Neural Combinatorial Optimization for Stochastic Flexible Job Shop Scheduling Problems

Igor G. Smit; Pavel Troubil; Wim P.M. Nuijten; Yaoxin Wu; Yingqian Zhang

REVIEW 2 major objections 3 minor 35 references

Neural Combinatorial Optimization for Stochastic Flexible Job Shop Scheduling Problems

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that an attention-based scenario processing module, SPM-DAN, learns flexible job shop schedules under stochastic processing times that beat both learning and non-learning baselines by feeding sampled scenarios into the…

desk verdict Useful SPM-DAN extension for stochastic FJSP, but the CP-stoch baseline in Appendix B optimizes the minimum scenario makespan rather than VaR95, so those non-learning comparisons need rework. read the letter →

arxiv 2412.14052 v1 pith:CMSZ5OOA submitted 2024-12-18 cs.AI cs.LGmath.OC

Igor G. Smit , Yaoxin Wu , Pavel Troubil , Yingqian Zhang , Wim P.M. Nuijten This is my paper · ORCID

classification cs.AIcs.LGmath.OC

keywords stochasticflexiblejobshopschedulingneuralcombinatorialoptimizationscenarioprocessingmoduledeepreinforcementlearningattentionmechanismValue-at-Riskmakespantimeuncertainty

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that a neural combinatorial optimization scheduler can be made effective for the flexible job shop scheduling problem with stochastic processing times by explicitly feeding it what the uncertainty looks like, not just training it on noisy rewards. To do this, the authors add a scenario processing module (SPM) to an existing dual attention network, forming SPM-DAN. The module takes several independently sampled processing-time scenarios, compresses them with attention into a single embedding, and concatenates that embedding with the deterministic instance features at every scheduling step. The same training loop is adapted so that rewards are computed from sampled scenarios under either the expected-makespan or the Value-at-Risk objective. Across synthetic and benchmark instances, SPM-DAN improves on dispatch rules, the deterministic base network, a stochastic-reward variant without the module, and the CP-SAT solver, which matters because real factories must commit to schedules before processing times are known.

What carries the argument

The central object is the scenario processing module (SPM), an attention block that maps a set of $n_{\mathrm{scn}}$ scenario state embeddings to a single stochastic embedding. It uses a trainable set of inducing points $I\in\mathbb{R}^{m\times d}$ so that the attention complexity stays linear in the number of scenarios: $\mathrm{SPM}(H)=\mathrm{Avg}(\mathrm{MHAB}(H,\mathrm{MHAB}(I,H)))$. The module is permutation invariant, so it treats the sampled scenarios as an unordered approximation of the underlying processing-time distribution. It is plugged into a base dual attention network as an additional input source for operation, machine, and operation-machine features, converting a deterministic scheduler into a stochastic one without restructuring the base network.

What would settle it

Train the same SPM-DAN architecture twice on an instance family with high processing-time variance: once with the paper's lower-bound reward and once with a reward computed from full Monte Carlo makespan simulations, then compare VaR95 on 1000 fresh scenarios. If the lower-bound-trained policy is worse, the lower-bound approximation is the component that limits the method.

Watch

Extended reading notes

Core claim

The discovery is that stochasticity can be incorporated into a learned scheduler as a state feature rather than only as reward noise. At each decision step, the SPM summarizes a set of sampled scenarios into an embedding via multi-head cross-attention with learnable inducing points, followed by permutation-invariant mean aggregation. This embedding is concatenated with the deterministic operation, machine, and operation-machine features before the base network scores actions. The reward is a scenario-level objective difference $f(C_t)-f(C_{t+1})$ computed from makespan lower bounds under the recursion $C(O_{ij},s_t)=C(O_{i(j-1)},s_t)+\min_k p^k_{ij}$, with $f$ equal to the expected makespan or $\mathrm{VaR}_{95\%}$. The paper argues, and its experiments show, that this arrangement lets the policy distinguish states that look identical deterministically but differ in their sampled processing-time distributions, and that the resulting schedules outperform both learning and non-learning baselines on a variety of instances.

Load-bearing premise

The load-bearing premise is that the makespan lower bound used as a reward signal approximates the true stochastic objective closely enough for the learned policy to stay aligned with that objective.

Editorial extensions

If this is right

On synthetic instances, greedy SPM-DAN schedules beat the CP-SAT solver on the largest tested sizes (20x10 and above), while sampling-based SPM-DAN beats CP-SAT on nearly all reported instance sets.
Policies trained on one objective (VaR95) and one distribution transfer to the expected-makespan objective and to beta, gamma, and mixed distributions without retraining.
The same scenario module can be attached to a different base network and problem: a preliminary SPM-L2D experiment improves over L2D on stochastic JSP.
Training with fewer scenarios (50 or 100) costs little performance, and inference can trade runtime for solution quality by increasing the number of inference scenarios.
The stochastic-aware module adds roughly 20% greedy-inference runtime and about three times sampling runtime compared with the deterministic base, while improving objective values.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A connection not explored in the paper is applying the same scenario-embedding idea to other stochastic combinatorial optimization problems with known distributions, such as stochastic vehicle routing or project scheduling, where sample approximations are already standard.
The paper does not analyze whether the attending embeddings capture higher-order moments such as skew or tail thickness or only mean shifts; a testable extension is to vary the tail thickness while holding the mean fixed and see whether SPM-DAN's improvement persists.
Because the module is permutation invariant over scenarios, it could in principle be trained to produce distributionally robust schedules by treating the set of scenarios as a synthetic ambiguity set, though the authors do not investigate this direction.
The reliance on lower-bound rewards suggests that a curriculum that starts with loose bounds and tightens them as training progresses might separate the module's contribution from the reward approximation's contribution; this is not explored in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. The scenario processing module (SPM) is a genuinely useful addition to the neural combinatorial optimization toolkit, and the experimental study is unusually thorough. But there is a real problem with the CP-stoch baseline: as written in Appendix B, the VaR95 constraint forces the selected makespan to be at most every other scenario makespan, i.e., the minimum, not the 95th percentile. For n=10 the constraint is infeasible because round(0.95n)=10 exceeds n-1=9. That invalidates the CP-stoch numbers in Tables 1, 3, 4, and 5 unless the released code does something different from the appendix. If the code matches the appendix, those comparisons should be discarded or redone; if it doesn't, the manuscript is not reproducible as written.

What is genuinely new here is the SPM itself: a Set-Transformer-style cross-attention module with inducing points that compresses nscn sampled scenarios into an embedding, concatenated with the deterministic DAN features. It is modular, transfers to L2D on the JSP, and works with both expected makespan and VaR objectives. The training reward telescopes to the terminal objective, so the lower-bound estimate is a legitimate potential-based shaping of the final makespan; the reader's concern about that is unfounded. The evaluation is broad: t-tests, generalization to larger instances, public benchmarks, sensitivity to nscn, and different distributions. That is solid work.

The CP-stoch flaw is the main soft spot. It is load-bearing for the claim of outperforming a scenario-based CP method, though not fatal to the overall story: SPM-DAN also beats deterministic CP-SAT and the DRL baselines on most larger instances. The abstract's "outperforms existing non-learning methods" is too strong if the strongest non-learning baseline is broken. Missing standard deviations in the main tables is a minor complaint; the t-tests partially compensate. The mk dataset overclaim is actually acknowledged in the text. The code is public but without a commit hash or data snapshot, which is a reproducibility nit.

This paper deserves a serious referee. The core contribution is sound and the evaluation is more careful than most in this area; the CP-stoch issue is fixable by rerunning with a correct VaR formulation or dropping those columns. I would send it to peer review with a clear request to address that baseline before acceptance.

Referee Report

2 major / 3 minor

Summary. The paper proposes SPM-DAN, a neural scheduler for stochastic flexible job shop scheduling (FJSP) with random processing times. It extends the deterministic dual attention network (DAN) with a scenario processing module (SPM) that attends over sampled processing-time scenarios and feeds the resulting embeddings into the base network. The training procedure uses a scenario-based reward that can optimize either expected makespan or Value-at-Risk (VaR95%). Experiments on synthetic instances, larger out-of-distribution sizes, public benchmarks (mk, rdata, edata, vdata), different processing-time distributions, and an alternative base network (L2D) aim to show that SPM-DAN outperforms dispatching rules, deterministic CP-SAT, a stochastic CP-SAT extension (CP-stoch), and DAN-based learning baselines. The manuscript includes code, ablations, t-tests in the appendix, and a sensitivity analysis of the number of scenarios.

Significance. If the results hold, the paper makes a useful contribution by showing how an attention-based scenario module can extend neural combinatorial optimization to stochastic scheduling with different objectives, and it provides a thorough experimental study with public benchmarks, generalization tests, and transferability to another network architecture. The availability of code and the inclusion of statistical significance tests are strengths. However, the central comparison against CP-stoch, a key non-learning stochastic baseline, is undermined by an error in the VaR formulation in Appendix B, so the claim of outperforming non-learning stochastic methods is not currently established.

major comments (2)

[Appendix B, Algorithm 2 (CP-stoch VaR95% branch)] The VaR95% objective in Algorithm 2 enforces Sum_{l' != l} [c_l <= c_l'] = round(0.95 n) to set is_obj_l. This counts the number of other scenarios whose makespan is at least c_l, so for n=25 the constraint is satisfied by the minimum makespan (for which all 24 other scenarios are >=), not by the 95th percentile, which has about one other scenario at least as large. For n=10, round(0.95*10)=10 exceeds the maximum possible count n-1=9, making the constraint infeasible or degenerate. Because this CP-stoch baseline is used in Tables 1, 3, 4, and 5, the reported CP-stoch results do not actually optimize the VaR95% objective as claimed.
[Abstract and Section 'Synthetic Datasets Results' (Tables 1, 3, 4, 5)] The abstract's claim that the approach 'outperforms existing learning and non-learning methods' relies in part on the CP-stoch baseline, which is not correctly formulated as a VaR95% optimizer (see preceding comment). The comparisons against CP-SAT and the learning baselines are unaffected, but the CP-stoch comparison must be corrected by re-implementing the VaR95% constraint (for instance, using an order-statistic or auxiliary-variable formulation) and re-running the experiments, or the claim must be narrowed to exclude CP-stoch.

minor comments (3)

[Section 'Reward' (paragraph on stochastic reward)] The reward uses lower-bound completion-time estimates for unscheduled operations; the authors should state explicitly that because the sum of rewards telescopes to f(C_0) - f(C_T) with C_T the exact terminal scenario makespans, the lower-bound approximation does not bias the terminal objective, which would preempt a common concern about reward shaping.
[Tables 1-6] The main tables report mean objective values and gaps without standard deviations; although t-tests are provided in Appendix D, reporting standard deviations in the tables would allow readers to assess the dispersion of the results more easily.
[Appendix A, Table 7] The transferability results for SPM-L2D are reported without statistical significance tests or run-to-run variation; a brief note on the number of seeds or instances would strengthen this preliminary evidence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the training objective is explicitly the telescoping stochastic objective, and all comparisons are empirical against external baselines.

full rationale

I examined the claimed derivation chain. The SPM-DAN policy is trained with reward r_t = f(C_t) - f(C_{t+1}), and the paper explicitly shows that the cumulative reward telescopes to f(C_0) - f(C_T), where C_T is the terminal makespan vector of independently sampled reward scenarios; maximizing this reward is the stochastic objective itself, not a fitted proxy renamed as a prediction. The SPM is constructed from cross-attention with inducing points and mean aggregation, explicitly credited to Lee et al. (2019), and no uniqueness theorem or load-bearing result is imported from the authors' own prior work. The only self-citation, 'We refer to a recent survey (Smit et al. 2025) for a complete outline,' is a literature pointer, not an argument that forces the paper's conclusions. Performance claims are empirical comparisons on held-out synthetic instances, larger unseen instance sizes, and public benchmarks (Brandimarte; Hurink et al.), so the outputs are not equivalent to training inputs by construction. The lower-bound makespan estimate used in the reward is a potential approximation concern, but the paper's own telescoping argument shows the terminal objective is exact; any looseness would be a correctness or efficiency issue, not circularity. The Appendix B CP-stoch VaR formulation, whatever its correctness, is a baseline-implementation matter and does not make any prediction reduce to its own fitted input. No step in the derivation exhibited the required equivalence-by-construction, so the appropriate finding is no circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim is an empirical performance claim; the assumptions that matter are about how scenarios are sampled and how the reward is computed. These are standard in stochastic optimization but are not proven, so they are listed as domain assumptions.

free parameters (5)

SPM attention heads = 4
Chosen based on preliminary tests; affects how scenario embeddings are aggregated.
SPM embedding dimension d = 32
Architecture choice for scenario and node embeddings.
Number of inducing points m = 16
Controls complexity of cross-attention in SPM; larger m increases expressiveness.
State scenarios nscn = 100 (training), 50-200 at inference
Sample size for approximating the stochastic state; paper tests trade-off.
Reward scenarios nrew = 1000
Number of Monte Carlo scenarios used to estimate expected makespan or VaR in the reward.

assumptions (4)

domain assumption Processing times are random variables with known distributions realized after scheduling
Core stochastic FJSP formulation stated in the problem definition section.
domain assumption A finite set of sampled scenarios approximates the true distribution (sample average approximation)
Used in state and reward; standard in stochastic optimization but not proven.
domain assumption The lower-bound completion time recursion is a valid training signal
The reward telescopes over lower bounds, not actual makespans; this proxy is assumed to align with the true objective.
domain assumption Log-normal / beta / gamma distributions adequately represent real processing time uncertainty
Used to create stochastic instances; borrowed from prior literature.

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Cite this review

Pith. "Pith review of Neural Combinatorial Optimization for Stochastic Flexible Job Shop Scheduling Problems." pith.science (2026). https://pith.science/paper/CMSZ5OOA

@misc{pith2026241214052,
  author       = {Pith},
  title        = {Pith review of: Neural Combinatorial Optimization for Stochastic Flexible Job Shop Scheduling Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMSZ5OOA}},
  note         = {Machine review of arXiv:2412.14052}
}

read the original abstract

Neural combinatorial optimization (NCO) has gained significant attention due to the potential of deep learning to efficiently solve combinatorial optimization problems. NCO has been widely applied to job shop scheduling problems (JSPs) with the current focus predominantly on deterministic problems. In this paper, we propose a novel attention-based scenario processing module (SPM) to extend NCO methods for solving stochastic JSPs. Our approach explicitly incorporates stochastic information by an attention mechanism that captures the embedding of sampled scenarios (i.e., an approximation of stochasticity). Fed with the embedding, the base neural network is intervened by the attended scenarios, which accordingly learns an effective policy under stochasticity. We also propose a training paradigm that works harmoniously with either the expected makespan or Value-at-Risk objective. Results demonstrate that our approach outperforms existing learning and non-learning methods for the flexible JSP problem with stochastic processing times on a variety of instances. In addition, our approach holds significant generalizability to varied numbers of scenarios and disparate distributions.

Figures

Figures reproduced from arXiv: 2412.14052 by the authors.

**Figure 1.** Overview of SPM-DAN. ity scales quadratically with nscn, making it too expensive for a large value nscn. Hence, we propose a trainable set of m d-dimensional inducing point vectors I ∈ R m×d to compute the cross-attention over H ∈ R nscn×d , thereby maintaining a linear complexity with nscn and resulting in the embeddings J ∈ R m×d . We then apply the original set H to perform cross-attention with J, to get H′ ∈ R… view at source ↗

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Reference graph

Works this paper leans on

35 extracted references · 26 canonical work pages

[1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...
[2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...
[3]

L.; Kiros, J

Ba, J. L.; Kiros, J. R.; and Hinton, G. E. 2016. Layer Normalization. arXiv:1607.06450

arXiv 2016
[4]

L.; and Nuijten, W

Baptiste, P.; Pape, C. L.; and Nuijten, W. 2001. Constraint-based scheduling: applying constraint programming to scheduling problems, volume 39. Springer Science & Business Media. ISBN 0792374088

work page 2001
[5]

Bengio, Y.; Lodi, A.; and Prouvost, A. 2021. Machine learning for combinatorial optimization: a methodological tour d’horizon. European Journal of Operational Research, 290(2): 405--421

2021
[6]

Brandimarte, P. 1993. Routing and scheduling in a flexible job shop by tabu search. Annals of Operations Research, 41: 157--183

work page 1993
[7]

L.; and Kozan, E

Burdett, R. L.; and Kozan, E. 2018. An integrated approach for scheduling health care activities in a hospital. European Journal of Operational Research, 264: 756--773

work page 2018
[8]

H.; and Gnanavelbabu, A

Caldeira, R. H.; and Gnanavelbabu, A. 2021. A simheuristic approach for the flexible job shop scheduling problem with stochastic processing times. SIMULATION, 97(3): 215--236

work page 2021

Show all 35 references

[9]

D.; and Teppan, E

Col, G. D.; and Teppan, E. C. 2022. Industrial-size job shop scheduling with constraint programming. Operations Research Perspectives, 9: 100249

2022
[10]

Flores-Gómez, M.; Borodin, V.; and Dauzère-Pérès, S. 2023. Maximizing the service level on the makespan in the stochastic flexible job-shop scheduling problem. Computers & Operations Research, 157: 106237

2023
[11]

Ghaedy-Heidary, E.; Nejati, E.; Ghasemi, A.; and Torabi, S. A. 2024. A simulation optimization framework to solve Stochastic Flexible Job-Shop Scheduling Problems—Case: Semiconductor manufacturing. Computers & Operations Research, 163: 106508

2024
[12]

Hurink, J.; Jurisch, B.; and Thole, M. 1994. Tabu search for the job-shop scheduling problem with multi-purpose machines. OR Spektrum, 15: 205--215

1994
[13]

Infantes, G.; Roussel, S.; Pereira, P.; Jacquet, A.; and Benazera, E. 2024. Learning to Solve Job Shop Scheduling Under Uncertainty. In Dilkina, B., ed., Integration of Constraint Programming, Artificial Intelligence, and Operations Research, 329--345. Cham: Springer Nature Sw...

2024
[14]

Joe, W.; and Lau, H. C. 2020. Deep Reinforcement Learning Approach to Solve Dynamic Vehicle Routing Problem with Stochastic Customers. Proceedings of the International Conference on Automated Planning and Scheduling, 30(1): 394--402

2020
[15]

Kwon, Y.-D.; Choo, J.; Kim, B.; Yoon, I.; Gwon, Y.; and Min, S. 2020. POMO: Policy Optimization with Multiple Optima for Reinforcement Learning. In Larochelle, H.; Ranzato, M.; Hadsell, R.; Balcan, M.; and Lin, H., eds., Advances in Neural Information Processing Systems, volum...

2020
[16]

Kwon, Y.-D.; Choo, J.; Yoon, I.; Park, M.; Park, D.; and Gwon, Y. 2021. Matrix encoding networks for neural combinatorial optimization. In Ranzato, M.; Beygelzimer, A.; Dauphin, Y.; Liang, P.; and Vaughan, J. W., eds., Advances in Neural Information Processing Systems, volume ...

2021
[17]

Lee, J.; Lee, Y.; Kim, J.; Kosiorek, A.; Choi, S.; and Teh, Y. W. 2019. Set Transformer: A Framework for Attention-based Permutation-Invariant Neural Networks. In Chaudhuri, K.; and Salakhutdinov, R., eds., Proceedings of the 36th International Conference on Machine Learning, ...

2019
[18]

Lei, K.; Guo, P.; Wang, Y.; Zhang, J.; Meng, X.; and Qian, L. 2023. Large-scale dynamic scheduling for flexible job-shop with random arrivals of new jobs by hierarchical reinforcement learning. IEEE Transactions on Industrial Informatics

2023
[19]

Lei, K.; Guo, P.; Zhao, W.; Wang, Y.; Qian, L.; Meng, X.; and Tang, L. 2022. A multi-action deep reinforcement learning framework for flexible Job-shop scheduling problem. Expert Systems with Applications, 205: 117796

2022
[20]

H.; Kim, Y.; and Park, J

Park, J.; Chun, J.; Kim, S. H.; Kim, Y.; and Park, J. 2021. Learning to schedule job-shop problems: representation and policy learning using graph neural network and reinforcement learning. International Journal of Production Research, 59(11): 3360--3377

2021
[21]

Perron, L.; Didier, F.; and Gay, S. 2023. The CP-SAT-LP Solver. In Yap, R. H. C., ed., 29th International Conference on Principles and Practice of Constraint Programming (CP 2023), volume 280 of Leibniz International Proceedings in Informatics (LIPIcs), 3:1--3:2. Dagstuhl, Ger...

2023
[22]

Reijnen, R.; van Straaten, K.; Bukhsh, Z.; and Zhang, Y. 2023. Job Shop Scheduling Benchmark: Environments and Instances for Learning and Non-learning Methods. arXiv:2308.12794

2023 arXiv
[23]

Rooyani, D.; and Defersha, F. M. 2019. An Efficient Two-Stage Genetic Algorithm for Flexible Job-Shop Scheduling. IFAC-PapersOnLine, 52: 2519--2524

2019
[24]

Schmitt-Ulms, F.; Hottung, A.; Sellmann, M.; and Tierney, K. 2022. Learning to Solve a Stochastic Orienteering Problem with Time Windows. In Simos, D. E.; Rasskazova, V. A.; Archetti, F.; Kotsireas, I. S.; and Pardalos, P. M., eds., Learning and Intelligent Optimization, 108--...

2022
[25]

Schulman, J.; Wolski, F.; Dhariwal, P.; Radford, A.; and Klimov, O. 2017. Proximal Policy Optimization Algorithms. arXiv:1707.06347

2017 arXiv
[26]

Sels, V.; Gheysen, N.; and Vanhoucke, M. 2012. A comparison of priority rules for the job shop scheduling problem under different flow time- and tardiness-related objective functions. International Journal of Production Research, 50: 4255--4270. Doi: 10.1080/00207543.2011.611539

2012
[27]

G.; Zhou, J.; Reijnen, R.; Wu, Y.; Chen, J.; Zhang, C.; Bukhsh, Z.; Zhang, Y.; and Nuijten, W

Smit, I. G.; Zhou, J.; Reijnen, R.; Wu, Y.; Chen, J.; Zhang, C.; Bukhsh, Z.; Zhang, Y.; and Nuijten, W. 2025. Graph Neural Networks for Job Shop Scheduling Problems: A Survey. Computers & Operations Research, 176: 106914

2025
[28]

Song, W.; Chen, X.; Li, Q.; and Cao, Z. 2022. Flexible Job-Shop Scheduling via Graph Neural Network and Deep Reinforcement Learning. IEEE Transactions on Industrial Informatics, 19: 1600--1610

2022
[29]

Tamssaouet, K.; Dauzère-Pérès, S.; Knopp, S.; Bitar, A.; and Yugma, C. 2022. Multiobjective optimization for complex flexible job-shop scheduling problems. European Journal of Operational Research, 296: 87--100

2022
[30]

N.; Kaiser, L

Vaswani, A.; Shazeer, N.; Parmar, N.; Uszkoreit, J.; Jones, L.; Gomez, A. N.; Kaiser, L. u.; and Polosukhin, I. 2017. Attention is All you Need. In Guyon, I.; Luxburg, U. V.; Bengio, S.; Wallach, H.; Fergus, R.; Vishwanathan, S.; and Garnett, R., eds., Advances in Neural Infor...

2017
[31]

Veličković, P.; Cucurull, G.; Casanova, A.; Romero, A.; Liò, P.; and Bengio, Y. 2018. Graph Attention Networks. arXiv:1710.10903

2018 arXiv
[32]

Wang, R.; Wang, G.; Sun, J.; Deng, F.; and Chen, J. 2023. Flexible job shop scheduling via dual attention network-based reinforcement learning. IEEE Transactions on Neural Networks and Learning Systems

2023
[33]

S.; and Chi, X

Zhang, C.; Song, W.; Cao, Z.; Zhang, J.; Tan, P. S.; and Chi, X. 2020. Learning to Dispatch for Job Shop Scheduling via Deep Reinforcement Learning. In Larochelle, H.; Ranzato, M.; Hadsell, R.; Balcan, M.; and Lin, H., eds., Advances in Neural Information Processing Systems, v...

2020
[34]

Zhang, J.; Yang, J.; and Zhou, Y. 2016. Robust scheduling for multi-objective flexible job-shop problems with flexible workdays. Engineering Optimization, 48: 1973--1989. Doi: 10.1080/0305215X.2016.1145216

2016
[35]

Zhang, L.; Feng, Y.; Xiao, Q.; Xu, Y.; Li, D.; Yang, D.; and Yang, Z. 2023. Deep reinforcement learning for dynamic flexible job shop scheduling problem considering variable processing times. Journal of Manufacturing Systems, 71: 257--273

2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

L.; Kiros, J

Ba, J. L.; Kiros, J. R.; and Hinton, G. E. 2016. Layer Normalization. arXiv:1607.06450

arXiv 2016

[4] [4]

L.; and Nuijten, W

Baptiste, P.; Pape, C. L.; and Nuijten, W. 2001. Constraint-based scheduling: applying constraint programming to scheduling problems, volume 39. Springer Science & Business Media. ISBN 0792374088

work page 2001

[5] [5]

Bengio, Y.; Lodi, A.; and Prouvost, A. 2021. Machine learning for combinatorial optimization: a methodological tour d’horizon. European Journal of Operational Research, 290(2): 405--421

2021

[6] [6]

Brandimarte, P. 1993. Routing and scheduling in a flexible job shop by tabu search. Annals of Operations Research, 41: 157--183

work page 1993

[7] [7]

L.; and Kozan, E

Burdett, R. L.; and Kozan, E. 2018. An integrated approach for scheduling health care activities in a hospital. European Journal of Operational Research, 264: 756--773

work page 2018

[8] [8]

H.; and Gnanavelbabu, A

Caldeira, R. H.; and Gnanavelbabu, A. 2021. A simheuristic approach for the flexible job shop scheduling problem with stochastic processing times. SIMULATION, 97(3): 215--236

work page 2021