A Complete Algebraic Solution to the Optimal Dynamic Rationing Policy in the Stock-Rationing Queue with Two Demand Classes

Heng-Li Liu; Jing-Yu Ma; Quan-Lin Li; Yi-Meng Li

REVIEW 4 major objections 4 minor 95 references

A Complete Algebraic Solution to the Optimal Dynamic Rationing Policy in the Stock-Rationing Queue with Two Demand Classes

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that in a two-class stock-rationing queue the optimal dynamic rationing policy is always a single-threshold policy after inventory levels are reordered by policy-dependent penalty breakpoints.

desk verdict The 'complete algebraic solution' is not proven: Theorem 9 assembles the optimum from thresholds of a fixed reference policy, and the numerical experiment quietly omits the best threshold baseline. read the letter →

arxiv 1908.09295 v3 pith:WPSUSNEL submitted 2019-08-25 math.OC cs.CEcs.SYeess.SY

Quan-Lin Li , Yi-Meng Li , Jing-Yu Ma , Heng-Li Liu This is my paper · ORCID

classification math.OCcs.CEcs.SYeess.SY MSC 90B2290C4060J27

keywords stock-rationingqueuedynamicrationingpolicytwodemandclassessensitivity-basedoptimizationperformancedifferenceequationthresholdpenaltycosttransformational

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 studies a warehouse that stocks one product and faces two demand classes: high-priority demand is always served when inventory is positive, while low-priority demand can be served or rejected when inventory is low, with a penalty cost for serving it. It claims that the optimal dynamic rationing policy—the rule for when to reject low-priority demand—always has a threshold structure after a certain reordering of inventory levels. The reordering is by the penalty-cost values at which a marginal comparison for each inventory level changes sign. If true, this gives a complete and computable description of the optimal policy, including the difficult middle penalty range where earlier methods only suggested bang-bang control. The result matters for make-to-stock systems, supply chains, and other settings where inventory is rationed across customers with different priorities.

What carries the argument

The central object is the perturbation realization factor G^(d)(i)=g^(d)(i-1)-g^(d)(i), the difference between the performance potentials of adjacent inventory levels under policy d. Combined with the linear equation G^(d)(i)+b=0 in the penalty cost P, this produces the breakpoints P_i^(d) that order the low-stock states. The performance difference equation η_{d'}-η_d=μ_2 π^(d')(i)(d'_i-d_i)[G^(d)(i)+b], valid for policies differing at a single position, carries the argument: it reduces a profit comparison between policies to the sign of one factor per inventory level, and that sign is controlled by where the penalty cost P sits relative to P_i^(d).

What would settle it

Enumerate all policies for a small instance with K low-stock levels in the middle penalty region, for example K=5, N=100, $\lambda$=3, mu_1=4, mu_2=2, C1=1, C2,1=4, C2,2=1, C3=5, C4=1, R=15, with P strictly between PL(d) and PH(d). Compute the long-run average profit of every policy from the birth-death stationary distribution, and also compute the best policy that becomes a block of zeros followed by ones after sorting levels by that policy's own breakpoints P_i^(d); if the unrestricted best profit exceeds the transformed-threshold best profit, the paper's central existence result fails.

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Extended reading notes

Core claim

The paper proves that for any given policy d, the sign of G^(d)(i)+b, where G^(d)(i) is the perturbation realization factor between adjacent inventory levels and b=R+C_{2,2}-P, determines whether changing the service decision at level i improves the long-run average profit. The unique penalty value P_i^(d) where G^(d)(i)+b=0 orders the low-stock levels. Sorting those levels by the P_i^(d), the optimal dynamic rationing policy takes the transformed form d*(Transfer)=(0;0,...,0,1,...,1;1,...,1): reject Class 2 demand at the transformed levels with the smallest breakpoints and serve it at the rest. When the sorted order is the natural order 1,2,...,K, the original policy itself is a threshold policy; when it is not, the optimal policy is of what the paper calls transformational threshold type. The paper also gives three penalty regions—high, low, and middle—with explicit sufficient conditions under which the original policy is of threshold type.

Load-bearing premise

The core assumption is that a policy can be assembled coordinate by coordinate: an improving change at one stock level stays improving after changes are made at other stock levels, even though those changes alter the system's steady-state probabilities and the marginal values that justify the first change.

Editorial extensions

If this is right

If the penalty cost P is at least PH(d), the optimal policy rejects every Class 2 demand while stock is low: d*=(0;0,...,0;1,...,1).
If 0<P<=PL(d) and PL(d)>0, the optimal policy serves Class 2 at every low-stock level: d*=(0;1,...,1;1,...,1).
In the middle range PL(d)<P<PH(d), the optimal policy becomes a threshold after sorting inventory levels by the breakpoints P_i^(d); when that sorted order is the natural order 1,2,...,K, the original policy itself is a threshold policy.
The long-run average profit is linear in the penalty cost for each fixed policy, and the paper derives closed-form profit expressions for the extreme optimal policies.
Restricting to static threshold policies is suboptimal in the middle penalty region, so a manager who insists on a fixed critical rationing level can lose profit relative to the optimal dynamic policy.

Reading between the lines

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

The same sign-of-G^(d)(i)+b criterion suggests a natural extension to more than two demand classes: the single transformed threshold would likely become a staircase of cuts ordered by the analogous breakpoints, one cut per additional priority class.
Because the breakpoints P_i^(d) depend on the reference policy, the characterization suggests an iterative policy-improvement loop—sort states by current breakpoints, apply the transformed threshold update, recompute, and repeat—that could find the optimum without enumerating 2^K policies.
The algebra uses only the birth-death generator and the linearity of profit in P, so the threshold-after-reordering structure may persist under phase-type service times or Markovian arrival processes, although the paper does not claim that extension.

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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

Two things you should know. First, the central structural result (Theorems 9–10) is not established. The proof takes a reference policy d, sorts states by the roots P_i^(d) of G^(d)(i)+b=0, and claims that because P lies above all roots on one block and below all roots on another, the optimal sub-policies are all-zero and all-one. But P_i and G change when you change the policy; the roots for a modified policy are not P_i^(d). The performance difference equation only compares policies that differ at one coordinate, and the induction in Theorem 3 tries to transfer sign information across coordinates after policy changes. Step one proves G^(d(1))(j1)+b <= 0 only at the one coordinate where d(1) differs from d; it does not give the sign at other coordinates for the same d(1), so Step two's use of the induction premise is unjustified. The wildcard sub-policies in Theorem 9 are not elements of D, so no stationary distribution or profit is defined for them. Once this goes, Theorem 10 goes with it.

Second, the numerics do not demonstrate what they claim. In Example 2, K=15 and they scan theta=1..15, omitting theta=16, which is the all-zero threshold policy. They report a static optimum theta*=9 with value 21.4, below the all-zero value 22.3, and conclude the dynamic optimum is non-threshold. But all-zero is itself in the static family; with theta=16 included it would be the best static policy. The example is internally inconsistent.

The paper is not without merit. The policy-based birth-death model and explicit stationary probabilities are clean, the general solution of the Poisson equation with two free constants is a nice observation, and the performance difference equation is standard. The idea of a 'transformational threshold' is a reasonable way to think about permuting state coordinates. But the central theorem is load-bearing and unproven, and the numerical evidence for the main claim does not hold up.

Who is this for? Readers working on inventory rationing may find the modeling and computational framework useful, but they should not rely on the main theorem. This paper deserves a serious referee because the claims are formal and checkable, and a referee report could pinpoint the exact gap. My recommendation would be reject or major revision; the current version should not be accepted.

Referee Report

4 major / 4 minor

Summary. This paper studies a single-product, two-class stock-rationing queue with Poisson arrivals, exponential service times, lost sales, and a penalty cost P for serving Class 2 at low inventory. Using sensitivity-based optimization, the authors derive a performance difference equation (Lemma 2) and define P_i^(d) as the root of G^(d)(i)+b=0 for each policy d and state i. They divide the penalty range into P ≥ PH(d), 0 ≤ P ≤ PL(d), and PL(d) < P < PH(d). The paper claims that in the first two regions the optimal policy is the all-0 or all-1 threshold policy, and in the middle region the optimal policy is of 'transformational threshold type': after permuting states by the order of P_i^(d), the optimal policy is a single threshold. This is summarized as Theorem 10 and advertised as a complete algebraic solution to the optimal dynamic rationing problem.

Significance. If the main theorem were correct, it would give a structural characterization of optimal rationing policies that goes beyond the threshold results obtained by MDP/submodularity methods, and it would demonstrate the usefulness of sensitivity-based optimization for inventory rationing. The paper does contain some correct and useful building blocks: the policy-based birth-death description, the explicit stationary distribution, the linearity of the long-run average profit in P, and the single-coordinate performance difference equation are standard but carefully derived. The explicit formulas for the profit of static policies are also a useful computational reference. However, the central structural theorem is not established, and the numerical experiment intended to support it omits a key policy, so the claimed significance is not currently realized.

major comments (4)

[Section 7.3, Theorem 9 (and Theorem 8)] The proof of the middle-region result assumes that coordinates can be optimized independently. The objects ~d_a = (0; d1, ..., d_{n0-1}, *, ..., *; 1, ..., 1) and ~d_b = (0; *, ..., *, d_{n0}, ..., d_K; 1, ..., 1) are not policies in the policy space D: no stationary distribution, perturbation realization factor, or long-run average profit is defined for them. Moreover, P_i^(d) is defined as the root of G^(d)(i)+b=0 for the fixed reference policy d, and once any other coordinate is changed, both pi(d') and G^(d')(i) change; hence the sign of G^(d')(i)+b at the same P is not controlled by P_i^(d). The 'it is easy to see from the proof of Theorem 4' step is therefore not a valid derivation, and Theorems 9 and 10 are left without proof.
[Section 6.3, Theorem 3] The induction in the proof of Theorem 3 is invalid because the sign-transfer step (the equation relating G^(d(1))(j1)+b to G^(d)(j1)+b) applies only at the single coordinate where the two adjacent policies differ. In Step one, the argument shows only that for the policy d(1) that differs from d at coordinate j1, the sign at that same coordinate j1 is nonpositive. In Step two, to conclude G^(d(2))(j2)+b <= 0, the proof needs G^(d(1))(j2)+b <= 0 at the next coordinate j2, which was never established because d(1) and d differ only at j1. The phrase 'for each j1' does not turn this into a property of one policy at all coordinates of S(d,c). Theorem 3(1) and (2), which are used in Theorems 4 and 6, therefore do not follow.
[Section 9, Example 2] The numerical scan restricts theta to 1 <= theta <= 15, but the static policy set D_Delta defined in Section 8 includes theta = K+1 = 16. The omitted policy d_Delta,16 is exactly (0; 0, ..., 0; 1, ..., 1), which is the policy d* of Example 1 with eta = 22.3. Since theta = 16 is not included, the observation that eta_d_Delta,9 = 21.4 < eta_d* = 22.3 does not establish that the optimal static rationing policy is suboptimal in D, and it cannot support the conclusion that the optimal dynamic policy is not of threshold type. The numerical demonstration of the paper's central claim is therefore incomplete.
[Section 7.3, Theorem 10 and Section 6.3, Eq. (45)] The 'transformational threshold' characterization is self-referential and non-constructive. The permutation used to define d*(Transfer) is determined by sorting the values P_i^(d), but P_i^(d) is a function of the stationary distribution pi(d) of the very policy being sorted; thus the theorem describes a property that an optimal policy would satisfy after being sorted by its own P-values, rather than providing an algebraic rule that determines the optimal policy or its permutation without enumerating the 2^K policies. The title's claim of a 'complete algebraic solution' is therefore stronger than what is actually shown.

minor comments (4)

[Section 6.3, proof of Lemma 4] PL(d) is written with a max instead of a min in the proof; the displayed definition in Eq. (47) uses min, so the proof should be corrected.
[Section 9, Example 2, Figures 3 and 4] The vertical axis is labeled eta_d* in both figures, but the plotted quantity appears to be eta_d_Delta,theta, the long-run average profit of the static policy indexed by theta; the labels should distinguish the two quantities.
[Section 4, Eq. (7)] The reward function uses indicator notation 1_{i<N} and 1_{i=N}; this is fine, but the same symbol 1 is later used for both an indicator and the numeral one, which makes equations such as (17) harder to read.
[Section 5, Theorem 1] The claim that this is 'the first' general solution of Poisson equations with two free constants is not substantiated by a comparison with the cited references; please soften the claim or provide a precise novelty statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's structural results are derived from the performance-difference equation and explicit sign analysis, not assumed through the definitions.

full rationale

The derivation chain is self-contained rather than circular. The perturbation factor G^(d)(i) and the root P_i^(d) are explicitly computed from the Poisson equation and the stationary distribution of a given policy d, and the sign statements in Section 6.3 follow from the linearity of G^(d)(i)+b in P. Lemma 2 then gives a local pairwise comparison, and the threshold/transformational-threshold conclusions in Theorems 4, 6, 8, 9, and 10 are consequences of applying that comparison to the sign of G^(d)(i)+b. The policy-dependent permutation used to define the transformational policy is a fixed-point self-consistency property of an optimal policy, not an input that already contains the output: the theorem states that after ordering states by the optimal policy's own P_i^(d*), the decisions are monotone, which is a substantive consequence of the sign analysis rather than a definitional identity. The main proof gaps, such as the use of wildcard sub-policies that are not admissible policies in D, are correctness concerns about the induction argument, not circular reductions of the claimed result to its own inputs. Self-citations in the paper, including references to Cao [11], Li [56], and Ma et al. [60], are used for standard background results such as the performance-difference equation and the stationary distribution of birth-death processes; these are externally established tools, and the paper's central sign and threshold arguments do not rely on an unverified self-citation chain. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. Therefore the appropriate circularity score is 0.

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

The central claim rests on standard MDP machinery plus one ad hoc separability assumption in the middle penalty region. No parameters are fitted to data, and no new entities are postulated.

assumptions (4)

domain assumption The policy-based birth-death process is irreducible, aperiodic and positive recurrent for every policy in D.
Used in Section 4 to guarantee a unique stationary distribution pi(d) for the performance potential and Poisson equation; holds for finite birth-death processes with strictly positive rates.
standard math The performance difference equation (Lemma 1) from Cao [11] applies to this CTMC.
Restated without proof in Section 6.2; standard in sensitivity-based optimization for finite Markov processes.
standard math The perturbation realization factor G^(d)(i) has the explicit expression in Theorem 2, and its root P_i^(d) is unique and real.
Derived from the linear recursion (36); the uniqueness follows from linearity in P, but the denominator positivity is not discussed.
ad hoc to paper In the middle region, the optimal policy can be found by independent optimization over the sets Lambda1 and Lambda2 (coordinate-wise separability).
This is the unproven assumption in Theorem 9; the paper asserts it is 'easy to see' from Theorems 4 and 6 without showing that interactions through pi(d') preserve the sign conditions.

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Cite this review

Pith. "Pith review of A Complete Algebraic Solution to the Optimal Dynamic Rationing Policy in the Stock-Rationing Queue with Two Demand Classes." pith.science (2026). https://pith.science/paper/WPSUSNEL

@misc{pith2026190809295,
  author       = {Pith},
  title        = {Pith review of: A Complete Algebraic Solution to the Optimal Dynamic Rationing Policy in the Stock-Rationing Queue with Two Demand Classes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WPSUSNEL}},
  note         = {Machine review of arXiv:1908.09295}
}

read the original abstract

In this paper, we study a stock-rationing queue with two demand classes by means of the sensitivity-based optimization, and develop a complete algebraic solution to the optimal dynamic rationing policy. We show that the optimal dynamic rationing policy must be of transformational threshold type. Based on this finding, we can refine three sufficient conditions under each of which the optimal dynamic rationing policy is of threshold type (i.e., critical rationing level). To do this, we use the performance difference equation to characterize the monotonicity and optimality of the long-run average profit of this system, and thus establish some new structural properties of the optimal dynamic rationing policy by observing any given reference policy. Finally, we use numerical experiments to demonstrate our theoretical results of the optimal dynamic rationing policy. We believe that the methodology and results developed in this paper can shed light on the study of stock-rationing queues and open a series of potentially promising research.

Figures

Figures reproduced from arXiv: 1908.09295 by the authors.

**Figure 2.** State transition relations of the policy-based Ma [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** The optimal long-run average profit η d ∗ vs. the threshold θ Case two: A lower penalty cost From [PITH_FULL_IMAGE:figures/full_fig_p055_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The optimal long-run average profit η d ∗ vs. the threshold θ queue depends on the arrival rate. Our observation focuses on the higher penalty cost P = 10 and the lower penalty cost P = 0.1, respectively. To do this, we further take the system parameters: µ1 = 30, µ2 =…

**Figure 5.** Figure 5: η d ∗ vs. λ under three different thresholds K Case two: A lower penalty cost 56 [PITH_FULL_IMAGE:figures/full_fig_p056_5.png]

**Figure 6.** Figure 6: η d ∗ vs. λ under three different thresholds K Example 4. Our observation is to focus on how the penalty cost P influences the long-run average profit η d for any given policy d. From (17), it is easy to see that for any given policy d, the long-run average profit η d …

**Figure 7.** Figure 7: The long-run average profit η d ∗ vs. the penalty cost P find the optimal dynamic rationing policy and compute the maximal long-run average profit from three different areas of the penalty costs. Therefore, we provide an algebraic method to set up a complete algebraic …

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