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arxiv: 2605.03785 · v2 · submitted 2026-05-05 · 🧮 math.OC

The Distributionally Robust Cyclic Inventory Routing Problem

Menglei Jia , Albert H. Schrotenboer , Ahmadreza Marandi , Feng Chen This is my paper

Pith reviewed 2026-05-08 18:46 UTC · model grok-4.3

classification 🧮 math.OC
keywords distributionally robust optimizationcyclic inventory routingmoment-based ambiguity setchance constraintbranch-and-priceinfinite-horizon planningvehicle routinginventory replenishment
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The pith

The worst-case expected inventory cost in cyclic routing is attained at a multi-point demand distribution identifiable by linear programming, allowing a deterministic reformulation of the distributionally robust problem.

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

The paper establishes a distributionally robust model for the cyclic inventory routing problem on an infinite horizon, where retailer demands are uncertain and belong to a moment-based ambiguity set. It minimizes the worst-case expected total cost while enforcing service reliability via a distributionally robust chance constraint. The central results show that the worst-case inventory cost is achieved under a multi-point distribution that can be recovered after the fact by solving a linear program, and that the chance constraint admits a near-exact deterministic equivalent. This combination produces a fully deterministic reformulation of the original stochastic problem that can be solved by a nested branch-and-price algorithm.

Core claim

We prove that the worst-case expected inventory cost is attained under a multi-point distribution, which can be identified a posteriori via linear programming, and that the distributionally robust chance constraint can be reformulated into near-equivalent deterministic forms. This yields a deterministic reformulation of the original problem.

What carries the argument

The moment-based ambiguity set on retailer demands together with the distributionally robust chance constraint, which together permit exact identification of the worst-case multi-point distribution via linear programming and deterministic reformulation of the robust model.

If this is right

  • The cyclic inventory routing problem admits an exact deterministic equivalent that can be solved by nested branch-and-price without sampling or scenario generation.
  • The first level of the algorithm partitions retailers into clusters; the second level solves routing and replenishment inside each cluster.
  • The same reformulation technique applies to both synthetic instances and real-world data drawn from a major automobile manufacturer.
  • Service reliability is maintained at the prescribed level for every distribution inside the moment-based ambiguity set.

Where Pith is reading between the lines

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

  • The approach may extend to other cyclic or periodic routing problems whose costs are convex in demand, because the multi-point worst-case property relies mainly on moment constraints rather than the specific routing structure.
  • Replacing the moment-based set with a Wasserstein ball would require a different reformulation argument, since the worst-case distribution need no longer be discrete with a small support.
  • The nested branch-and-price structure suggests that similar two-level decompositions could be useful for other distributionally robust vehicle-routing variants with infinite-horizon inventory costs.

Load-bearing premise

Retailer demands are random variables whose distributions belong to a moment-based ambiguity set, and service reliability is enforced via a distributionally robust chance constraint whose reformulation remains valid for the cyclic infinite-horizon setting.

What would settle it

An instance in which the linear program that identifies the candidate multi-point distribution returns a worst-case expected cost strictly lower than the true supremum over all distributions in the ambiguity set.

Figures

Figures reproduced from arXiv: 2605.03785 by Ahmadreza Marandi, Albert H. Schrotenboer, Feng Chen, Menglei Jia.

Figure 3
Figure 3. Figure 3: Illustration of a node with two infeasible intervals, leading to four child nodes, each with updated sets according to the branching conditions. Solving 2nd-PP2(𝑖) with incorporated lazy constraints. We now describe how to solve 2nd-PP2(𝑖) under given states 𝐴𝑌 and 𝐴𝑁 . Eariler, we introduced how to solve R-2nd-PP2(𝑖); with the additional restrictions imposed by the sets 𝐴𝑌 and 𝐴𝑁 , the solution process re… view at source ↗
read the original abstract

We study the cyclic inventory routing problem that involves joint decisions on vehicle routing and inventory replenishment on an infinite, cyclic horizon. It considers a single warehouse and a set of geographically dispersed retailers. We model retailer demand as random variables with uncertain distributions belonging to a moment-based ambiguity set. We develop a distributionally robust optimization formulation that minimizes the worst-case expected cost over the ambiguity set, while ensuring service reliability through a distributionally robust chance constraint. Our main results are that we prove that the worst-case expected inventory cost is attained under a multi-point distribution, which can be identified a posteriori via linear programming, and that the distributionally robust chance constraint can be reformulated into near-equivalent deterministic forms. This yields a deterministic reformulation of the original problem. To solve it, we design a nested branch-and-price framework, in which the first level partitions retailers into clusters, and the second level concerns routing and replenishment decisions within each cluster. Computational experiments on both synthetic instances and real-world data from SAIC Volkswagen Automobile Co., Ltd. demonstrate the effectiveness and efficiency of the proposed approach.

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

0 major / 2 minor

Summary. The paper studies the cyclic inventory routing problem (CIRP) over an infinite horizon with a single warehouse and dispersed retailers. Retailer demands are modeled as random variables belonging to a moment-based ambiguity set. The authors formulate a distributionally robust optimization problem that minimizes worst-case expected cost subject to a distributionally robust chance constraint for service reliability. They prove that the worst-case expected inventory cost is attained at a multi-point distribution identifiable a posteriori by linear programming, and that the chance constraint admits a near-equivalent deterministic reformulation. The resulting deterministic model is solved via a nested branch-and-price algorithm (outer level: retailer clustering; inner level: routing and replenishment within clusters). Computational experiments on synthetic instances and real data from SAIC Volkswagen are reported.

Significance. If the claimed reformulations and proofs hold, the work provides a tractable DRO approach to cyclic IRP that handles demand ambiguity while preserving periodicity and infinite-horizon structure. The multi-point distribution property, LP-based identification, and nested branch-and-price decomposition are concrete strengths that extend standard moment-based DRO techniques to this setting. The inclusion of real-world data strengthens practical relevance for periodic replenishment applications.

minor comments (2)
  1. [Abstract and main-results section] The abstract and the section stating the main results use the phrase 'near-equivalent deterministic forms' for the chance-constraint reformulation without specifying the precise sense of approximation (e.g., exact equivalence under additional conditions, or a bounded violation probability). Clarify this terminology and state any conditions required for validity in the cyclic setting.
  2. [Solution-method section] The description of the nested branch-and-price framework would benefit from an explicit pseudocode or flowchart that distinguishes the outer clustering partition from the inner routing/replenishment subproblems, including how inventory carry-over across cycles is handled in the pricing problems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our work on the distributionally robust cyclic inventory routing problem. The provided summary accurately reflects the problem setting, our moment-based ambiguity set, the DRO formulation with chance constraints, the multi-point worst-case distribution result, the deterministic reformulation, the nested branch-and-price algorithm, and the experiments on synthetic and SAIC Volkswagen data. We are grateful for the recommendation of minor revision.

Circularity Check

0 steps flagged

No significant circularity; standard DRO reformulations are self-contained

full rationale

The paper's core claims—that the worst-case expected inventory cost under a moment-based ambiguity set is attained at a multi-point distribution identifiable a posteriori by LP, and that the distributionally robust chance constraint admits a near-equivalent deterministic reformulation valid for the cyclic infinite-horizon setting—follow directly from standard duality and reformulation techniques in distributionally robust optimization applied to periodic inventory costs. These steps do not reduce by construction to fitted parameters, self-definitional equations, or load-bearing self-citations; the nested branch-and-price is a natural decomposition of the resulting deterministic model. The derivation remains independent of the paper's own inputs and is externally falsifiable against DRO literature benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from distributionally robust optimization rather than new invented entities or heavily fitted parameters.

axioms (2)
  • domain assumption Retailer demands are random variables whose distributions lie in a moment-based ambiguity set
    Invoked in the modeling section of the abstract as the foundation for the worst-case expectation and chance constraint.
  • domain assumption The infinite cyclic horizon allows steady-state analysis of inventory and routing decisions
    Implicit in the problem definition and required for the cyclic formulation to be well-posed.

pith-pipeline@v0.9.0 · 5492 in / 1488 out tokens · 33098 ms · 2026-05-08T18:46:11.260630+00:00 · methodology

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Reference graph

Works this paper leans on

72 extracted references · 3 canonical work pages

  1. [1]

    IISE Transactions , volume=

    Robust inventory routing problem considering budget violation under demand uncertainty , author=. IISE Transactions , volume=. 2025 , publisher=

    2025

  2. [2]

    Operations Research , volume=

    The distributionally robust chance-constrained vehicle routing problem , author=. Operations Research , volume=. 2020 , publisher=

    2020

  3. [3]

    International Journal of Production Economics , volume=

    Multi-vehicle stochastic cyclic inventory routing with guaranteed replenishments , author=. International Journal of Production Economics , volume=. 2021 , publisher=

    2021

  4. [4]

    Operations Research , volume=

    Robust optimization with ambiguous stochastic constraints under mean and dispersion information , author=. Operations Research , volume=. 2018 , publisher=

    2018

  5. [5]

    Management Science , volume=

    Closed-form solutions for robust inventory management , author=. Management Science , volume=. 2017 , publisher=

    2017

  6. [6]

    Operations research , volume=

    On the quasiconcavity of lost-sales inventory models with fixed costs , author=. Operations research , volume=. 2012 , publisher=

    2012

  7. [7]

    Journal of the Operational Research Society , volume=

    Robust distribution planning for supplier-managed inventory agreements when demand rates and travel times are stationary , author=. Journal of the Operational Research Society , volume=. 2008 , publisher=

    2008

  8. [8]

    Management Science , volume=

    The value of information design in supply chain management , author=. Management Science , volume=. 2025 , publisher=

    2025

  9. [9]

    Transportation Science , volume=

    Stochastic inventory routing for perishable products , author=. Transportation Science , volume=. 2018 , publisher=

    2018

  10. [10]

    Transportation Science , volume=

    Stochastic inventory routing: Route design with stockouts and route failures , author=. Transportation Science , volume=. 1992 , publisher=

    1992

  11. [11]

    Management Science , volume=

    Supply chain competition: A market game approach , author=. Management Science , volume=. 2020 , publisher=

    2020

  12. [12]

    Procedia-Soc

    Cyclic versus reactive planning for inventory routing , author=. Procedia-Soc. Behav. Sci , volume=. 2014 , publisher=

    2014

  13. [13]

    European Journal of Operational Research , volume=

    Stochastic inventory routing with time-based shipment consolidation , author=. European Journal of Operational Research , volume=. 2023 , publisher=

    2023

  14. [14]

    Operations Research , volume=

    Tailored base-surge policies in dual-sourcing inventory systems with demand learning , author=. Operations Research , volume=. 2025 , publisher=

    2025

  15. [15]

    Management Science , volume=

    Bandits atop reinforcement learning: Tackling online inventory models with cyclic demands , author=. Management Science , volume=. 2024 , publisher=

    2024

  16. [16]

    Transportation science , volume=

    A branch-and-cut algorithm for a vendor-managed inventory-routing problem , author=. Transportation science , volume=. 2007 , publisher=

    2007

  17. [17]

    INFORMS Journal on Computing , volume=

    A hybrid heuristic for an inventory routing problem , author=. INFORMS Journal on Computing , volume=. 2012 , publisher=

    2012

  18. [18]

    Transportation Science , volume=

    Deterministic order-up-to level policies in an inventory routing problem , author=. Transportation Science , volume=. 2002 , publisher=

    2002

  19. [19]

    Manufacturing & Service Operations Management , volume=

    Robust inventory management: A cycle-based approach , author=. Manufacturing & Service Operations Management , volume=. 2023 , publisher=

    2023

  20. [20]

    Computers & Operations Research , volume=

    Cyclic inventory routing with dynamic safety stocks under recurring non-stationary interdependent demands , author=. Computers & Operations Research , volume=. 2021 , publisher=

    2021

  21. [21]

    Transportation Science , year=

    Stochastic cyclic inventory routing with supply uncertainty: A case in green-hydrogen logistics , author=. Transportation Science , year=

  22. [22]

    Networks: An International Journal , volume=

    An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems , author=. Networks: An International Journal , volume=. 2004 , publisher=

    2004

  23. [23]

    Transportation Science , volume=

    Exact branch-price-and-cut algorithms for vehicle routing , author=. Transportation Science , volume=. 2019 , publisher=

    2019

  24. [24]

    Mathematical Programming , volume=

    Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations , author=. Mathematical Programming , volume=. 2018 , publisher=

    2018

  25. [25]

    Mathematical Programming , volume=

    Data-driven chance constrained stochastic program , author=. Mathematical Programming , volume=. 2016 , publisher=

    2016

  26. [26]

    Management Science , volume=

    Robust solutions of optimization problems affected by uncertain probabilities , author=. Management Science , volume=. 2013 , publisher=

    2013

  27. [27]

    Operations research , volume=

    Distributionally robust convex optimization , author=. Operations research , volume=. 2014 , publisher=

    2014

  28. [28]

    Operations research , volume=

    Distributionally robust optimization under moment uncertainty with application to data-driven problems , author=. Operations research , volume=. 2010 , publisher=

    2010

  29. [29]

    Mathematics of Operations Research , volume=

    Models for minimax stochastic linear optimization problems with risk aversion , author=. Mathematics of Operations Research , volume=. 2010 , publisher=

    2010

  30. [30]

    Operations Research , volume=

    Robust mean-covariance solutions for stochastic optimization , author=. Operations Research , volume=. 2007 , publisher=

    2007

  31. [31]

    European Journal of Operational Research , year=

    Cyclic stochastic inventory routing planning with reorder points and recourse decision for an application in medical supply , author=. European Journal of Operational Research , year=

  32. [32]

    European Journal of Operational Research , volume=

    A shortfall modelling-based solution approach for stochastic cyclic inventory routing , author=. European Journal of Operational Research , volume=. 2023 , publisher=

    2023

  33. [33]

    Mathematical Programming , volume=

    Quantitative stability analysis for minimax distributionally robust risk optimization , author=. Mathematical Programming , volume=. 2022 , publisher=

    2022

  34. [34]

    European Journal of Operational Research , volume=

    Nested branch-and-price-and-cut for vehicle routing problems with multiple resource interdependencies , author=. European Journal of Operational Research , volume=. 2019 , publisher=

    2019

  35. [35]

    2004 , publisher=

    Convex optimization , author=. 2004 , publisher=

    2004

  36. [36]

    Operations Research , volume=

    The inventory routing problem under uncertainty , author=. Operations Research , volume=. 2023 , publisher=

    2023

  37. [37]

    Available at SSRN 4101774 , year=

    Distributionally Robust Inventory Management with Advance Purchase Contracts , author=. Available at SSRN 4101774 , year=

  38. [38]

    Transportation Science , volume=

    The fixed-partition policy inventory routing problem , author=. Transportation Science , volume=. 2021 , publisher=

    2021

  39. [39]

    Operations Research , volume=

    A periodic inventory routing problem at a supermarket chain , author=. Operations Research , volume=. 2004 , publisher=

    2004

  40. [40]

    Computers & Operations Research , volume=

    Heuristics for dynamic and stochastic inventory-routing , author=. Computers & Operations Research , volume=. 2014 , publisher=

    2014

  41. [41]

    Journal of Mathematical Analysis and Applications , volume=

    On optimality criteria for dynamic programs with long finite horizons , author=. Journal of Mathematical Analysis and Applications , volume=. 1980 , publisher=

    1980

  42. [42]

    Transportation Research Part C: Emerging Technologies , volume=

    Consistency in multi-vehicle inventory-routing , author=. Transportation Research Part C: Emerging Technologies , volume=. 2012 , publisher=

    2012

  43. [43]

    Maritime Transport Research , volume=

    Cyclic solutions to a maritime inventory routing problem , author=. Maritime Transport Research , volume=. 2022 , publisher=

    2022

  44. [44]

    Transportation science , volume=

    Delivery cost approximations for inventory routing problems in a rolling horizon framework , author=. Transportation science , volume=. 2002 , publisher=

    2002

  45. [45]

    Operations research , volume=

    A column-generation based tactical planning method for inventory routing , author=. Operations research , volume=. 2012 , publisher=

    2012

  46. [46]

    arXiv preprint arXiv:2401.17787 , year=

    Scenario Predict-then-Optimize for Data-Driven Online Inventory Routing , author=. arXiv preprint arXiv:2401.17787 , year=

    work page arXiv

  47. [47]

    Discrete Optimization , volume=

    An integer optimality condition for column generation on zero--one linear programs , author=. Discrete Optimization , volume=. 2019 , publisher=

    2019

  48. [48]

    2000 , publisher=

    Foundations of Inventory Management , author=. 2000 , publisher=

    2000

  49. [49]

    Management Science , volume=

    Adaptive distributionally robust optimization , author=. Management Science , volume=. 2019 , publisher=

    2019

  50. [50]

    Optimal pricing, inflation, and the cost of price adjustment , pages=

    The optimality of (S, s) policies in the dynamic inventory problem , author=. Optimal pricing, inflation, and the cost of price adjustment , pages=. 1960 , publisher=

    1960

  51. [51]

    arXiv preprint arXiv:2203.10661 , year=

    On the sparsity of optimal linear decision rules in robust inventory management , author=. arXiv preprint arXiv:2203.10661 , year=

    work page arXiv

  52. [52]

    Iie Transactions , volume=

    Minimax analysis for finite-horizon inventory models , author=. Iie Transactions , volume=. 2001 , publisher=

    2001

  53. [53]

    Journal of the Operational Research Society , volume=

    The distribution free newsboy problem: review and extensions , author=. Journal of the Operational Research Society , volume=. 1993 , publisher=

    1993

  54. [54]

    Journal of business logistics , volume=

    Vendor-managed inventory in the retail supply chain , author=. Journal of business logistics , volume=. 1999 , publisher=

    1999

  55. [55]

    Management Science , volume=

    Stock replenishment and shipment scheduling for vendor-managed inventory systems , author=. Management Science , volume=. 2000 , publisher=

    2000

  56. [56]

    Management science , volume=

    The inventory benefit of shipment coordination and stock rebalancing in a supply chain , author=. Management science , volume=. 2002 , publisher=

    2002

  57. [57]

    Management Science , volume=

    Dynamic inventory control with stockout substitution and demand learning , author=. Management Science , volume=. 2020 , publisher=

    2020

  58. [58]

    Computers & Operations Research , volume=

    Supply interruptions in a lost-sales inventory system with random lead time , author=. Computers & Operations Research , volume=. 2003 , publisher=

    2003

  59. [59]

    European Journal of Operational Research , volume=

    An inventory control model for modal split transport: A tailored base-surge approach , author=. European Journal of Operational Research , volume=. 2018 , publisher=

    2018

  60. [60]

    Available at SSRN , year=

    Asymptotic Optimality of Simple Replenishment Policies for a Lost-Sales Inventory System with Delivery Lead Time and Purchase Returns , author=. Available at SSRN , year=

  61. [61]

    @FrickTobias , year =

  62. [62]

    2022 , note =

    IBM ILOG CPLEX Optimization Studio , author =. 2022 , note =

    2022

  63. [63]

    Management science , volume=

    One warehouse multiple retailer systems with vehicle routing costs , author=. Management science , volume=. 1990 , publisher=

    1990

  64. [64]

    Management Science , volume=

    On the effectiveness of direct shipping strategy for the one-warehouse multi-retailer R-systems , author=. Management Science , volume=. 1990 , publisher=

    1990

  65. [65]

    European Journal of Operational Research , volume=

    A two-phase algorithm for the cyclic inventory routing problem , author=. European Journal of Operational Research , volume=. 2016 , publisher=

    2016

  66. [66]

    European Journal of Operational Research , volume=

    A practical solution approach for the cyclic inventory routing problem , author=. European Journal of Operational Research , volume=. 2009 , publisher=

    2009

  67. [67]

    European Journal of Operational Research , volume=

    Route and fleet design for cyclic inventory routing , author=. European Journal of Operational Research , volume=. 2017 , publisher=

    2017

  68. [68]

    European Journal of Operational Research , volume=

    An iterated local search algorithm for the single-vehicle cyclic inventory routing problem , author=. European Journal of Operational Research , volume=. 2014 , publisher=

    2014

  69. [69]

    Operations Research , volume=

    Distributionally robust optimization under distorted expectations , author=. Operations Research , volume=. 2025 , publisher=

    2025

  70. [70]

    Distributionally robust optimization: A review

    Distributionally robust optimization: A review , author=. arXiv preprint arXiv:1908.05659 , year=

    work page arXiv 1908

  71. [71]

    Acta Numerica , volume=

    Distributionally robust optimization , author=. Acta Numerica , volume=. 2025 , publisher=

    2025

  72. [72]

    Operations Research , year=

    Optimized dimensionality reduction for moment-based distributionally robust optimization , author=. Operations Research , year=