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A neural policy that projects action targets into a feasible set and rounds via dual prices can learn hard-constrained inventory decisions end to end, beating classical baselines.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-04 08:57 UTC pith:SXDPRZQ4

load-bearing objection A strong DRL feasibility mechanism with real formal guarantees; the main caveats are downward-closure scope and some empirical self-reliance. the 3 major comments →

arxiv 2608.02343 v1 pith:SXDPRZQ4 submitted 2026-08-03 cs.AI cs.LG

Hard Constraints, Smooth Gradients: Learning Feasible Inventory Policies via Differentiable Projection

Patrick Helm , Jan-Niklas Doerr , Joren Gijsbrechts , Stefan Minner This is my paper
classification cs.AI cs.LG MSC 90B0590C2090C4068T07
keywords deep reinforcement learningdifferentiable optimizationquadratic programminghard constraintsmulti-echelon inventory planningpathwise gradientsinteger feasibilitycapacity constraints
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to show that deep reinforcement learning can handle sequential decision problems with hard, interdependent feasibility constraints—a regime where existing DRL methods either penalize violations or use feasibility mechanisms that break down when constraints interact. The proposed policy chains a neural network that proposes continuous action targets, a quadratic program that projects those targets onto the feasible set of the continuous relaxation, and a dual-informed integer mapping that rounds to feasible integer actions while preserving a usable gradient signal. The authors prove that this two-step mapping stays close to the exact integer projection, that every feasible integer action is reachable from a positive-measure set of targets, and that ambitious targets push onto the action frontier. Empirically, the policy reaches an average optimality gap below 1% on small three-item instances, cuts costs by up to 9.75% against the leading echelon base-stock policy and at least 7.7% against a rolling-horizon multi-stage stochastic program, and reduces costs by around 3% on an industry-scale high-tech planning benchmark. A sympathetic reader would take away that hard combinatorial feasibility and end-to-end learning are not mutually exclusive.

Core claim

The central claim is that hard, state-dependent, interdependent integer constraints can be pushed inside a smoothly differentiable policy without per-state integer programming. The neural network outputs continuous production targets; a strongly convex QP projects them onto the relaxed feasible set and, through implicit differentiation, supplies analytic Jacobians for both the target channel and the state-dependent right-hand side; a greedy integer mapping then walks from the componentwise floor to a feasible integer action, using dual prices to prioritize unit increments. The authors prove three properties: bounded error relative to the exact integer projection, completeness (every feasible

What carries the argument

The central object is the composite policy map z_t -> x*_t = P_t(z_t) -> x_t = IM_t(x*_t, z_t, λ*_t), where P_t is the strongly convex quadratic projection onto the relaxed feasible set and IM_t is the dual-informed greedy integer mapping. The workhorse is the QP's Jacobian structure: the target-channel sensitivity is a W-orthogonal projector, which is nonexpansive and therefore cannot amplify gradients, while the right-hand-side channel routes gradient information from binding resource constraints back to earlier decisions. The integer mapping's completeness and frontier properties differentiate it from naive flooring and justify the straight-through gradient estimator used in training.

Load-bearing premise

The load-bearing premise is that the relaxed feasible set is downward closed—so that flooring a feasible continuous action always yields a feasible integer action—which holds when all constraint coefficients and right-hand sides are nonnegative, but fails as soon as lower bounds, equality constraints, minimum lot sizes, or negative coefficients appear.

What would settle it

Construct a feasible region with a lower bound or equality constraint—for example, a single item with x≥1 or an assembly step requiring at least one unit—so the componentwise floor of a feasible continuous point is infeasible; if the proposed integer mapping without a repair step produces an infeasible action or the training diverges, the feasibility guarantee is falsified. A second check: on an instance with a unique fully-resource-utilizing action, verify that the learned policy can actually select it (non-zero probability of occurrence), since naive flooring provably cannot.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the central claim holds, constrained sequential decision problems no longer require solving a MILP at each step: the policy is a fixed differentiable architecture whose QP coefficients encode the constraints, so adding or changing resources requires editing the QP specification rather than the learning algorithm.
  • The non-expansive projection Jacobian implies that the feasibility layer does not inflate the policy's Lipschitz constant, so the projection module cannot by itself cause the gradient instability observed in unconstrained differentiable-policy training.
  • Because the integer mapping is complete on the active boundary, the policy can in principle select precisely the resource-utilizing actions that matter most in tightly capacitated systems; the reported cost savings concentrate in high-utilization, high-variability settings.
  • The method outperforms both optimized echelon base-stock policies and rolling-horizon stochastic programming on the studied networks, and its empirical cost distribution stochastically dominates the best benchmark, meaning lower realized costs are more likely.
  • The decomposition of the QP into connected components of the variable–constraint graph keeps the projection cost governed by the largest local cluster, so the architecture scales to large networks with local resource constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same architecture transfers to other constrained sequential decision problems with differentiable simulators—such as financial hedging with position limits or manufacturing process control—because only the QP coefficients change; this is the paper's own suggested direction but not demonstrated.
  • A testable extension is to replace the differentiable simulator with a learned critic (actor–critic style); the authors mention this, and the projection gradients would then flow through the critic rather than the simulator.
  • The projection sensitivities open a route to jointly optimize resource capacities as a partially learnable parameter: since ∂x*/∂b is available, capacity limits could be tuned alongside policy weights, an economic extension beyond fixed capacity.
  • The method's guarantees depend on downward closedness; for constraints like minimum lot sizes or lower bounds, the floor initialization can be infeasible, and if a repair step is added, completeness and error bounds must be re-derived—an explicit boundary of the approach.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a differentiable policy architecture for discrete-action constrained MDPs, targeting multi-echelon production-inventory planning with material/resource constraints. The policy consists of a neural network that proposes continuous action targets, a differentiable QP projection onto the relaxed feasible set, and a dual-informed integer mapping with straight-through gradients. The authors prove bounded error relative to an exact integer projection, completeness of the integer mapping, and frontier selection, and validate the method on small instances with exact optimal policies, larger Tempelmeier-Derstroff-based networks, and the 2025 ASML Research Challenge. Reported results include an average optimality gap below 1% on small instances, savings up to 9.75% over echelon base-stock policies, and 3.22%/2.54% cost reductions on the ASML benchmark.

Significance. The methodological combination of differentiable QP projection, dual-informed integer rounding, and HDPO-style pathwise training is novel and well-motivated. The theoretical guarantees in Section 4.2 are nontrivial, and the proofs in Appendix E are internally consistent for the stated assumptions. The small-instance comparison against an exact optimal policy is credible and gives a strong sanity check. The ablation showing that removing the projection KKT gradients leads to training divergence is informative and supports the central architectural claim. If the empirical claims survive a more independent benchmark comparison, this would be a meaningful contribution to DRL for problems with interdependent hard constraints. The paper is currently overstated in its generality and has one empirically load-bearing design choice that needs clarification before publication.

major comments (3)
  1. [Abstract; §3.1, Assumption (i); Propositions 1–3] The abstract's claim that the method handles hard constraints 'with similar flexibility to MILPs' is not supported. Feasibility of the floor initialization and hence Propositions 1–3 all rely on Assumption (i): A_rel(s_t) is downward closed. This holds for nonnegative A,C and nonnegative right-hand sides, as in the inventory application, but excludes lower bounds, equality constraints, fixed charges, minimum lot sizes, and negative coefficients. In those settings the componentwise floor of the QP projection can be infeasible and the integer mapping has no repair step. The conclusion acknowledges this limitation, but the abstract and introduction do not. Please revise the general claims to state the resource-consumption scope, or add a repair mechanism for non-downward-closed sets.
  2. [§6.4; Appendix D.1; Table 5] The headline ASML improvement over the best-known policy is not independent of the baseline. Appendix D.1 states that the neural network's output is scaled by the SBS base-stock levels. The ablation variant −BS, which removes this scaling, increases cost by +5.81% (Setting 1) and +7.06% (Setting 2) relative to the full method (Table 5). Since the full method's advantage over DBS is only 3.22% and 2.54%, the −BS policy would likely be worse than DBS on both settings. Thus the claimed reduction over the best-known benchmark depends on injecting tuned SBS levels into the learned policy. Please report the absolute costs of −BS against DBS, state whether the SBS levels used for scaling are the same tuned levels used in the baselines, and clarify what information is available to each method in a fair deployment.
  3. [§2.1; §6.4] The 'best-known' baseline DBS is taken from van Dijck et al. (2026), which is coauthored by one of the current authors, and the paper uses that paper's baseline configurations without modification. This is not by itself invalid, but combined with the SBS-scaling issue it makes the empirical comparison less independent than the text suggests. The authors should disclose this relationship in the experimental section and, if possible, add an independent benchmark or at least report the sensitivity of the conclusions to the baseline configuration.
minor comments (5)
  1. [§4.1.1] The sentence about linearly dependent active constraints is ambiguous: Lemma 1 assumes an independent active set. If the target-channel Jacobian is invariant under choice of a basis of the active row space, please say so explicitly and add a proof or reference; otherwise the statement appears unsupported.
  2. [Appendix D.1] Please clarify whether the SBS base-stock scaling and the cost scaling by 'the costs of the SBS policy' use training-set, validation-set, or test-set statistics, and whether these normalizations are recomputed per instance. This is important for assessing information leakage.
  3. [Table 5] Report absolute costs and significance tests for the ablation variants, not only percentage changes relative to the full method. In particular, the comparison of −BS to DBS is central and should be shown directly.
  4. [§2.1] DBS is repeatedly referred to as 'best-known policy'; given that the reference is coauthored by a current author, consider wording such as 'the best policy in the benchmark literature available to us' or cite independent replications if they exist.
  5. [§6.3] The claim 'MSP outperforms DET in all but two instances' is verifiable in Table 2 but would benefit from a marker or a supplementary table identifying the exceptions.

Circularity Check

0 steps flagged

No significant circularity: the feasibility guarantees are proved from stated assumptions; the empirical comparison, though it uses a self-cited baseline, does not reduce to a fit.

full rationale

The paper's derivation chain is self-contained. Lemma 1 derives the QP Jacobians from the KKT system; Corollary 1 is a structural property of Euclidean projections; Propositions 1–3 are proved directly from Assumption (i) (downward closedness) and the explicit acceptance rule of the integer mapping. None of these results imports a fitted parameter or a prediction as an input: the bounded-error bound follows from the floor point being feasible and the integer mapping never increasing the projection objective; completeness is shown by constructing a positive-measure preimage for every feasible integer action; frontier selection follows from the mapping rejecting an increment only when it is infeasible. The self-citation of van Dijck et al. (2026), coauthored by Helm, is used to label DBS/SBS as 'best-known' and to obtain baseline configurations, and Appendix D.1 scales the network output by SBS base-stock levels; however, this is an inductive-bias/normalization choice, not a fitted parameter that is then reported as a prediction. The baseline policies are re-simulated in the same evaluation harness, and the QP-plus-integer-mapping allocation is not identical to the shortfall-allocation rule, so the reported savings are not forced by construction. The downward-closure limitation (excluding lower bounds, equalities, fixed charges) is a real scope restriction acknowledged in the conclusion, but it is an assumption of the proofs, not circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 8 axioms · 0 invented entities

No new physical entities are introduced. The method relies on standard convex-optimization assumptions, the stated HDPO assumptions, and the inventory-specific domain assumptions listed above. The most fragile are downward closedness and the use of baseline-derived scaling.

free parameters (3)
  • QP projection weights w_i = w_i ∝ 1/(long-run average gross requirement of item i), normalized to sum to 1
    Section 6, 'Our Policy': set by hand as a heuristic. These weights define the projection metric and therefore which continuous actions are favoured, though Proposition 2 shows they do not restrict reachability.
  • Neural network hyperparameters (width, learning rate, batch size, gradient clip, epochs) = width=32/128, η=1e-3, batch=1024, clip=10, epochs≤1000
    Section 6 and Appendix D.1. Chosen without per-instance tuning but they affect all reported results.
  • SBS base-stock scaling levels = Optimized SBS safety-stock factors from the baseline configuration, used as output scaling
    Appendix D.1: 'the echelon base-stock network uses softplus as the output activation and then scales by the base-stock levels of SBS.' This injects the comparator policy into the learned policy's output scale and is not reported as a benchmark-dependent prior.
axioms (8)
  • domain assumption A_rel(s_t) is downward closed for all states; A,C≥0; componentwise floor of every relaxed-feasible point is integer-feasible.
    Section 3.1, Assumption (i). Required by the integer mapping and used in Propositions 1–3. Violated by lower bounds, equality constraints, fixed charges, and negative coefficients.
  • domain assumption A(s_t) is nonempty for every reachable state; b(s_t), k are nonnegative.
    Section 3.1, Assumption (ii). Needed for the projection to be well-defined; combined with (i) the zero action is feasible.
  • domain assumption The per-period cost and transition functions are differentiable almost everywhere in state and action for fixed demand realizations.
    Section 3.1, Assumption (iii). Required for pathwise gradients through the simulator.
  • domain assumption The stochastic process Ξ_{1:T} is exogenous: its law may depend on covariates in the state but not on actions or the endogenous part of the state.
    Section 3.1, Assumption (iv). Standard HDPO assumption; critical for unbiased pathwise gradients.
  • domain assumption The BOM graph is acyclic and material/resource consumption coefficients are nonnegative.
    Section 5. Ensures the echelon base-stock recursion is well-defined and downward closure holds.
  • domain assumption No setup times or setup costs; constant setup time is subtracted from available capacity.
    Section 5. Used to avoid endogenous fixed-charge decisions, which would break downward closure.
  • standard math The QP active set is locally fixed and linearly independent for differentiability; on measure-zero active-set changes, a directional subgradient is used.
    Lemma 1 and its proof; standard QP sensitivity result (Amos and Kolter 2017).
  • standard math The pathwise gradient estimator is unbiased under mild regularity conditions.
    Section 3.2, citing Suh et al. (2022). Needed for HDPO to optimize the policy correctly.

pith-pipeline@v1.3.0-daily-deepseek · 30710 in / 16032 out tokens · 132112 ms · 2026-08-04T08:57:59.765837+00:00 · methodology

0 comments
read the original abstract

Many operational problems are constrained sequential decision processes with large, combinatorial action spaces and interdependent feasibility constraints. Mixed-integer linear programs (MILPs) handle such constraints flexibly but scale poorly in stochastic environments. Deep reinforcement learning (DRL) promises scalable decision rules, but existing methods either penalize constraints rather than enforce them, or rely on feasibility mechanisms that break down once constraints interact. We bridge this gap by embedding a differentiable convex optimization module inside the policy: a neural network proposes continuous action targets, a quadratic program projects them onto the relaxed feasible set, and a dual-informed integer mapping restores integrality while preserving feasibility. Given a differentiable simulator, the policy trains end to end from sampled trajectories using pathwise gradients, while handling hard constraints with similar flexibility to MILPs. We show that our feasibility enforcement has bounded error relative to an exact integer projection and ensures the entire feasible action space is reachable. We apply the method to multi-echelon production-inventory planning under shared resource and material constraints. Our policy attains an average optimality gap below 1% on small instances. It further outperforms state-of-the-art echelon base-stock policies by up to 9.75% and a rolling-horizon multi-stage stochastic program by at least 7.7% in larger networks. On an industry-scale case study from ASML, it reduces average cost by up to 3.22% relative to the best-known benchmark policy. The savings are largest where planning is hardest: in tightly capacitated systems with high demand variability. More broadly, our work shows that DRL can deliver economically significant savings in sequential decision problems with interdependent hard constraints, which are widespread in practice.

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Reference graph

Works this paper leans on

50 extracted references · 6 linked inside Pith

  1. [1]

    Proceedings of the 33rd International Conference on Neural Information Processing Systems

    Agrawal A, Amos B, Barratt S, Boyd S, Diamond S, Kolter JZ (2019) Differentiable convex optimization layers. Proceedings of the 33rd International Conference on Neural Information Processing Systems

    2019

  2. [2]

    arXiv preprint arXiv:2306.11246

    Alvo M, Russo D, Kanoria Y, Lee M (2025) Deep reinforcement learning for inventory networks: Toward reliable policy optimization. arXiv preprint arXiv:2306.11246

    Pith/arXiv arXiv 2025

  3. [3]

    Proceedings of the 34th International Conference on Machine Learning, volume 70 of PMLR, 136--145

    Amos B, Kolter JZ (2017) O pt N et: Differentiable optimization as a layer in neural networks. Proceedings of the 34th International Conference on Machine Learning, volume 70 of PMLR, 136--145

    2017

  4. [4]

    Bauschke HH, Combettes PL (2017) Convex Analysis and Monotone Operator Theory in H ilbert Spaces (Cham: Springer), 2nd edition

    2017

  5. [5]

    arXiv preprint arXiv:1308.3432

    Bengio Y, L \'e onard N, Courville AC (2013) Estimating or propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432

    Pith/arXiv arXiv 2013

  6. [6]

    Journal of Machine Learning Research 21(35):1--69

    Blondel M, Martins AFT, Niculae V (2020) Learning with fenchel--young losses. Journal of Machine Learning Research 21(35):1--69

    2020

  7. [7]

    INFORMS Journal on Computing 35(6):1308--1328

    B \"o ttcher L, Asikis T, Fragkos I (2023) Control of dual-sourcing inventory systems using recurrent neural networks. INFORMS Journal on Computing 35(6):1308--1328

    2023

  8. [8]

    Boyd S, Vandenberghe L (2004) Convex Optimization (Cambridge University Press)

    2004

  9. [9]

    Quantitative Finance 19(8):1271--1291

    Buehler H, Gonon L, Teichmann J, Wood B (2019) Deep hedging. Quantitative Finance 19(8):1271--1291

    2019

  10. [10]

    arXiv preprint arXiv:2409.03740

    Che E, Dong J, Namkoong H (2024) Differentiable discrete event simulation for queuing network control. arXiv preprint arXiv:2409.03740

    Pith/arXiv arXiv 2024

  11. [11]

    Management S cience 6(4):475--490

    Clark AJ, Scarf H (1960) Optimal policies for a multi-echelon inventory problem. Management S cience 6(4):475--490

    1960

  12. [12]

    CoRR abs/1801.08757

    Dalal G, Dvijotham K, Vecer \' k M, Hester T, Paduraru C, Tassa Y (2018) Safe exploration in continuous action spaces. CoRR abs/1801.08757

    Pith/arXiv arXiv 2018

  13. [13]

    de Kok A, Graves S, eds., Design and Analysis of Supply Chains, 597--675, Handbooks in Operations Research and Management Science (North-Holland Publishing Company)

    de Kok A, Fransoo J (2003) Planning supply chain operations: definition and comparison of planning concepts. de Kok A, Graves S, eds., Design and Analysis of Supply Chains, 597--675, Handbooks in Operations Research and Management Science (North-Holland Publishing Company)

    2003

  14. [14]

    European Journal of Operational Research 269(3):955--983

    de Kok T, Grob C, Laumanns M, Minner S, Rambau J, Schade K (2018) A typology and literature review on stochastic multi-echelon inventory models. European Journal of Operational Research 269(3):955--983

    2018

  15. [15]

    International Conference on Learning Representations (ICLR)

    Donti PL, Rolnick D, Kolter JZ (2021) DC3 : A learning method for optimization with hard constraints. International Conference on Learning Representations (ICLR)

    2021

  16. [16]

    Journal of Process Control 116:80--92

    Drgoňa J, Kiš K, Tuor A, Vrabie D, Klaučo M (2022) Differentiable predictive control: Deep learning alternative to explicit model predictive control for unknown nonlinear systems. Journal of Process Control 116:80--92

    2022

  17. [17]

    arXiv preprint arXiv:2410.02817

    Eisenach C, Ghai U, Madeka D, Torkkola K, Foster D, Kakade S (2024) Neural coordination and capacity control for inventory management. arXiv preprint arXiv:2410.02817

    Pith/arXiv arXiv 2024

  18. [18]

    Management Science 68(1):9--26

    Elmachtoub AN, Grigas P (2022) Smart ``predict, then optimize''. Management Science 68(1):9--26

    2022

  19. [19]

    International Journal of Production Economics 288:109687

    Fleuren T, Merzifonluoglu Y, Sotirov R, Hendriks M (2025) Production–inventory planning in high-tech low-volume manufacturing supply chains. International Journal of Production Economics 288:109687

    2025

  20. [20]

    Manufacturing & Service Operations Management 24(3):1349--1368

    Gijsbrechts J, Boute RN, van Mieghem JA, Zhang DJ (2022) Can deep reinforcement learning improve inventory management? performance on lost sales, dual-sourcing, and multi-echelon problems. Manufacturing & Service Operations Management 24(3):1349--1368

    2022

  21. [21]

    Cohen MC, Dai T, eds., AI in Supply Chains: Perspectives from Global Thought Leaders, 137--148 (Cham: Springer Nature Switzerland)

    Gijsbrechts J, Boute RN, van Mieghem JA, Zhang DJ (2026) AI in inventory management: The disruptive era of DRL and beyond. Cohen MC, Dai T, eds., AI in Supply Chains: Perspectives from Global Thought Leaders, 137--148 (Cham: Springer Nature Switzerland)

    2026

  22. [22]

    Management Science 41(2):263--281

    Glasserman P, Tayur S (1995) Sensitivity analysis for base-stock levels in multiechelon production-inventory systems. Management Science 41(2):263--281

    1995

  23. [23]

    Hansen N (2023) The CMA evolution strategy: A tutorial

    2023

  24. [24]

    Manufacturing & Service Operations Management 27(2):369--388

    Harsha P, Jagmohan A, Kalagnanam J, Quanz B, Singhvi D (2025) Deep policy iteration with integer programming for inventory management. Manufacturing & Service Operations Management 27(2):369--388

    2025

  25. [25]

    The 39th Annual Conference on Neural Information Processing Systems

    Hoppe H, Baty L, Bouvier L, Parmentier A, Schiffer M (2025) Structured reinforcement learning for combinatorial decision-making. The 39th Annual Conference on Neural Information Processing Systems

    2025

  26. [26]

    Manufacturing & Service Operations Management 18(4):570--584

    Huh WT, Janakiraman G, Nagarajan M (2016) Capacitated multiechelon inventory systems: Policies and bounds. Manufacturing & Service Operations Management 18(4):570--584

    2016

  27. [27]

    Management Science 33(3):409--418

    Karmarkar US (1987) Lot sizes, lead times and in-process inventories. Management Science 33(3):409--418

    1987

  28. [28]

    International Journal of Production Economics 267:109088

    Kaynov I, van Knippenberg M, Menkovski V, van Breemen A, van Jaarsveld W (2024) Deep reinforcement learning for one-warehouse multi-retailer inventory management. International Journal of Production Economics 267:109088

    2024

  29. [29]

    International Conference on Learning Representations (ICLR)

    Kingma DP, Ba J (2015) Adam: A method for stochastic optimization. International Conference on Learning Representations (ICLR)

    2015

  30. [30]

    arXiv preprint arXiv:2507.14446

    Liu D, Liu Y, Eisenach C (2025) Deep RL dual sourcing inventory management with supply and capacity risk awareness. arXiv preprint arXiv:2507.14446

    arXiv 2025

  31. [31]

    human buyers: A study of alibaba’s inventory replenishment system

    Liu J, Lin S, Xin L, Zhang Y (2023) AI vs. human buyers: A study of alibaba’s inventory replenishment system. INFORMS Journal on Applied Analytics 53(5):372--387

    2023

  32. [32]

    arXiv preprint arXiv:2210.03137

    Madeka D, Torkkola K, Eisenach C, Luo A, Foster DP, Kakade SM (2022) Deep inventory management. arXiv preprint arXiv:2210.03137

    Pith/arXiv arXiv 2022

  33. [33]

    European Journal of Operational Research 329(2):447--459

    Malicki S, Minner S (2026) The impact of initial inventory assumptions on cost performance in lot-sizing models. European Journal of Operational Research 329(2):447--459

    2026

  34. [34]

    Journal of Artificial Intelligence Research 80:1623--1701

    Mandi J, Kotary J, Berden S, Mulamba M, Bucarey V, Guns T, Fioretto F (2024) Decision-focused learning: Foundations, state of the art, benchmark and future opportunities. Journal of Artificial Intelligence Research 80:1623--1701

    2024

  35. [35]

    Markgraf H, Sawant S, Krasowski H, Sch \"a fer L, Gros S, Althoff M (2026) Safe reinforcement learning using action projection: Safeguard the policy or the environment? Transactions on Machine Learning Research Expert Certification

    2026

  36. [36]

    Manufacturing & Service Operations Management 24(1):285--304

    Oroojlooyjadid A, Nazari M, Snyder LV, Tak\' a c M (2022) A deep Q -network for the beer game: Deep reinforcement learning for inventory optimization. Manufacturing & Service Operations Management 24(1):285--304

    2022

  37. [37]

    Proceedings of the 33rd International Conference on Neural Information Processing Systems

    Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L, Desmaison A, K\" o pf A, Yang E, DeVito Z, Raison M, Tejani A, Chilamkurthy S, Steiner B, Fang L, Bai J, Chintala S (2019) Pytorch: an imperative style, high-performance deep learning library. Proceedings of the 33rd International Conference on Neural Info...

    2019

  38. [38]

    Proceedings of the 38th International Conference on Machine Learning, volume 139 of PMLR, 8443--8453

    Paulus A, Rol \'i nek M, Musil V, Amos B, Martius G (2021) C omb O pt N et: Fit the right NP -hard problem by learning integer programming constraints. Proceedings of the 38th International Conference on Machine Learning, volume 139 of PMLR, 8443--8453

    2021

  39. [39]

    Management Science 69(2):759--773

    Qi M, Shi Y, Qi Y, Ma C, Yuan R, Wu D, Shen ZJM (2023) A practical end-to-end inventory management model with deep learning. Management Science 69(2):759--773

    2023

  40. [40]

    Manufacturing & Service Operations Management 22(1):36--46

    Song JS, van Houtum GJ, van Mieghem JA (2020) Capacity and inventory management: Review, trends, and projections. Manufacturing & Service Operations Management 22(1):36--46

    2020

  41. [41]

    Suh HJ, Simchowitz M, Zhang K, Tedrake R (2022) Do differentiable simulators give better policy gradients? Proceedings of the 39th International Conference on Machine Learning, volume 162 of PMLR, 20668--20696

    2022

  42. [42]

    Management Science 42(5):738--757

    Tempelmeier H, Derstroff M (1996) A L agrangean-based heuristic for dynamic multilevel multiitem constrained lotsizing with setup times. Management Science 42(5):738--757

    1996

  43. [43]

    Production and Operations Management 30(2):475--493

    Thevenin S, Adulyasak Y, Cordeau JF (2021) Material requirements planning under demand uncertainty using stochastic optimization. Production and Operations Management 30(2):475--493

    2021

  44. [44]

    International Journal of Production Economics 49(2):157--174

    van der Heijden M, Diks E, de Kok A (1997) Stock allocation in general multi-echelon distribution systems with (r, s) order-up-to-policies. International Journal of Production Economics 49(2):157--174

    1997

  45. [45]

    https://www.isir2025.com/research-challenge/information

    van Dijck T, Fleuren T, Hendriks M, Temizoz T, van Jaarsveld W (2025) ASML supply chain planning challenge. https://www.isir2025.com/research-challenge/information

    2025

  46. [46]

    SSRN 4843271

    van Dijck T, Fleuren T, Temizoz T, Merzifonluoglu Y, Hendriks M, van Jaarsveld W (2024) Inventory planning in capacitated high-tech assembly systems under non-stationary demand. SSRN 4843271

    2024

  47. [47]

    van Dijck T, Helm PP, Fleuren T, Temizoz T, van Jaarsveld W (2026) Production-inventory planning in high-tech supply chains: One policy to rule them all? SSRN 6735067

    2026

  48. [48]

    Proceedings of the AAAI Conference on Artificial Intelligence 33(01):1658--1665

    Wilder B, Dilkina B, Tambe M (2019) Melding the data-decisions pipeline: Decision-focused learning for combinatorial optimization. Proceedings of the AAAI Conference on Artificial Intelligence 33(01):1658--1665

    2019

  49. [49]

    Machine Learning 8(3-4):229--256

    Williams RJ (1992) Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning 8(3-4):229--256

    1992

  50. [50]

    arXiv preprint arXiv:2603.19621

    Xie Y, Hao X, Liu J, Ma W, Xin L, Cao L, Zhang Y (2026) Deepstock: Reinforcement learning with policy regularizations for inventory management. arXiv preprint arXiv:2603.19621

    arXiv 2026