formation of job batches, all under a series of constraints including machine capacity and job priorities. However, due to the interdependence between job sequencing and batch formation decisions, coupled with the combinatorial nature of the problem, it has been proven to be NP-hard [2]. Given the inherent computational complexity of UPBPM, the design and application of efficient dispatching rules has consequently emerged as a new focus of research.

In the UPBPM research area, efficient scheduling rules are crucial for solving batch processor scheduling problems. These rules not only provide high-quality starting points for complex scheduling heuristics, but also directly affect the merit of the final solution and the heuristic’s solution efficiency. The current research has fully proved the necessity of heuristic innovation. For the UPBPM scheduling problem, Elias and Joseph constructed a hierarchical heuristic by combining classical rules (First Fit (FF)/Best Fit (BF)/Longest Processing Time (LPT)/Earliest Due Date (EDD)) with dynamic policies (rolling window) [3]. When addressing batch processing machine scheduling problems that consider energy consumption, Ju and Wu et al. identified the introduction of resource constraints as a key strategy for obtaining superior solutions [4]. Similarly, in the context of complex constraints in multi-stage batch processing, Jia and Huo et al. emphasized the importance of dispatching rules for generating feasible solutions, citing assignment rules like FAM and NQM and the batching rule FCFS as examples [5]. In conclusion, these studies collectively indicate that the effectiveness of the chosen dispatching rules directly impacts the quality of the final solution.

However, traditional dispatching rules (e.g., FF/SPT) in UPBPM applications are often confined to phased solution generation strategies [6]. The architectural fragmentation of this method weakens the correlation of complex scheduling decisions [7], triggers the failure of resource coordination, and thereby suppresses the global optimization of the solution space quality. To overcome this limitation, this study proposes a Dual-List Feature-Driven (DLFD) heuristic, aiming to provide a more efficient and coordinated method for solving this class of complex scheduling problems.

2. Overview of the Problem

The problem studied in this paper can be described as the assignment and batching of $\mathrm{n}$ jobs $\{\mathrm{i}=1,2,\dots ,\mathrm{n}$} on $\mathrm{m}\{\mathrm{i}=1,2,\dots ,\mathrm{m}\}$ parallel batch processing machines. The machines are considered unrelated, meaning that the capacity (${\mathrm{Q}}_{\mathrm{i}}$), processing power (${\mathrm{P}}_{\mathrm{i}}^{\mathrm{p}}$), and idle power (${\mathrm{P}}_{\mathrm{i}}^{\mathrm{i}}$) can differ for each machine ${\mathrm{M}}_{\mathrm{i}}$. Furthermore, the processing time of a job ${\mathrm{J}}_{\mathrm{j}}$ on a machine ${\mathrm{M}}_{\mathrm{i}}$ is also machine-dependent, denoted as ${\mathrm{p}}_{\mathrm{i}\mathrm{j}}$. Each job ${\mathrm{J}}_{\mathrm{j}}$ is characterized by a size ${\mathrm{s}}_{\mathrm{j}}$, an arrival time ${\mathrm{r}}_{\mathrm{j}}$, and a due date ${\mathrm{d}}_{\mathrm{j}}$. Considering the actual machining process, the batch machining start time needs to be greater than or equal to the maximum arrival time of the jobs in the batch. In order to reduce production cost and increase customer satisfaction, the optimization objective is to minimize the makespan (${\mathrm{C}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$), total tardiness ($\mathrm{T}\mathrm{T}$), and total energy consumption ($\mathrm{T}\mathrm{E}\mathrm{C}$).

This section clearly presents the main variables, parameters used in the model, and their corresponding mathematical symbols, as shown in

Table 1. Symbols and definitions.

Symbols	Definitions
${\mathrm{x}}_{\mathrm{i}\mathrm{j}\mathrm{k}}$	If the job $\mathrm{j}$ is assigned to the $\mathrm{k}$th batch of processing on machine $\mathrm{i}$, the value is 1; otherwise, it is 0.
$S_{ik}$	The starting processing time of the $\mathrm{k}$th batch on machine $\mathrm{i}$
$P_{ik}$	The processing time of the $\mathrm{k}$th batch on machine $\mathrm{i}$
$C_{ik}$	The completion time of the $\mathrm{k}$th batch on machine $\mathrm{i}$
$C_j$	Completion time of job $\mathrm{j}$
$T_j$	The delay time of job $\mathrm{j}$

. It also elaborates on the core mathematical model, as presented in

Table 2. Objective function and constraint definitions.

Objective Function	Mathematical Expression
Makespan	$\mathrm{m}\mathrm{i}\mathrm{n}{\mathrm{C}}_{\mathrm{i}}$
Total tardiness	$\mathrm{T}\mathrm{T}={\sum }_{\mathrm{j}=1\mathrm{n}}{\mathrm{T}}_{\mathrm{j}}$
Total energy consumption	$\mathrm{T}\mathrm{E}\mathrm{C}$= ${\sum }_{\mathrm{i}=1}^{\mathrm{m}}(\sum ({\mathrm{P}}_{\mathrm{i}\mathrm{k}}\cdot {\mathrm{P}}_{\mathrm{i}\mathrm{p}})+({\mathrm{C}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-\sum {\mathrm{P}}_{\mathrm{i}\mathrm{k}})\cdot {\mathrm{P}}_{\mathrm{i}\mathrm{s}})$
Constraint definitions	Mathematical expression
Machine capacity	${\displaystyle \sum _{\mathrm{j}=1}^{\mathrm{n}}}{\mathrm{s}}_{\mathrm{j}}\cdot {\mathrm{x}}_{\mathrm{i}\mathrm{j}\mathrm{k}}\le {\mathrm{Q}}_{\mathrm{i}},\forall \mathrm{j}\in \{1,\dots ,\mathrm{n}\},\mathrm{k}$
Batch processing time	${\mathrm{P}}_{\mathrm{i}\mathrm{k}}=\max \left\{{\mathrm{p}}_{\mathrm{i}\mathrm{j}}\cdot {\mathrm{x}}_{\mathrm{i}\mathrm{j}\mathrm{k}}\right\},\forall \mathrm{j}\in \{1,\dots ,\mathrm{n}\},\mathrm{k}$
Batch start time	${\mathrm{S}}_{\mathrm{i}\mathrm{k}}\ge \max \{{\mathrm{r}}_{\mathrm{j}}\cdot {\mathrm{x}}_{\mathrm{i}\mathrm{j}\mathrm{k}}\},\forall \mathrm{i}\in \{1,\dots ,\mathrm{m}\},\forall \mathrm{k}$ ${\mathrm{S}}_{\mathrm{i}\mathrm{k}}\ge {\mathrm{C}}_{\mathrm{i},\mathrm{k}-1},\forall \mathrm{i}\in \{1,\dots ,\mathrm{m}\},\forall \mathrm{k}>1$
Batch completion time	${\mathrm{C}}_{\mathrm{i}\mathrm{k}}={\mathrm{S}}_{\mathrm{i}\mathrm{k}}+{\mathrm{P}}_{\mathrm{i}\mathrm{k}},\forall \mathrm{i}\in \{1,\dots ,\mathrm{m}\},\forall \mathrm{k}$
Completion time of the jobs	${\mathrm{C}}_{\mathrm{j}}={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}}{\displaystyle \sum _{\mathrm{k}}}{\mathrm{C}}_{\mathrm{i}\mathrm{k}}\cdot {\mathrm{x}}_{\mathrm{i}\mathrm{j}\mathrm{k}},\forall \mathrm{j}\in \{1,\dots ,\mathrm{n}\}$
Job delay time	${\mathrm{T}}_{\mathrm{j}}=\max \{0,{\mathrm{C}}_{\mathrm{j}}-{\mathrm{d}}_{\mathrm{j}}\},\forall \mathrm{j}\in \{1,\dots ,\mathrm{n}\}$
Maximum completion time	${\mathrm{C}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\ge {\mathrm{C}}_{\mathrm{j}},\forall \mathrm{j}\in \{1,\dots ,\mathrm{n}\}$

.

Given the uncorrelated nature of parallel batch machines and the dynamic arrival characteristics of the jobs, the research problem involves searching for a solution to multi-objective collaborative optimization in a complex and large decision space, i.e., the combined space of machine assignment of the jobs, grouping of jobs into batches on the machine, and batch machining sequencing. In order to efficiently search for a Pareto solution set with better diversity and convergence, the optimization characteristics of the research problem were analyzed, and two optimization principles based on the DLFD heuristic were obtained as follows:

(1) According to the $\mathrm{T}\mathrm{E}\mathrm{C}$ objective, the machine power is constant, and the total energy consumption of the system is positively related to the total number of processing batches and idle time. When the capacity of the machine is larger, the more jobs can be put into a batch, the total processing energy consumption is smaller. Thus, the jobs are prioritized for the machine with lower processing energy power.

(2) To optimize the ${\mathrm{C}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $\mathrm{T}\mathrm{T}$ targets, batches can be constructed with a preference for joining jobs with processing durations smaller than the batch capacity. This strategy helps to form a batch structure with a more uniform processing time, which significantly reduces the blocking effect of individual long-duration jobs and effectively reduces the idle waiting time of the machine. This optimization mechanism contributes directly to shortening the overall completion cycle and reducing the accumulation of delays due to operational delays.

3. Dual-List Feature-Driven Heuristic

In UPBPM problems, there is often an intrinsic conflict between the optimization objectives: on the one hand, the core claim of minimization is to reduce the number of batches as much as possible (in order to reduce the additional energy overhead caused by equipment startup or idling); on the other hand, the pursuit of the shortest completion times, ${\mathrm{C}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $\mathrm{T}\mathrm{T}$, e.g., to minimize the total delay or to maximize the number of jobs meeting the delivery date, is highly dependent on the creation of batches with consistent processing durations. The inherent contradiction between ${\mathrm{C}}_{\mathrm{m}\mathrm{a}\mathrm{x}}/\mathrm{T}\mathrm{T}/\mathrm{T}\mathrm{E}\mathrm{C}$ goals makes scheduling strategies that rely on a single traditional principle often lose sight of the other and limit overall efficiency. In order to effectively reconcile this conflict and achieve synergistic optimization of the dual objectives, the DLFD heuristic is designed in this study. The core mechanism of this heuristic aims to dynamically trade off the batch size with the dispersion of the processing time of the jobs within the batch so as to achieve efficient synergies between the two competing features of minimizing the number of batches and maintaining the consistency of the batch processing time.

The heuristic injects controllable randomness perturbations through a probabilistic selection mechanism to expand the solution space exploration capability. Firstly, the eigenvalues of machine ${\mathrm{M}}_{\mathrm{i}}$ are determined based on the capacity and processing energy consumption of the machine in Equation (1), and the selection probability of each machine is calculated based on Equation (2). After the target machine ${\mathrm{M}}_{\mathrm{i}}$ is selected using a roulette strategy, an empty batch is initialized when the batch creation condition is met (current machine candidate artifact set is empty). The dynamic candidate artifact filtering mechanism is then activated to filter the set of suitable artifacts to be added to the batch, i.e., the highest prioritized candidate list, based on the current batch status. Finally, taking into account the job information of batch processing time, batch arrival time, delivery time, etc., a roulette strategy is used to select a job to be added to the batch, remove the information of the job that has already been put into the batch and update the candidate list. Repeat the process of job selection and machine selection until all jobs have been dispensed.

$${MFI}_i=Q_i\times {\displaystyle \frac{1}{P_i^p}}$$

(1)

$${PM}_i={\displaystyle \frac{{MFI}_i}{{\sum }_{i=1}^m{MFI}_i}}$$

(2)

In order to optimize the quality of batch artifact combinations, a two-tier screening candidate list mechanism is constructed based on Equations (3) and (4). This mechanism significantly improves the search efficiency and convergence quality by incrementally shrinking the feasible solution space, which shows that the priority setting of the candidate list is negatively correlated with the size of the feasible solution space: the stronger the constraints, the higher the convergence of the feasible solutions, and the higher the preference.

$$L_{ib}^1=\{J_j|s_j\le Q_i-{\sum }_{J_j\in B_{ib}}{s}_j,J_j\in J\}$$

(3)

$$L_{ib}^2=\{J_j|P_{ib}-p_{ij}\ge 0,J_j\in L_{ib}^2\}$$

(4)

Candidate list ${\mathrm{L}}_{\mathrm{i}\mathrm{b}}^1$ limits the size of the jobs added to the current batch to be smaller than the remaining batch capacity of the machine. Candidate list ${\mathrm{L}}_{\mathrm{i}\mathrm{b}}^2$ further limits the jobs in candidate list ${\mathrm{L}}_{\mathrm{i}\mathrm{b}}^1$ to those jobs for which the processing time of the batch does not increase after the addition of the current batch.

Selecting the most suitable job from the candidate list, and considering the problem characteristics and optimization objectives, the eigenvalue and selection probability calculation of batch ${\mathrm{B}}_{\mathrm{i}\mathrm{b}}$ for job arrangement to machine ${\mathrm{M}}_{\mathrm{i}}$ are designed as shown in Equations (5) and (6). Jobs are selected to be placed in the batch based on the selection probability of the jobs using a roulette strategy. In Equation (5), if there is no job in the batch, then ${|\mathrm{P}}_{\mathrm{i}\mathrm{b}}-{\mathrm{p}}_{\mathrm{i}\mathrm{j}}|=0$, ${|\mathrm{R}}_{\mathrm{i}\mathrm{b}}-{\mathrm{r}}_{\mathrm{j}}|=0$, ${\mathrm{J}\mathrm{F}\mathrm{I}}_{\mathrm{i}\mathrm{j}\mathrm{b}}={\displaystyle \frac{{\mathrm{T}\mathrm{D}}_{\mathrm{j}}+1}{{\mathrm{d}}_{\mathrm{j}}}}$. Suppose the job is not assigned to the machine with the minimum machining time, then ${\mathrm{T}\mathrm{D}}_{\mathrm{j}}=0$.

$${JFI}_{ijb}={\displaystyle \frac{{TD}_j+1}{{(|P}_{ib}-p_{ij}|+1){(|R}_{ib}-r_j|+1)\times d_j}}$$

(5)

$$P{J}_{ijb}={\displaystyle \frac{{JFI}_{ijb}}{\sum {JFI}_{ijb}}},J_j\in L_{bi}$$

(6)

In order to more intuitively clarify the specific operation process and logical steps of the DLFD algorithm, this paper has drawn a flowchart to clearly demonstrate the entire process of the algorithm from initialization to the key decision points, and finally to the generation of a feasible scheduling plan. Figure 1 illustrates the specific steps.

4. Experimentation and Analysis

In order to evaluate the effectiveness and robustness of the heuristic, problem instances close to realistic scenarios are generated to analyze the experimental results. The experimental running environment is Intel(R) Core (TM) i7-8700 CPU@3.20GHz 3.19GHz and the programming language is Python 3.12.5 64-bit.

4.1. Design of Experiments

In order to solve the UPBPM problem, the comparison experiment will randomly assign the jobs, introduce the shortest processing time (SPT) and the longest processing time (LPT) for sorting, the first fit (FF), and the best fit (BF) for grouping batches, and use them in combination to obtain the four scheduling rules LPTFF/LPTBF [8]/SPTFF/SPTBF/[9].

According to the actual application scenarios of UPBPM, the problem sizes are designed as small, medium, and large sizes. The machine and job parameters are generated by Monte Carlo simulation, where the machine capacity is ${\mathrm{Q}}_{\mathrm{i}}$, the machining time of the job is ${\mathrm{p}}_{\mathrm{i}\mathrm{j}}$, the size of the job is ${\mathrm{s}}_{\mathrm{j}}$, the arrival time of the job is ${\mathrm{r}}_{\mathrm{j}}$, the delivery time of the job is ${\mathrm{d}}_{\mathrm{j}}$, and the machine machining power is ${\mathrm{P}}_{\mathrm{i}}^{\mathrm{p}}$. Since the UPBPM problem studied in this paper has specific physical constraints and model assumptions, after investigation, the currently available standard scheduling problem test sets cannot directly match the core characteristics of this problem, making it difficult to apply directly. However, by generating self-generated instances, we can systematically control the key parameters of the problem, thereby conducting a more comprehensive and targeted evaluation of the performance and robustness of the proposed algorithm on problems of different scales and characteristics. To ensure the rationality of the instances, the generation methods of each parameter follow the common settings in the literature of the scheduling field. The value ranges of the experimental parameters are shown in

Table 3. Range of experimental parameters.

Size	$\mathbf{m}$	$\mathbf{n}$	$\mathbf{Q}$	${\mathbf{p}}_{\mathbf{i}\mathbf{j}}$	${\mathbf{s}}_{\mathbf{j}}$	${\mathbf{r}}_{\mathbf{j}}$	${\mathbf{d}}_{\mathbf{j}}$	${\mathbf{P}}_{\mathbf{i}}^{\mathbf{p}}$
small	2	15	U(20,41)	U(5,51)	U(1,20)	U(1,51)	U(70,301)	U(3,11)
medium	4	30	U(20,41)	U(5,51)	U(1,20)	U(1,51)	U(70,301)	U(3,11)
large	6	60	U(20,41)	U(5,51)	U(1,20)	U(1,51)	U(70,301)	U(3,11)

. In order to fairly evaluate the performance of the heuristics, each of the four heuristics was executed 10 times within the same experimental environment.

In order to provide a comprehensive and in-depth evaluation of the performance of the heuristics in solving multi-objective optimization problems, two evaluation metrics that are widely used within the field of multi-objective optimization are selected: the number of non-dominated solutions (NR) and the inverse generation distance (IGD). According to Equation (7), the NR metric visualizes the convergence ability of the heuristic to search for and approximate the ideal Pareto frontiers. However, the NR metric does not measure the uniformity of the distribution of these solutions and their overall proximity to the optimal frontier. To compensate for this limitation, the IGD metric is introduced, which can comprehensively evaluate the convergence and diversity of the solution set by calculating the average of the minimum distances from the points on the ideal Pareto front to the solution set generated by the heuristic according to Equation (8). Focusing on the combination of “quantitative” NR indicators and “qualitative” IGD indicators, a complementary evaluation system is constructed to verify the effectiveness and robustness of the heuristic.

$$NR(A)={\displaystyle \frac{\left|\left\{a\in A|\exists a\in A\cap a\in S\right\}\right|}{\left|S\right|}}$$

(7)

$$IGD(A,\Omega )={\displaystyle \frac{{\sum }_{x\in \Omega }\underset{a\in A}{\mathit{min}} d_{xa}}{\left|S\right|}}$$

(8)

where $\left|\left\{\mathrm{a}\in \mathrm{A}|\exists \mathrm{a}\in \mathrm{A}\cap \mathrm{a}\in \mathrm{S}\right\}\right|$ denotes the number of solutions in the solution set A that are also in the Pareto optimal frontier solution set, and $\left|\mathrm{S}\right|$ denotes the number of solutions in the Pareto optimal frontier solution set.${\mathrm{d}}_{\mathrm{x}\mathrm{a}}$ denotes the minimum Euclidean distance from a solution on the Pareto solution set $\mathsf{\Omega }$ to an individual in the heuristic’s solution set.

4.2. Comparison of Solution Effectiveness of Different Heuristics

In order to fairly evaluate the performance of the heuristics, each of the four heuristics was executed 10 times within the same experimental environment, and the average NR and IGD metrics were calculated. The results are shown in

Table 4. NR metrics for the heuristic at different scales (Table 4 (NR): data shown are 10-run averages).

$\mathbf{n}\times \mathbf{m}$	DLFD	SPTFF	SPTBF	LPTFF	LPTBF
$15\times 2$	0.6667	0	0	0	0.3333
$30\times 4$	0.6667	0	0	0	0.3333
	0.6667	0	0	0	0.3333
$60\times 6$	0.8182	0	0	0	0.1818
	0.7500	0	0	0.0833	0.1667
	0.8000	0	0	0	0.2000

and

Table 5. IGD metrics of the heuristic at different scales (Table 5 (IGD): data shown are 10-run averages).

$\mathbf{n}\times \mathbf{m}$	DLFD	SPTFF	SPTBF	LPTFF	LPTBF
$15\times 2$	0.0408	0.2938	0.2631	0.2439	0.1738
$30\times 4$	0.0735	0.2796	0.2878	0.2128	0.1746
	0.0533	0.2942	0.2842	0.1886	0.1430
$50\times 5$	0.1137	0.6619	0.6335	0.5061	0.5134
	0.1858	0.5577	0.5255	0.5108	0.4593
	0.1435	0.6252	0.6114	0.5202	0.5334

, respectively.

As shown in Table 1 and Table 2, the DLFD heuristic exhibits significant and consistent performance benefits on all three scheduling problems of different sizes. In terms of NR metrics, DLFD achieves 0.6667 at small scale, which is significantly better than the highest value of 0.3333 of the other heuristics, and stabilizes at 0.6667 in both experiments at medium scale, which is also ahead. At large scale, its average value of 0.7893 for the three experiments is much higher than the highest value of 0.2000 for the other heuristics. In terms of IGD metrics, DLFD also excels, with 0.0408 at small scale, much lower than the lowest value of 0.1738 of the other heuristics. The maximum value of 0.0735 of the two experimental values at medium scale is significantly lower than the lowest value of 0.1430 of the comparison heuristics. On large-scale problems, the maximum value of its three experiments, 0.1858, is smaller than the lowest value of all the compared heuristics, 0.4593. These data fully demonstrate the effectiveness of the DLFD heuristic, especially its superior performance in complex, large-scale problems and its effective balancing of multi-objective conflicts.

4.3. Robustness Comparison of Different Heuristics

This experiment not only verifies the relative advantages of the DLFD heuristic in specific metrics but also works to reveal its robustness advantages. This validation session provides a quantifiable empirical basis for the long-term reliable operation of the DLFD heuristic in industrial scenarios.

According to the boxplots of Figure 2 and Figure 3, the DLFD heuristic shows remarkable robustness in repeated tests, with consistently high NR (median ≈ 0.8), compact box distribution, and no outliers, which proves that the heuristic is able to output high-quality non-dominated solutions stably at different scales. The IGD values (median ≈ 0.15) are significantly lower than those of the comparison heuristics, with distributional concentration up to more than three times that of the other heuristics, and with an upper value that is still lower than the lower limit of most of the comparison heuristics. This low fluctuation characteristic verifies the high immunity of the heuristic to parameter perturbations and ensures the long-term reliable operation of the actual scheduling scheme.

5. Conclusions

Comprehensive cross-scale experimental validation shows that the DLFD heuristic achieves a double breakthrough in the unrelated parallel batch processor scheduling problem in terms of both the quality and robustness of the solution set through the double-list synergy mechanism and the dynamic probabilistic perturbation strategy. The results show that the heuristic consistently maintains a high level on the NR metric, while the IGD value is significantly lower than that of the control group, and the dual metrics synergistically validate the close approximation of its solution set to the ideal Pareto front. The box plots further show that the distribution of the NR and IGD metrics are boxed compactly and free of outliers, systematically verifying that the heuristics have strong robustness in the face of parameter perturbations. The DLFD heuristic shows significant performance advantages in all types of problems of various scales and improves the IGD metrics by more than 14% compared to the classical dispatching rules, which fully validates its effectiveness. In summary, the DLFD heuristic provides an efficient and stable optimization strategy for the UPBPM problem. Based on the results of this paper, further research can be carried out in the future by considering more realistic constraints such as preventive maintenance, sequence-dependent set-up time, etc.