pith. machine review for the scientific record. sign in

arxiv: 2604.13478 · v1 · submitted 2026-04-15 · 🧮 math.OC · cs.CE· econ.GN· q-fin.EC· stat.AP

Recognition: unknown

Deepbullwhip: An Open-Source Simulation and Benchmarking for Multi-Echelon Bullwhip Analyses

Mansur M. Arief
Authors on Pith no claims yet

Pith reviewed 2026-05-10 12:58 UTC · model grok-4.3

classification 🧮 math.OC cs.CEecon.GNq-fin.ECstat.AP
keywords bullwhip effectmulti-echelon supply chainsinventory simulationordering policiesMonte Carlo benchmarkingdemand amplificationsemiconductor supply chainpolicy tradeoffs
0
0 comments X

The pith

A modular open-source simulator shows demand variability amplifying 427 times through four supply chain tiers and exposes tradeoffs no single metric captures.

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

The paper creates deepbullwhip to supply missing modular simulation tools and standardized benchmarks for the bullwhip effect, in which order variability grows upstream in multi-echelon chains. It packages a vectorized Monte Carlo engine with interchangeable demand generators, ordering policies, and cost models, then ships a registry of policies, forecasting methods, six metrics, and real datasets such as WSTS semiconductor billings. Experiments on a four-echelon semiconductor chain produce a mean cumulative amplification of 427 times across 1,000 paths, upstream filtering to a coefficient of variation of 0.01, super-exponential lead-time sensitivity, and a 155-fold gap between synthetic and real demand severity under the order-up-to policy. The results also show that different ordering policies trade off across the six metrics, so no lone number ranks them reliably. These capabilities matter because the bullwhip effect still drives excess inventory and shortages, and prior progress has been slowed by the lack of shared, scalable testbeds.

Core claim

Deepbullwhip supplies a simulation engine for serial multi-echelon chains that uses abstract base classes for demand generators, ordering policies, and cost functions together with a vectorized Monte Carlo engine for 50- to 90-fold speedups. On a four-echelon semiconductor model the engine produces a Monte Carlo mean cumulative amplification of 427 times, a stochastic filtering effect that reduces upstream coefficient of variation to 0.01, super-exponential lead-time sensitivity, and scalability to 20.8 million cells in under seven seconds. Benchmark runs further show a 155 times disparity in bullwhip severity between synthetic AR(1) and real WSTS data under the order-up-to policy, plus BWR-

What carries the argument

The vectorized Monte Carlo engine with pluggable abstract base classes for demand generators, ordering policies, and cost functions, plus a registry of policies, metrics, and datasets.

If this is right

  • Demand variability accumulates to a mean of 427 times its original level across four echelons.
  • Upstream tiers exhibit stochastic filtering that lowers the coefficient of variation to 0.01.
  • Amplification grows super-exponentially as lead times lengthen.
  • Real WSTS demand produces 155 times higher bullwhip severity than synthetic AR(1) data under standard policies.
  • Ordering policies display clear tradeoffs across the six bullwhip metrics, so single-metric rankings are incomplete.

Where Pith is reading between the lines

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

  • The open registry structure allows quick addition and testing of new ordering policies beyond those initially catalogued.
  • Benchmark results imply that future studies should favor real demand datasets over synthetic ones for policy evaluation.
  • Demonstrated scalability to millions of cells supports extending the framework to optimize larger or more complex networks.
  • The metric disagreements suggest developing composite scores that balance multiple bullwhip measures for policy selection.

Load-bearing premise

The abstract simulation components and the chosen four-echelon semiconductor model accurately reproduce the inventory dynamics and asymmetric costs of real multi-echelon supply chains.

What would settle it

Direct observation of amplification factors far below 427 times or absence of upstream filtering in empirical data from actual semiconductor supply chains would show the reported results do not generalize.

Figures

Figures reproduced from arXiv: 2604.13478 by Mansur M. Arief.

Figure 1
Figure 1. Figure 1: Architecture of the deepbullwhip simulation engine. At each period t, the en￾gine receives demand from the DemandGenerator, obtains forecasts from the Forecaster, queries the OrderingPolicy for an order quantity Ok(t), and evaluates inventory cost via the CostFunction. The dashed arrow shows the demand cascade: orders at echelon k become demand at echelon k+1. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Experiment 1 summary dashboard. (a) AR(1) demand with structural shock [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: BWR distributions across 1,000 Monte Carlo paths. E1 shows high stochastic [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Cumulative BWR vs. echelon for three lead time scenarios (logarithmic [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Computational scalability: serial vs. vectorized engine across (a) paths [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Policy comparison at E4: (a) cumulative BWR (logarithmic scale) and (b) total [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: POUT smoothing parameter tradeoff at E4: (a) cumulative BWR vs. NSAmp [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Forecaster comparison at E4 under OUT policy ( [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Cumulative BWR at E4 under OUT: synthetic AR(1) vs. real WSTS demand. [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
read the original abstract

The bullwhip effect remains operationally persistent despite decades of analytical research. Two computational deficiencies hinder progress: the absence of modular open-source simulation tools for multi-echelon inventory dynamics with asymmetric costs, and the lack of a standardized benchmarking protocol for comparing mitigation strategies across shared metrics and datasets. This paper introduces deepbullwhip, an open-source Python package that integrates a simulation engine for serial supply chains (with pluggable demand generators, ordering policies, and cost functions via abstract base classes, and a vectorized Monte Carlo engine achieving 50 to 90 times speedup) with a registry-based benchmarking framework shipping a curated catalog of ordering policies, forecasting methods, six bullwhip metrics, and demand datasets including WSTS semiconductor billings. Five sets of experiments on a four-echelon semiconductor chain demonstrate cumulative amplification of 427x (Monte Carlo mean across 1,000 paths), a stochastic filtering phenomenon at upstream tiers (CV = 0.01), super-exponential lead time sensitivity, and scalability to 20.8 million simulation cells in under 7 seconds. Benchmark experiments reveal a 155x disparity between synthetic AR(1) and real WSTS bullwhip severity under the Order-Up-To policy, and quantify the BWR-NSAmp tradeoff across ordering policies, demonstrating that no single metric captures policy quality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Deepbullwhip, an open-source Python package for simulating and benchmarking multi-echelon bullwhip dynamics. It provides a modular engine with abstract base classes for demand generators, ordering policies, and cost functions, a vectorized Monte Carlo simulator (50-90x speedup), and a registry-based benchmarking framework with policies, six metrics, and datasets including WSTS semiconductor data. Five experiments on a four-echelon semiconductor chain report 427x mean cumulative amplification (1,000 paths), upstream CV=0.01 filtering, super-exponential lead-time sensitivity, scalability to 20.8 million cells in <7s, a 155x synthetic AR(1) vs. real WSTS disparity under Order-Up-To, and BWR-NSAmp tradeoffs showing no single metric suffices.

Significance. If the implementation is sound, the work supplies a much-needed open, reproducible platform for multi-echelon bullwhip research that integrates real datasets and complex policies beyond what closed-form analysis can reach. The vectorized engine, multiple-metric benchmarking, and explicit demonstration that policy quality cannot be captured by one number are concrete strengths that could accelerate empirical progress in the field.

major comments (2)
  1. [Simulation engine and experimental results sections] The headline quantitative claims (427x amplification, 155x disparity, CV=0.01 filtering, lead-time sensitivity) rest entirely on the vectorized Monte Carlo engine correctly implementing serial inventory balance, lead-time delays, and policy logic with asymmetric costs. No cross-validation against closed-form bullwhip ratios is reported for standard base cases (e.g., two-echelon AR(1) demand under Order-Up-To policy, as in Lee et al. 1997). This verification is load-bearing for trusting the reported numbers rather than possible coding artifacts in demand propagation or vectorization.
  2. [Methods and benchmarking framework] The claim that the reported amplification, filtering, and policy tradeoffs generalize rests on the untested assumption that the abstract base classes and cost-asymmetry handling faithfully reproduce real multi-echelon dynamics; the manuscript provides no empirical calibration or sensitivity checks against observed semiconductor supply-chain data beyond the WSTS billings input.
minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly state the GitHub repository URL and installation command so readers can immediately access the claimed open-source code and reproduce the 20.8-million-cell scalability timing.
  2. [Benchmarking framework] Definitions and exact formulas for the six bullwhip metrics (including BWR and NSAmp) and the precise handling of cost asymmetry should be moved from supplementary material into the main text or an appendix table to support independent replication.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive evaluation of Deepbullwhip's potential contribution. We address each major comment below with clarifications and commitments to revision where the points identify verifiable gaps.

read point-by-point responses

  1. Referee: [Simulation engine and experimental results sections] The headline quantitative claims (427x amplification, 155x disparity, CV=0.01 filtering, lead-time sensitivity) rest entirely on the vectorized Monte Carlo engine correctly implementing serial inventory balance, lead-time delays, and policy logic with asymmetric costs. No cross-validation against closed-form bullwhip ratios is reported for standard base cases (e.g., two-echelon AR(1) demand under Order-Up-To policy, as in Lee et al. 1997). This verification is load-bearing for trusting the reported numbers rather than possible coding artifacts in demand propagation or vectorization.

    Authors: We agree that cross-validation against analytical results is necessary to substantiate the simulation engine. In the revised manuscript we will insert a new verification subsection (likely in Section 3 or an appendix) that replicates the two-echelon AR(1) case under the Order-Up-To policy and directly compares simulated mean amplification and variance against the closed-form bullwhip ratio derived in Lee et al. (1997). This will confirm correct serial inventory balance, lead-time handling, and vectorized sampling before presenting the four-echelon results. revision: yes

  2. Referee: [Methods and benchmarking framework] The claim that the reported amplification, filtering, and policy tradeoffs generalize rests on the untested assumption that the abstract base classes and cost-asymmetry handling faithfully reproduce real multi-echelon dynamics; the manuscript provides no empirical calibration or sensitivity checks against observed semiconductor supply-chain data beyond the WSTS billings input.

    Authors: We partially agree. The WSTS dataset serves as the primary real-world input to demonstrate the 155x disparity, and the four-echelon parameterization follows documented semiconductor lead times and cost structures. However, we acknowledge that additional sensitivity checks on cost-asymmetry parameters and explicit discussion of calibration limits would strengthen the generalization argument. In revision we will add a dedicated sensitivity subsection and a limitations paragraph clarifying that the abstract base classes are designed for user-supplied calibrated models; we will not claim broader empirical validation beyond the reported WSTS experiments. revision: partial

Circularity Check

0 steps flagged

No circularity: forward Monte Carlo simulation on explicit rules and external data

full rationale

The paper's core claims consist of numerical outcomes generated by running a vectorized Monte Carlo engine on explicitly defined ordering policies, demand generators, and external datasets (including WSTS semiconductor billings). No equations derive the reported amplification factors, CV values, or policy tradeoffs by fitting back to those same quantities, and no load-bearing step reduces to a self-citation or ansatz that presupposes the results. The derivation chain is the forward application of inventory-balance and lead-time logic to independent inputs, making the reported statistics independent of the metrics they are later summarized by.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard multi-echelon inventory modeling assumptions and the representativeness of the chosen datasets and metrics; no new free parameters, axioms, or invented entities are introduced beyond the software implementation itself.

axioms (1)
  • domain assumption Pluggable demand generators, ordering policies, and cost functions via abstract base classes faithfully represent real serial supply-chain dynamics including asymmetric costs.
    Invoked throughout the simulation engine and experiment design sections.

pith-pipeline@v0.9.0 · 5555 in / 1452 out tokens · 45073 ms · 2026-05-10T12:58:45.157847+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. ISOMORPH: A Supply Chain Digital Twin for Simulation, Dataset Generation, and Forecasting Benchmarks

    stat.ML 2026-05 accept novelty 8.0

    ISOMORPH is a modular digital twin simulator for supply chain networks that releases datasets exhibiting variance amplification and regime shifts for benchmarking forecasting models and performing forward uncertainty ...

Reference graph

Works this paper leans on

42 extracted references · 22 canonical work pages · cited by 1 Pith paper

  1. [1]

    H. L. Lee, V. Padmanabhan, S. Whang, Information distortion in a supply chain: The bullwhip effect, Management Science 43 (1997) 546–

    1997

  2. [2]

    doi:10.1287/mnsc.43.4.546

    work page doi:10.1287/mnsc.43.4.546

  3. [3]

    F. Chen, Z. Drezner, J. K. Ryan, D. Simchi-Levi, Quantifying the bullwhip effect in a simple supply chain: The impact of forecasting, lead times, and information, Management Science 46 (2000) 436–443. doi:10.1287/mnsc.46.3.436.12069. 31

    work page doi:10.1287/mnsc.46.3.436.12069 2000

  4. [4]

    S. M. Disney, D. R. Towill, On the bullwhip and inventory variance produced by an ordering policy, Omega 31 (2003) 157–167. doi:10. 1016/S0305-0483(03)00028-8

    2003

  5. [5]

    Dejonckheere, S

    J. Dejonckheere, S. M. Disney, M. R. Lambrecht, D. R. Towill, Mea- suring and avoiding the bullwhip effect: A control theoretic approach, European Journal of Operational Research 147 (2003) 567–590. doi:10. 1016/S0377-2217(02)00369-7

    2003

  6. [6]

    S. M. Disney, Z. Michna, P. Nielsen, The bullwhip effect with correlated lead times and autocorrelated demand, International Journal of Produc- tion Research (2025). doi:10.1080/00207543.2025.2566958, published online 9 October 2025

    work page doi:10.1080/00207543.2025.2566958 2025

  7. [7]

    Jannelli, S

    P. Saoud, N. Kourentzes, J. E. Boylan, The importance of forecast uncertainty in understanding the bullwhip effect, International Journal of Production Research (2025). doi:10.1080/00207543.2025.2527957, published online 9 July 2025

    work page doi:10.1080/00207543.2025.2527957 2025

  8. [8]

    Brauch, M

    M. Brauch, M. Mohaghegh, A. Größler, Causes of the bullwhip effect: A systematic review and categorization, Management Research Review 47 (2024) 1127–1149. doi:10.1108/MRR-05-2023-0392

    work page doi:10.1108/mrr-05-2023-0392 2024

  9. [9]

    Bain & Company, Prepare for the coming AI chip shortage,https://www.bain.com/insights/ prepare-for-the-coming-ai-chip-shortage-tech-report-2024/, 2024

    2024

  10. [10]

    X. Wang, S. M. Disney, The bullwhip effect: Progress, trends and directions, European Journal of Operational Research 250 (2016) 691–

    2016

  11. [11]

    doi:10.1016/j.ejor.2015.07.022

    work page doi:10.1016/j.ejor.2015.07.022 2015

  12. [12]

    IISE Transactions52(11), 1204–1217 (2020) https://doi.org/10.1080/24725854.2019.1701753

    A. Oroojlooyjadid, L. V. Snyder, M. Takáč, Applying deep learning to the newsvendor problem, IISE Transactions 52 (2020) 444–463. doi:10. 1080/24725854.2019.1632502

    work page arXiv 2020

  13. [14]

    Dominguez, S

    R. Dominguez, S. Cannella, J. M. Framinan, On bullwhip-limiting strategies in divergent supply chain networks, Computers & Industrial Engineering 73 (2014) 85–95. doi:10.1016/j.cie.2014.04.008

    work page doi:10.1016/j.cie.2014.04.008 2014

  14. [15]

    Pattern Recognition , author =

    J.K.Sagawa, R.F.S.Borges, Bullwhipeffectanalysisunderdemandun- certainty with correlated demands across retailers, Computers & Indus- trial Engineering 193 (2024) 110309. doi:10.1016/j.cie.2024.110309

    work page doi:10.1016/j.cie.2024.110309 2024

  15. [16]

    G.-Y. Ban, C. Rudin, The big data newsvendor: Practical insights from machine learning, Operations Research 67 (2019) 90–108. doi:10.1287/ opre.2018.1757

    work page arXiv 2019

  16. [17]

    R. R. Corsini, A. Costa, S. Fichera, J. M. Framinan, Digital twin model with machine learning and optimization for resilient production– distribution systems under disruptions, Computers & Industrial Engi- neering 191 (2024) 110145. doi:10.1016/j.cie.2024.110145

    work page doi:10.1016/j.cie.2024.110145 2024

  17. [18]

    J. D. Sterman, Modeling managerial behavior: Misperceptions of feed- back in a dynamic decision making experiment, Management Science 35 (1989) 321–339. doi:10.1287/mnsc.35.3.321

    work page doi:10.1287/mnsc.35.3.321 1989

  18. [19]

    Fasusi, supplychainpy: Python library for supply chain anal- ysis, modelling and simulation,https://github.com/KevinFasusi/ supplychainpy, 2018

    K. Fasusi, supplychainpy: Python library for supply chain anal- ysis, modelling and simulation,https://github.com/KevinFasusi/ supplychainpy, 2018. V0.0.5

    2018

  19. [20]

    Open-source production scheduling and capacity planning platform

    frePPLe, frePPLe: Open source supply chain planning,https:// frepple.com, 2024. Open-source production scheduling and capacity planning platform

    2024

  20. [21]

    L. Chen, H. L. Lee, Bullwhip effect measurement and its implications, Operations Research 60 (2012) 771–784. doi:10.1287/opre.1120.1074

    work page doi:10.1287/opre.1120.1074 2012

  21. [22]

    Makridakis, E

    S. Makridakis, E. Spiliotis, V. Assimakopoulos, M5 accuracy com- petition: Results, findings, and conclusions, International Journal of Forecasting 38 (2022) 1346–1364. doi:10.1016/j.ijforecast.2021. 11.013

    work page doi:10.1016/j.ijforecast.2021 2022

  22. [23]

    Monthly billings by region and product category, freely downloadable

    World Semiconductor Trade Statistics, WSTS blue book: His- torical semiconductor billings data,https://www.wsts.org/67/ Historical-Billings-Report, 2025. Monthly billings by region and product category, freely downloadable. Data through December 2025. 33

    2025

  23. [24]

    J.W.Forrester, IndustrialDynamics, MITPress, Cambridge, MA,1961

    1961

  24. [25]

    Mönch, J

    L. Mönch, J. W. Fowler, S. J. Mason, Production Planning and Schedul- ing for Semiconductor Wafer Fabrication, Springer, 2013. doi:10.1007/ 978-1-4614-4472-5

    2013

  25. [26]

    Zhang, Evolution of ARMA demand in supply chains, Manufactur- ing & Service Operations Management 6 (2004) 195–198

    X. Zhang, Evolution of ARMA demand in supply chains, Manufactur- ing & Service Operations Management 6 (2004) 195–198. doi:10.1287/ msom.1040.0042

    work page arXiv 2004

  26. [27]

    Gilbert, An ARIMA supply chain model, Management Science 51 (2005) 305–310

    K. Gilbert, An ARIMA supply chain model, Management Science 51 (2005) 305–310. doi:10.1287/mnsc.1040.0308

    work page doi:10.1287/mnsc.1040.0308 2005

  27. [29]

    Cannella, B

    S. Cannella, B. Ponte, R. Dominguez, J. M. Framinan, Proportional order-up-to policies for closed-loop supply chains: The dynamic effects of inventory controllers, International Journal of Production Research 59 (2021) 3323–3337. doi:10.1080/00207543.2020.1867924

    work page doi:10.1080/00207543.2020.1867924 2021

  28. [30]

    Q. Li, G. Gaalman, S. M. Disney, On the equivalence of the proportional and damped trend order-up-to policies, International Journal of Produc- tion Economics 265 (2023) 109008. doi:10.1016/j.ijpe.2023.109008

    work page doi:10.1016/j.ijpe.2023.109008 2023

  29. [31]

    Wright, X

    D. Wright, X. Yuan, Mitigating the bullwhip effect by ordering poli- cies and forecasting methods, International Journal of Production Eco- nomics 113 (2008) 587–597. doi:10.1016/j.ijpe.2007.10.032

    work page doi:10.1016/j.ijpe.2007.10.032 2008

  30. [32]

    Borshchev, The Big Book of Simulation Modeling: Multimethod Modeling with AnyLogic 6, AnyLogic North America, 2013

    A. Borshchev, The Big Book of Simulation Modeling: Multimethod Modeling with AnyLogic 6, AnyLogic North America, 2013

    2013

  31. [33]

    W. D. Kelton, R. P. Sadowski, N. B. Zupick, Simulation with Arena, 6th ed., McGraw-Hill, 2015

    2015

  32. [34]

    E. A. Silver, D. F. Pyke, D. J. Thomas, Inventory and Production Man- agement in Supply Chains, 4th ed., CRC Press, Boca Raton, FL, 2017. doi:10.1201/9781315374406. 34

    work page doi:10.1201/9781315374406 2017

  33. [35]

    Cachon, C

    G. Cachon, C. Terwiesch, Matching Supply with Demand: An Intro- duction to Operations Management, 3rd ed., McGraw-Hill, New York, 2012

    2012

  34. [36]

    P. H. Zipkin, Foundations of Inventory Management, McGraw- Hill/Irwin, Boston, MA, 2000

    2000

  35. [37]

    Gamma, R

    E. Gamma, R. Helm, R. Johnson, J. Vlissides, Design Patterns: Ele- ments of Reusable Object-Oriented Software, Addison-Wesley, 1994

    1994

  36. [38]

    Spyros Makridakis, Evangelos Spiliotis, and Vassilios Assimakopoulos

    S. Makridakis, E. Spiliotis, V. Assimakopoulos, The M4 competition: 100,000 time series and 61 forecasting methods, International Journal of Forecasting36(2020)54–74.doi:10.1016/j.ijforecast.2019.04.014

    work page doi:10.1016/j.ijforecast.2019.04.014 2020

  37. [39]

    K. Chen, J. Wang, J. Pang, Y. Cao, Y. Xiong, X. Li, S. Sun, W. Feng, Z. Liu, J. Xu, Z. Zhang, D. Cheng, C. Zhu, T. Cheng, Q. Zhao, B. Li, X. Lu, R. Zhu, Y. Wu, J. Dai, J. Wang, J. Shi, W. Ouyang, C. C. Loy, D. Lin, MMDetection: Open MMLab detection toolbox and bench- mark, 2019. Software available athttps://github.com/open-mmlab/ mmdetection

    2019

  38. [40]

    Y. Wu, A. Kirillov, F. Massa, W.-Y. Lo, R. Girshick, Detectron2,

  39. [41]

    Software available athttps://github.com/facebookresearch/ detectron2

  40. [42]

    R. C. Martin, Agile Software Development: Principles, Patterns, and Practices, Prentice Hall, Upper Saddle River, NJ, 2003

    2003

  41. [43]

    semiconductor industry 2025,https://www.semiconductors.org, 2025

    Semiconductor Industry Association, State of the U.S. semiconductor industry 2025,https://www.semiconductors.org, 2025

    2025

  42. [44]

    Confinement and the center of the gauge group,

    R. N. Boute, J. Gijsbrechts, W. van Jaarsveld, N. Vanvuchelen, Deep reinforcement learning for inventory control: A roadmap, European Journal of Operational Research 298 (2022) 401–412. doi:10.1016/j. ejor.2021.07.016. 35 Appendix A. Proof of Proposition 1 Proof.LetX= Var(O k |ω)andY= Var(O k−1 |ω)be positive random variables (over the probability space o...

    work page doi:10.1016/j 2022