A 5/3 Guarantee for Echelon Stock (R,nQ) Policies in Two-Stage Stochastic Serial Systems

1. Introduction

• Fixed charges, holding costs, backlog, lead times, and upstream material availability make serial replenishment decisions interdependent, while the classical rule links upstream lot size to an integer multiple of the downstream lot.The rule uses two reorder points and transfers complete downstream lots when material is available, limiting flexibility when stages prefer incompatible lot sizes.Source: §1 · pp. 1–2
• The classical integer (R,nQ) policy costs at most 5/3 of the optimal admissible system cost, even when it pays per lot and the benchmark pays once per aggregate dispatch.The guarantee holds for unit Poisson demand, full backlogging, nonnegative setup and holding costs, and nonnegative deterministic lead times; it is independent of their values.Source: §1 · pp. 2–3
• In constructed instances, even the best classical policy costs over 36% more than a feasible alternative outside the classical class, which orders one unit at a time and combines available units into varying-size downstream shipments.The sharpness of a guarantee remains open.Source: §1 · p. 3
• The proof upper-bounds the best integer-ratio policy using independently chosen replenishment cycles, then compares that value with a lower bound of 3/5 on optimal system cost.A grouping argument converts cycle choices into feasible integer-ratio lots without increasing cost; weighted lower bounds cancel the setup-free minimum inventory cost.Source: §1 · pp. 3–4
• Two single-stage cost allocations combined with the setup-free system’s minimum inventory cost yield a practical lower bound for assessing a policy’s instance-specific cost ratio.Optimizing the allocation and scale can tighten this bound for particular system parameters.Source: §1 · pp. 3–4
• 22.7% was the maximum gap between best-found classical policies and the evaluated lower bound across three broad grids, with a 12.1% median gap and a stress-grid ratio reaching about 1.60.The stress grid was deliberately asymmetric.Source: §1 · p. 4

• About 8.7%, the best evaluated classical cost improves when setup charges shift from per lot to per shipment; a finite search allowing incomplete downstream lots found no improvement.The best classical policy found uses unit downstream lots, which cannot be split further because shipment quantities are integers.Source: §1 · p. 4
• The management framing distinguishes standardizing replenishment quantities, coordinating standardized quantities across stages, and pricing shipments that carry several lots.The theorem quantifies a guarantee for standardization, while an impossibility example shows its potential cost; common lots or larger upstream multiples coordinate quantities, with reorder points retaining timing flexibility.Source: §1 · pp. 4–5
• A fixed charge per dispatch and a fixed charge per lot can yield different preferred policies even for identical physical flows.This makes shipment-pricing convention a distinct operating decision from lot sizes and replenishment timing.Source: §1 · p. 5
• Common lots use the same quantity at both stages, while a larger upstream multiple spreads one upstream setup over several downstream lots.In either arrangement, reorder points remain available to adjust inventory protection and replenishment timing.Source: §1 · p. 5, Figure 1 · p. 5

2. Related Research

• Clark and Scarf (1960) introduce the echelon-stock approach, and Zheng (1994a) formalizes the classical (R,nQ) class and derives its evaluation formulas.The paper retains integer ratios and complete-lot transfers.Source: §2 · p. 5
• Chen and Zheng (1994b) provide the serial-system lower-bound foundation, using induced penalties to represent downstream shortages upstream while including setup costs at both stages.The paper also draws on structural results for single-stage (r,Q) models.Source: §2 · p. 6
• The lower bound combines allocated single-stage costs with the zero-setup serial value; the 5/3 argument uses equal allocation and two setup scales.The numerical study evaluates a broader family of allocations and setup scales.Source: §2 · p. 6
• Hu and Yang (2014) study modified echelon policies with unrestricted lot ratios and partial transfers, while this theorem retains integer quantities, complete-lot transfers, both lead times, and a setup-ratio-independent constant.The theorem also charges the selected policy for each lot.Source: §2 · p. 6
• Other stochastic ordering-interval studies use service-level and safety-stock formulations and compare within those formulations or against their best policy class.Chu and Shen (2010) and Najy et al. (2025) are named examples.Source: §2 · pp. 6–7
• Unlike lost-sales serial-system models, this paper assumes unmet demand is fully backlogged.The passage identifies Huh and Janakiraman (2010) as studying lost demand.Source: §2 · p. 7

3.1. Material Flow and Decisions

• Stage 1 serves customers and replenishes from Stage 2, while Stage 2 replenishes from an external supplier after deterministic lead times.Demand arrives one unit at a time as a Poisson process; unmet demand waits in backlog, and shipment size is limited by material availability and integrality.Source: §3.1 · p. 7
• A Stage-1 request does not make material immediately available: it must be available upstream and then traverse the downstream lead time.The actual Stage-1 inventory position cannot exceed available net Stage-2 inventory.Source: §3.1 · pp. 7–8
• The model allows nonnegative demand, holding, backlog, and setup rates, and formulates the guarantee using cost infima when finite policies may approach but not attain an optimum.Policies must be feasible, use only current and past information, and satisfy long-run flow balance and negligible terminal effects.Source: §3.1 · pp. 7–8

3.2. Shipment-Based and Lot-Based Setup Costs

• Shipment accounting charges once per positive aggregate dispatch, whereas lot accounting charges for every lot, even when several lots travel together.A dispatch containing three lots incurs one setup charge under shipment accounting and three under lot accounting.Source: §3.2 · p. 9
• The classical (R,nQ) class is contained within the broader admissible fixed-lot class, and its guarantee remains valid against the smaller benchmark under either shared setup convention.The impossibility result is narrower: it is measured against OPTsh, which permits variable shipment quantities outside the fixed-lot class.Source: §3.2 · p. 9

4.1. Four Controls with Distinct Operational Roles

• The classical rule sends complete lots when reorder points are crossed, with downstream requests waiting until a lot is available and shipments constrained by arrived material.The four controls are fixed before future demands are observed.Source: §4.1 · p. 10
• For fixed n, the remaining decisions are downstream lot size Q, cycle endpoint s = R1 + Q, and reorder-point difference ℓ = R2 − R1.Q controls the downstream request rate, s sets the cycle's upper inventory-position limit, and ℓ shifts upstream replenishment timing relative to downstream requests.Source: §4.1 · p. 10

4.2. Averaging the Positions Within a Replenishment Cycle

• The inventory calculation accounts for both lead-time demand and material unavailable upstream, which can shift a downstream cycle to lower positions when a request must wait.The position averages preserve integer controls because the relevant convex cost functions are linear between consecutive integer points.Source: §4.2 · p. 12
• The exact cost calculation averages inventory and backlog costs over replenishment-cycle positions and includes demand during each transportation lead time.Its setup terms follow demand balance: every Q units require a downstream lot, and every nQ units require an upstream lot.Source: §4.2 · p. 12
• For positive holding and backlog rates, the cheapest cycle endpoint is an integer quantile at probability p/(h1 + h2 + p).At fixed Q, n, and ℓ, only the unavailable-upstream-material term depends on s, reducing endpoint selection to a one-dimensional calculation.Source: §4.2 · p. 13
• The multiplier n spreads upstream setups across more downstream lots but can increase inventory waiting upstream, so choosing it requires accounting for material timing as well as setup savings.The theorem does not require common lots (n = 1) in every environment.Source: §4.2 · p. 13

5. A Combined Lower Bound on the Optimal System Cost

• The combined lower-bound method joins allocated single-stage costs with the zero-setup minimum inventory cost, then connects the bound to the joint inventory calculation used to construct a classical policy.Source: §5 · p. 13

5.1. The Cost of Operating Without Setup Charges

• With setup charges set to zero, the minimum inventory cost V0 still lower-bounds the inventory and backlog component of every admissible policy.The zero-setup system permits ordering and transferring individual units, but demand uncertainty and transportation time still generate holding and backlog costs.Source: §5.1 · pp. 13–14

5.2. Dividing Cost Without Counting It Twice

• The allocation comparison assigns shares of physical holding and backlog costs to two single-stage problems, each retaining its own stage's setup charge.Assigning each physical charge only once avoids double counting when the single-stage costs are added.Source: §5.2 · pp. 14–15
• The single-stage cost CF is a lower bound for arbitrary admissible shipment decisions, including variable shipment quantities.The proof compares costs over completed cycles and accounts for the scaled setup charge within each cycle.Source: §5.2 · p. 14

5.3. Combining the Allocation and Zero-Setup Bounds

• Proposition 2 gives two sufficient lower bounds by combining the allocation benchmark with V0 at different setup-cost scales.The proof uses equal cost-allocation shares and the choices u = 1 and u = 1/2; the relative usefulness of the bounds depends on V0 compared with replenishment-cycle costs.Source: §5.3 · pp. 15–16

6. A 5/3 Guarantee for the Classical Policy

• 5/3: under unit Poisson demand, finite deterministic lead times, and nonnegative finite rates, the best classical-policy cost is at most (5/3)Ltwo.For positive λ, h1, h2, and p, a finite deterministic policy attains the guarantee; at other nonnegative boundaries, policies approach it arbitrarily closely. The same guarantee holds with Csh replacing Clot.Source: §6 · p. 16
• The guarantee allows arbitrary imbalances in setup charges, lead times, and holding rates, while retaining exact integer lot sizes.It bounds the best classical policy in the stated system, not every selection of its four controls.Source: §6 · p. 16

6.1. Choosing Cycles Before Coordinating Lot Sizes

• Independent integer cycle sizes define an auxiliary joint setup-and-inventory value J that bounds the best classical-policy cost and is bounded by allocated single-stage costs.Remainder groups convert the independent cycle sizes into feasible integer-ratio lots: their weighted inventory cost matches the auxiliary calculation, and their weighted setup cost is no greater.Source: §6.1 · pp. 17–18

6.2. Comparing Small and Large Cycles Together

• Parity-splitting integer cycle positions and joint-inventory convexity yield a smaller-cycle comparison; concavity of J in setup rates then gives inequality (24).The comparison relates J(k1,k2), J(4k1,4k2), and V0, the inventory value when setup costs are zero.Source: §6.2 · pp. 18–19

6.3. The Cancellation That Gives the Constant 3/5

• The shared joint inventory calculation combines two cost comparisons whose maximum covers at least 3/5 of J, yielding the 5/3 guarantee through cancellation of V0.The proof retains the zero-setup cost in both comparisons so its contribution cancels.Source: §6.3 · pp. 19–20

6.4. Interpreting the Two Cost Bounds

• For v = V0/J, the bounds are 2/3−v/6 and 1/2+v/4; their maximum is at least 3/5, meeting at v = 2/5, but this crossing does not establish an instance attaining 5/3.The analysis does not determine the best uniform factor obtainable from the full bound family; its impossibility construction establishes an actual loss with a factor below 5/3.Source: §6.4 · pp. 20–21

6.5. A Guaranteed Policy-Selection Procedure

• The finite-search procedure selects a classical policy whose lot-accounted and shipment costs are each at most 5/3 OPT_sh.The guarantee follows from Clot(bΠ) ≤ J(k1,k2) ≤ (5/3)Ltwo ≤ (5/3)OPT_sh and Csh(bΠ) ≤ Clot(bΠ).Source: §6.5 · p. 22

7. A Limit to Uniform Improvement

• For every r < ρ◦, no uniform factor of 1.366 or below holds for the classical policy class against OPTsh under either numerator convention.The finite instances can be chosen with all cost coefficients, demand rate, and both lead times strictly positive.Source: §7 · p. 22
• The flexible policy replenishes upstream at every demand and transfers stock when backlog warrants it, separating frequencies that the classical Q2=nQ1 link ties together.The fixed link leaves additional cycle inventory that reorder-point optimization cannot eliminate.Source: §7 · pp. 22–23
• In the S=1000 example, exact bounds give Jimp(k)/h2 ≥1729 and flexible-policy cost/h2 <1269, a ratio above 1.36.The example uses analytic Poisson bounds and integer arithmetic.Source: §7 · p. 24
• Small positive changes to zero coefficients preserve a strict gap, extending the construction to strictly positive primitives.The argument retains the competing policy’s dispatch decisions and bounds the added costs.Source: §7 · p. 24
• The lower limit concerns OPTsh, not fixed-batch OPTlot, and the exact worst-case factor remains between 1.366... and 1.6667....The construction does not identify the exact factor for the classical class.Source: §7 · p. 24

8. Using the Result in an Operating Decision

• For a feasible policy cost C and valid optimal-cost lower bound L, C −L bounds the fraction of current cost that any policy could save.The evaluated policy need not have optimal classical controls.Source: §8 · pp. 24–25
• If a feasible classical policy satisfies the 5/3 factor, any policy can save at most 40% of its cost; the explicit flexible example saves about 26.8% of the best classical policy’s cost.Savings in a particular operation require its own evaluation.Source: §8 · p. 25
• Evaluate retuned controls, shared dispatch charges, and variable shipment quantities as distinct changes, using comparisons matched to the activity being changed.A new upstream multiplier can change material availability, so reorder points should be retuned jointly with lot sizes.Source: §8 · pp. 25–26
• The comparison assumes stationary Poisson unit demand, deterministic lead times, full backlogging, and modeled costs; changing lead time, transport capacity, or service promise requires a new comparison.The bound measures policy improvement within a fixed system.Source: §8 · p. 26
• Recurring administrative or implementation costs above C −L cannot be justified by modeled operating savings; when the margin is small, extensive redesign has limited value.The bound can screen implementation expense measured on the same time basis as C and L.Source: §8 · p. 26

9.1. Evaluation Method and Interpretation

• The reported ratio divides the best-found finite-menu policy cost by bLSP, with bLSP ≤ LBSP ≤ OPT_sh; it is a numerical estimate, not an identified policy gap.The policy search does not globally exclude larger upstream multipliers, and the floating-point calculations do not provide rigorous rounding-error bounds.Source: §9.1 · pp. 26–27

9.2. Effects of Costs and Lead Times on Policy Controls

• The reorder-point difference can be positive or negative, and governs relative replenishment timing and inventory held to cover transportation.The examples show that common lots can pair with equal or different reorder points, while larger upstream lots can be useful with either sign of the difference.Source: §9.2 · p. 27
• Under the fixed-total normalization, small ρK tends to favor a smaller downstream lot and larger upstream multiplier, while small ρL tends to increase the reorder-point difference.At the symmetric setting, the selected controls are (R1,R2,Q1,Q2) = (1,2,3,3); the patterns are conditional on the normalization, not global monotonicity rules.Source: §9.2 · p. 28

9.3. Comparisons with the Combined Lower Bound

• For the best-found lot-accounted cost divided by the evaluated combined lower bound, the computed ratio had a maximum of 1.2265, median 1.1208, and 90th percentile 1.1928.These comparisons are substantially below 5/3 and show that the evaluated bound can provide useful assurance beyond the uniform theorem on these grids.Source: §9.3 · p. 28, Figure 6 · p. 29

9.4. An Asymmetric Stress Test and the Role of Shipment Accounting

• In the asymmetric stress grid, ratios to the evaluated combined lower bound ranged from 1.5445 to 1.6004, with median 1.5912.The test places almost all holding cost downstream, makes backlog inexpensive, and makes upstream setups nearly free.Source: §9.4 · p. 29
• With shipment accounting, controls (-1,9,1,33) cost 0.055303, about 8.7% less than the common-lot policy’s 0.060576, and yield ratio 1.4610.The common-lot policy has multiplier one, so both setup conventions give it the same cost; the other policy accumulates complete unit lots into one charged physical shipment.Source: §9.4 · pp. 29–30
• High policy-to-bound ratios do not reveal how much of the gap reflects a weak lower bound versus policy inefficiency.The stress experiment complements the analytical impossibility example rather than resolving that decomposition.Source: §9.4 · p. 30
• A particular shipment-rule expansion need not improve every difficult-looking environment, although flexible shipment quantities can substantially improve some environments.The numerical stress experiment complements the analytical impossibility example; searches beyond those considered may achieve further savings.Source: §9.4 · p. 30
• Evaluating shipment-cost convention, controls, and benchmark separately gives a more useful diagnosis than interpreting the policy-to-bound ratio alone.Source: §9.4 · p. 30

10. Concluding Remarks

• The classical echelon stock (R,nQ) policy has optimized cost at most 5/3 of the optimal admissible shipment cost, even when it pays per lot and the benchmark pays per shipment.The guarantee holds over the stated nonnegative parameter space.Source: §10 · p. 30
• The exact worst-case factor remains between 1.366... and 1.6667..., so the lower-limit construction does not determine the classical class’s exact factor.The result applies to strictly positive finite primitives as well as the simpler boundary construction.Source: §10 · pp. 30–31
• Shipment consolidation can improve performance within the classical class, while variable deliveries can separate upstream replenishment and downstream delivery frequencies.The latter flexibility can be valuable when the stages should operate at different frequencies.Source: §10 · p. 31
• Extensions to more than two stages, compound or renewal demand, and lost sales require additional analysis; the theorem concerns a two-stage Poisson system.These changes affect integer-multiple feasibility, demand-window dependence, or inventory balances and decision information.Source: §10 · p. 31

A. Policy Costs and Admissibility

• The appendix’s policy and lower-bound arguments assume positive demand, finite nonnegative lead times, and finite nonnegative cost coefficients.With zero demand, empty initial inventories and pipelines, and no orders, cost is zero.Source: A. Policy Costs and Admissibility · p. 33

A.1. The Admissible Comparison Class

• The admissible comparison class uses nonanticipating policies with nondecreasing integer dispatches, finite expected costs, stability conditions, and an unfinished-path condition.A sufficient criterion for the latter is an expected inventory-position range of o(T); stationary integrable inventory-position processes satisfy it.Source: §A.1 · pp. 33–34

A.2. Stationary Complete-Lot Construction

• The stationary complete-lot construction couples downstream requests and upstream orders through a common demand counter and matches only available complete lots, maintaining nonnegative physical inventories.Its expected inventory cost is E[Gadj(IP 1) + h2IL2], equivalently E[G0(IP 1) + h2(IL2 − IP 1)]; Proposition A.1 establishes the exact stationary cost.Source: §A.2 · pp. 34–35
• Stationary request and upstream-order rates are λ/Q and λ/(nQ), while matched lots occur at λ/Q; per-lot charges bound each positive dispatch’s single shipment charge pathwise.The request, upstream-order, and matched-lot rates give the two setup terms; every positive dispatch contains at least one lot.Source: §A.2 · p. 36
• Every finite integer construction satisfies (A.5) and the unfinished-path condition; its inventory, pending lots, and pipelines have finite second moments in stationarity.Terminal expectations and fixed-window costs are consequently O(1).Source: §A.2 · p. 36
• An arbitrary fixed initial remainder of the common demand counter yields the same long-run cost when the counters retain their relative timing.The Poisson counter modulo nQ converges to uniform, and independent future increments give convergence of bounded state costs to stationary values.Source: §A.2 · p. 37
• Uniformly bounded second moments after initial lead-time histories clear make expected inventory costs converge to stationary values by truncation, while setup rates and pathwise range estimates remain unchanged.The expected contribution from costs above a threshold a is bounded by a^-1 times the second moment.Source: §A.2 · p. 37

A.3. An Initial Remainder of Upstream Stock

• Removing ρ units of upstream stock reduces inventory cost by h2ρ while preserving the complete-lot count and operational decisions.The same demand realization and matching decisions are used; the difference in physical stock is only the removed ρ units.Source: §A.3 · pp. 37–38
• Allowing an initial upstream remainder leaves both optimized classical cost infima unchanged.Each initialization with remainder ρ can be mapped to a zero-remainder rule by removing ρ units and replacing R2 with R2−ρ, without changing requests, orders, dispatches, or deliveries.Source: §A.3 · p. 37
• Every compatible initialization with fixed remainder has the canonical long-run cost, and randomized initializations cannot lower the policy infimum.The canonical class includes the shifted reorder points, and the comparison maps other initializations to policies of weakly smaller cost.Source: §A.3 · pp. 38–39

B.1. The Integer-Cycle Inequality

• The cycle bound remains valid when physical shipment sizes are arbitrary positive integers and states lie on a restricted integer lattice.Allowing real placements in CF can only lower the cycle infimum.Source: §B.1 · p. 39
• Each erased simple cycle has one upward jump and m unit-demand decreases, and its cost is at least mCF0(k).The result applies to an adaptive policy's chronological path under the stated demand and unfinished-path conditions.Source: §B.1 · pp. 39–40

B.2. Lead-Time Shifts and the Cost Split

• Lead-time shifts allocate inventory and backlog costs across two components, to which the cycle lemma supplies separate lower bounds.The first component's lower limiting rate is at least ηH0 + CFη,ζ; the combined component bounds hold for every admissible policy.Source: §B.2 · pp. 40–41
• For any admissible policy, the allocation supremum A(k1,k2) is at most OPTsh.The bound permits zero allocated slopes and setups without requiring either separate cost component to converge.Source: §B.2 · p. 41

C. The Zero-Setup Value and Setup-Preserving Rescaling

• For all nonnegative cost rates, V0 bounds every admissible policy's cost when both setup charges are zero.Finite integer unit-lot policies approach V0 arbitrarily closely in that zero-setup system.Source: C. The Zero-Setup Value and… · p. 41
• Setup-preserving rescaling yields LBSP ≤ OPTsh ≤ OPTlot, while removing lot labels preserves material flows and weakly decreases setup charges.The rescaling multiplies setup coefficients by 1/u and then restores the original charges.Source: C. The Zero-Setup Value and… · p. 42

D. The Joint-Cycle Comparison

• The joint-cycle comparison independently selects replenishment frequencies, then constructs feasible classical policies including one no costlier than the auxiliary calculation.The upstream order remains a whole number of downstream lots.Source: D. The Joint-Cycle Comparison · pp. 42–43

D.1. A Common Inventory Calculation for Both Cycles

• The common inventory calculation constrains downstream inventory by material availability and retains inventory and backlog costs.Its downstream choice may respond to D2 and Ξ2, but not to D1.Source: §D.1 · p. 43
• The calculation is invariant to constant shifts of both cycle positions, and its value is convex under simultaneous scaling.At Ξ1 = Ξ2 = 0, its value is V0.Source: §D.1 · p. 44

D.2. Converting Independent Cycle Sizes into Feasible Policies

• For every independent integer size pair (m1,m2), some nonempty remainder group yields a deterministic classical policy with lot-accounted cost at most k1/m1 + k2/m2 + V_m1,m2.The construction preserves complete-lot dispatches and makes the upstream lot an integer multiple of the downstream lot.Source: §D.2 · p. 45

D.3. A Stagewise Upper Bound for the Joint Value

• The joint inventory value is bounded by H0 + H_m1(P) + H_m2(Fup), and J(k1,k2) by B1(k1) + B2(k2).The separate setup-rate minimizations remain eligible for the group construction.Source: §D.3 · p. 47

D.4. Even and Odd Positions and the Two-Scale Comparison

• For every nonnegative setup vector, joint cycle doubling gives J(4k) ≥ 2J(k) − V0.The proof compares even and odd cycle positions and shows each candidate defining J(4k) costs at least 2J(k) − V0.Source: §D.4 · pp. 47–48
• The cycle-size-three/eight comparison gives V3,8 ≥ 2[(2/3)V2,4 + (1/3)V1,4] − V0; each candidate is at least J(k), yielding the lower estimate 2J(k) − V0 at setup vector 4k.The comparison also covers unit lots under nonnegative setup costs.Source: §D.4 · pp. 48–49
• The two-scale estimates combine with equal allocation to prove the cost-infimum comparison, which also holds under shipment accounting because each constructed policy has Csh ≤ Clot.A 3/5–2/5 weighted combination cancels the zero-setup inventory term, avoiding cases based on the relative importance of setup and inventory costs.Source: §D.4 · p. 49

D.5. Finite Attainment and Zero-Cost Cases

• The independent-cycle infimum has a minimizing pair and yields a finite deterministic policy satisfying the bound, even when setup coefficients are zero.Lower bounds that grow with each cycle size restrict all candidates below a finite reference cost to finitely many integer pairs.Source: §D.5 · p. 50
• A finite search over cycle sizes, thresholds, and endpoints suffices to find an exact minimizing candidate and exclude all omitted choices.The selected threshold is characterized by a distribution quantile, while endpoint bounds control backlog at low placements and holding at high placements.Source: §D.5 · pp. 50–51
• For nonnegative costs, the comparison holds for cost infima, with finite policies approaching the bound even when the optimum is zero and is not attained.For every positive ε, a finite construction achieves the stated comparison within ε; attainment is not claimed when only a sequence of finite lot sizes or placements approaches the bound.Source: §D.5 · p. 53

E. Proof of Theorem 2: the Policy-Class Impossibility Result

• On the stated boundary, physical running cost is at least φ(Y), with equality when available upstream material is used to remove downstream backlog.Here Y is total net echelon inventory; the comparison permits any allocation of physical stock between locations.Source: E. Proof of Theorem 2: the… · p. 53

E.1. The Exact Classical Numerator Under Shipment Accounting

• The shipment-accounted and lot-accounted classical cost infima both equal Jimp(k) on this boundary.Source: §E.1 · p. 53
• The shipment cap includes aggregate dispatches and consolidation savings, while the inventory lower bound uses upstream lot size Q2 regardless of downstream reorder point or retained upstream inventory.The upstream cycle position is uniform across the finite demand-residue chain, so its long-run averages do not depend on the initial residue.Source: §E.1 · p. 54

E.2. A Feasible Policy with Frequent Upstream Replenishment

• For integer S ≥10, ordering one unit per demand and dispatching available stock when X < 0 gives an adapted policy with finite inventory and cost moments.The flow balances imply upstream ordering and downstream delivery rates of one, and the inventory-position paths satisfy the unfinished-path condition for admissibility.Source: §E.2 · pp. 54–55

E.3. Poisson Bounds and the Full Classical Policy Optimization

• For the Poisson bounds, Sν →1 as S grows, and every lot size m ≥10S costs more than the reference candidate.The argument uses a geometric tail bound for Poisson demand and compares blocks of integer positions against a candidate based on lot size S.Source: §E.3 · p. 57

D.2. Converting Independent Cycle Sizes into Feasible Policies

• Excluded placements cost more than the reference, while the displayed lower bound is below 2S when α ≤ 1/2, covering the infinite optimization domain.The remaining blocks satisfy σ ≥ S − 2.Source: §E.3 · p. 58
• Writing setup and cycle-inventory terms as a square yields a nonnegative tradeoff for every positive integer lot size, without substituting a continuous optimizer.The square measures the cost of choosing a lot size above or below the scale balancing the two terms.Source: §E.3 · p. 58
• The probability mass at S − 2 prevents avoiding cycle inventory by shifting blocks to low positions, and a separate large-lot exclusion prevents spreading the shortage contribution across arbitrarily large cycles.Although the mass at S − 2 is small, its ratio to h2 grows as a polynomial of degree four in S.Source: §E.3 · p. 58

E.4. The Ratio, a Finite Example, and Positive Primitives

• A finite example with rational parameters yields a bound greater than 1.3637647, and therefore greater than 1.36, showing that the threshold is exceeded without identifying the example as a boundary case.The stated comparison is 1729/1269 > 34/25 = 1.36, with positive cross-multiplication difference 79.Source: §E.4 · pp. 58–59
• Strict comparisons on the boundary extend to positive primitives by adding small positive holding, setup, and downstream lead-time values.The perturbed competitor retains the shadow policy’s upstream orders and dispatches, and its added cost is bounded by the perturbation terms in (E.8).Source: §E.4 · p. 59
• The construction compares against OPTsh, because its feasible competitor uses variable downstream shipment quantities.The same competitor cost does not upper-bound the fixed-batch optimum OPTlot, so this benchmark ordering alone proves no impossibility bound for OPTlot.Source: §E.4 · p. 60

F.1. Policy Evaluation and the Denominator

• An integer quantile of effective demand minimizes downstream mismatch cost, and the resulting controls and setup charges give the lot-accounted policy cost.The evaluated denominator is the largest feasible value found for the stated lower-bound objective, with selected allocations and scales reevaluated using integer-cycle searches.Source: §F.1 · p. 60
• Finite cycle menus alone cannot certify a component infimum; stopping bounds exclude larger cycles when slopes are positive.The placement search also uses lower bounds to check unsearched downstream lot sizes at the broad-grid points.Source: §F.1 · pp. 60–61

F.2. Components with a Zero Holding or Backlog Rate

• When an allocated holding or backlog slope is zero, the optimized single-stage component has exact infimum zero, so a finite-menu positive value would overstate the lower bound.If both slopes are zero, only the setup term k/m remains and tends to zero as the cycle size increases.Source: §F.2 · pp. 61–62

F.3. Numerical Summaries

• 1.2265 is the maximum ratio of best-found policy cost to an evaluated LBSP candidate across 1,801 distinct broad-grid points.The median is 1.1208 and the 90th percentile is 1.1928; the separate 625-point stress grid reaches 1.6004.Source: §F.3 · p. 62

F.4. Infinite Poisson Tails

• Computed summaries use ordinary floating-point arithmetic, and the analytic guarantee does not depend on these numerical evaluations.Probability-tail targets alone do not guarantee cost accuracy when loss slopes are large.Source: §F.4 · pp. 62–63

F.5. Shipment Accounting

• For the largest-ratio stress instance, direct evaluation over 22,200 control pairs gives a minimum cost of 0.055303 at (−1,9,1,33).This is a separate evaluation of the shipment-accounted cost for those controls.Source: §F.5 · p. 63

F.6. Shipment Simulations

• The classical simulation search selects (−1,9,1,32) with mean 0.055482, while the incomplete-lot search selects (−2,8,2,33) with mean 0.055608.The incomplete-lot replications contain 55 to 82 incomplete dispatches each; the comparison is within the evaluated ranges.Source: §F.6 · p. 64
• Simulation intervals are conditional on fixed controls and do not measure uncertainty for policies outside the search.The reported flow check screens for implementation errors or severe transient effects; it is not a statistical confidence statement.Source: §F.6 · p. 64