Closed-loop network design has been studied with every major uncertainty formalism. An early robust counterpart [6] set the template that much of the literature follows. A two-stage stochastic program covering demand, return rates, and returned-product quality [7] found that demand uncertainty had the strongest effect on the value of the stochastic solution, through its influence on facility location and capacity installation, and return-rate uncertainty the weakest. Later work adds a distributionally robust formulation [8], a data-driven robust model for a dual-channel network under carbon cap-and-trade [9], and a three-stage hybrid combining demand and return scenarios with an uncertainty set for the carbon tax rate [10].

Recent work estimates uncertainty sets from data, with balanced sets whose boundaries sit equidistant from uncovered observations [11]. That addresses calibration of the set, not whether the resulting protection changes any decision.

The SCODM literature shows the same breadth, and is the audience for the diagnostic proposed here: closed-loop spare parts with machine-learning return forecasting [12], opportunistic maintenance in a closed loop [13], robust empty-container positioning [14], bi-objective robust blood supply design [15], food-security chains solved by a genetic algorithm [16], fourth-party logistics in off-site construction [17], [18], and energy-integrated sustainable networks [19]. Treatments of uncertainty in the same venue vary widely: antifragility rather than worst-case cost [20], fuzzy multi-criteria supplier prioritization [21], combined knowledge- and data-driven resilient sourcing [22], and probabilistic arc availability in maximum flow [23]. Each formalism reports differently, and none, including the robust studies above, reports whether the protection purchased changes the decision taken. That is the reporting gap this paper addresses.

Bertsimas and Sim [1], [2] introduced the budget of uncertainty to control conservatism. For a constraint indexed by i with uncertain coefficient set Ji, the budget Γi ranges over the interval from zero to the cardinality of Ji, and limits the number of coefficients that may deviate simultaneously. The robust counterpart remains linear, which is why the approach has been adopted so widely.

Two limitations are established. Kim et al. [3] observed that when the cardinality of J subscript i equals one, the robust counterpart with a budget is equivalent to the counterpart with a box uncertainty set, so the budget is ineffective and amounts only to a reduction of the box. They proposed a reformulation in which decision variables express the proportion of available supply used rather than absolute quantities, which places all uncertain parameters in a common expression and restores a meaningful budget. Kim and Lee [4] extended this to collected-material quality uncertainty and reported comparative conditions under which quantity-based and proportion-based formulations are preferable. Notably, in both studies, the budget is applied to the reverse-logistics side, and demand uncertainty is handled with a box set.

Separately, Dehghani Filabadi and Mahmoudzadeh [5] studied problems with resource uncertainty on the right-hand side and showed that the conventional budget may remain over-protective: beyond a threshold, increasing the budget does not change the optimal solution, so the additional budget is ineffective. They proposed a two-stage method that identifies the admissible portion of the uncertainty set and protects only against an effective worst case, demonstrating it on power dispatch and calling for extension to other application domains. Related analyses of right-hand-side uncertainty appear in Minoux [24] and Ouorou [25], and two-stage robust network flow and design under demand uncertainty is treated by Atamturk and Zhang [26].

A third concern in the literature is when decisions are made. Ben-Tal et al. [27] introduced the adjustable robust counterpart, separating variables that must be fixed before the uncertainty is realized from those that may adapt afterwards, and showed that the adjustable counterpart is markedly less conservative than the static one, though generally intractable unless the adaptive variables are restricted to affine functions of the data. Atamturk and Zhang [26] developed two-stage robust formulations for network flow and design under demand uncertainty, and Bertsimas et al. [28] studied robust and adaptive network flows. Zeng and Zhao [29] gave the column-and-constraint generation scheme that makes two-stage robust integer problems solvable in practice, which is the method used in Section 5.4.

The distinction matters for the question posed here, because a static formulation chooses openings and flows together against one protected demand vector, whereas a two-stage formulation commits the openings first and lets the flows adapt. The two can differ: A first stage that must hedge across a whole uncertainty set may install capacity that a static model, optimizing against a single worst case, does not. Recent closed-loop work reflects this awareness, with two-stage distributionally robust designs [8] and multi-stage hybrid formulations combining scenarios for demands and returns with uncertainty sets for policy parameters [10]. Because our central claim concerns whether facility decisions respond to the budget, we report both formulations.

Both limitations reviewed above are established, so the gap must be stated narrowly. Solution invariance under increasing protection is not new: it is the finding of [5], and the intellectual basis for the admissibility mechanism examined here. Nor is the cardinality problem, which [3] and [4] identify and reformulate away. What remains insufficiently examined is how these two mechanisms interact with the decision structure of closed-loop network design. Research [5] is demonstrated on economic dispatch, an operating problem in which no facility is sited; research [3] and [4] are production and sourcing plans in which the discrete decision is collector selection rather than network configuration. Network design differs in that its decisions are heterogeneous and are committed at different times: strategic facility openings, operational flows across six arc families, regional as against aggregate service, and shortfall penalties. Whether a budget that leaves aggregate cost rising but induces no additional service is equally inert in each of those layers, whether addressing the cardinality problem restores a strategic response, and where the boundary between the responsive and unresponsive regimes lies, are open questions. Reviews of the field [30], [31] catalog the uncertainty types modeled but do not report the effectiveness of the protection applied.

This study addresses that gap. It examines both mechanisms in a single network design model on measured data, decomposes the response by layer, distinguishes the service policies that the two mechanisms imply, and derives the condition separating the regimes, using the numerical experiment as confirmation.

The recovery network has six echelons. Collected equipment originates at return sources K, each a disposal-responsible body with a reported annual quantity, and flows to consolidation hubs I, where it splits into a recoverable fraction routed through recycling J and redistribution M to secondary-market regions L, and a residual fraction routed to disposal N. A forward flow from depot P to the sources represents the equipment placed on the market that later returns.

Hubs, recycling facilities, and redistribution centers are siting decisions; disposal sites and market regions are fixed. Table 1 lists the notation.

The model is stated in full so that it can be reconstructed exactly. The objective minimizes annual fixed cost, transport cost on all six arc families, and shortfall penalty:

min Z = ∑i∈I fiYi + ∑j∈J gjZj + ∑m∈M hmWm + ∑p∈P CpRp + ∑p∈P ∑k∈K cpk PKVpk +

∑k∈K ∑i∈I cki KIXki + ∑i∈I ∑j∈J cij IJUij + ∑j∈J ∑m∈M cjm JMPjm + ∑m∈M ∑l∈L cml MLQml + (1)

∑i∈I ∑n∈N cin INTin + ∑l∈L πlδl.

subject to Eqs. (2)-(15). The service requirement is soft. The variable δl is a service slack: realized demand must be covered either through delivered recovered material or through explicitly penalized shortfall, at πl per tonne. The model therefore never becomes infeasible for want of recovered material, and every constraint below should be read as an accounting requirement on demand rather than a guarantee of physical satisfaction. This distinction is central to what follows, because the penalty layer of the decomposition measures exactly the demand that is accounted for but not served:

∑m∈M Qml + δl ≥ dl, ∀ l ∈ L. (2)

Delivery is bounded above by what each market can absorb. A region cannot take more than the greatest demand its uncertainty interval admits, so

∑m∈M Qml ≤ dl + d ̂ l, ∀ l ∈ L. (3)

Without Eq. (3), the service rows are pure lower bounds, and because Eqs. (8) and (9) are equalities, recovered material that must leave the recycling echelon can be pushed onto markets beyond any demand they could realize. Surplus recovered material is instead routed to disposal at the hub, which the yield ceiling (6) permits and which the Disposal Capacity (13) must accommodate; in this case, disposal runs at 84,167 tonnes of 100,000 at the lowest ratio tested.

Constraint (3) is a market-absorption bound rather than a protection requirement, and the distinction matters for reading the results. The uncertainty budget determines the demand level against which shortfall is penalized, whereas dl + d̂l represents the largest volume that market l is assumed able to absorb physically. Delivered volume may therefore exceed the protected requirement at low budget levels, when placing recoverable material is cheaper than disposing of it. Still, it can never exceed the market's assumed absorption envelope. The case instance shows this directly: at ρ = 0.2 and Γ = 0, the model delivers 18,572 tonnes against a nominal demand of 17,975 tonnes, because the cheapest route through recycling and redistribution costs 6.23 euros per tonne against 7.20 euros per tonne for the cheapest disposal route, while disposal capacity is not binding at 87,166 tonnes of 100,000. Accordingly, we refer to Q throughout as delivered recovered volume rather than as served demand, and the service layer of the decomposition measures changes in that delivered volume.

The model requires complete processing of the municipal return quantities observed in the case dataset:

∑i∈I Xki = rk, ∀ k ∈ K. (4)

Constraint (4) is a modeling choice about the scope of the instance, not a derived legal requirement. Article 3 of the Bavarian Waste Act designates the Landkreise and kreisfreie Städte as public-law waste-disposal authorities and assigns them disposal responsibility, and the reported quantities are those they collected; that regulatory framing motivates treating the observed tonnage as a quantity the system must handle rather than a decision variable. It does not establish that every tonne must traverse the particular echelon structure modeled here, and we do not claim it does.

Material is conserved at each consolidation hub, splitting between the recycling and disposal streams:

∑j∈J Uij + ∑n∈N Tin = ∑k∈K Xki, ∀ i ∈ I. (5) The recovery yield is bounded by the technically recoverable fraction. At most (1 − s) of the material entering a hub can be routed to recycling, where s is the physically determined non-recoverable share:

∑j∈J Uij ≤ (1 − s) ∑k∈K Xki, ∀ i ∈ I. (6)

Constraints (5) and (6) imply disposal of at least s times the collected quantity, and permit more. This design is deliberate: material that is technically recoverable but that recycling capacity cannot absorb is landfilled rather than left unaccounted for, whereas fixing disposal at exactly sR would make the model infeasible whenever recycling capacity binds. The relaxation is not vacuous here, and its magnitude should be reported: 17,738 tonnes are disposed against a technical minimum of 15,861, an excess of 1,877 tonnes or 1.78 percent of collections, at 2 of the 12 hubs and reaching 990 tonnes at the largest. These are identical at every budget level under both policies, so the excess reflects a structural shortfall in recycling capacity, not a response to protection.

Returned material cannot exceed what was previously shipped to the market, which links the forward depot flow to the reverse network:

∑p∈P Vpk ≥ ∑i∈I Xki, ∀ k ∈ K. (7)

Without Eq. (7), the forward variables would be free to take the value zero at no cost, and the depot echelon would be disconnected from the rest of the model.

Material is conserved at recycling facilities and at redistribution centers. Recovery losses are modeled once, upstream at the hub through the Yield Ceiling (6); no second loss or inventory process is introduced downstream, so both balances are equalities and material cannot leave the network at a recycler:

∑m∈M Pjm = ∑i∈I Uij, ∀ j ∈ J. (8)

∑j∈J Pjm = ∑l∈L Qml, ∀ m ∈ M. (9)

Throughput at each echelon is bounded by installed capacity, which is available only at opened sites:

∑k∈K Xki ≤ cciYi, ∀ i ∈ I. (10)

∑i∈I Uij ≤ crjZj, ∀ j ∈ J. (11)

∑j∈J Pjm ≤ cemWm, ∀ m ∈ M. (12)

∑i∈I Tin ≤ cdn, ∀ n ∈ N. (13)

∑k∈K Vpk ≤ cspRp, ∀ p ∈ P. (14)

Disposal sites in Eq. (13) carry no opening variable because they are existing permitted facilities in the case setting. Finally, the variable domains are:

Xki, Uij, Vpk, Pjm, Qml, Tin, δl ≥ 0, YI, Zj, Wm, Rp ∈ {0, 1}. (15)

In the case study, a single depot with zero fixed cost is used, so R is fixed at one and Eq. (14) reduces to a capacity bound on the forward flow; the general form is retained above because it is the form implemented in the code described under Data Availability.

Uncertainty enters through secondary-market demand, with the demand in region l lying in the interval of half-width d̂l around its nominal value dl. Writing the realized demand as dl + ξld̂l with ξl ∈ [0,1], the budgeted uncertainty set is the set of ξ vectors whose entries sum to at most Γ.

We now formulate two models. They are not two counterparts of the same problem. They express different service policies, they carry different feasibility guarantees, and Section 3.3.3 shows that neither is obtained from the other by reformulation. We set them out separately for that reason.

The motivation for network design under uncertainty is that facility decisions are committed before the quantities they must handle are observed, so the timing of decisions has to be stated rather than left implicit. We use two formulations, and we are explicit about the role of each.

The static formulation of Sections 3.2 and 3.3 determines openings and flows simultaneously against a single protected demand vector. It is not a claim about how decisions are actually taken. It is common in the applied robust closed-loop literature, and we retain it as the diagnostic benchmark precisely because it is what a reader of that literature will recognize. Claims derived from it are stated as properties of that formulation.

The two-stage adjustable formulation assigns the facility-opening variables Y, Z, and W to the first stage, taken before the demand realization is known, and all flow and shortfall variables to recourse, taken after it. Writing UΓ for the budgeted uncertainty set, the problem is

minY,Z,W { ∑i∈I fiYi + ∑j∈J gjZj + ∑m∈M hmWm + maxd∈UΓ minx∈X(Y,Z,W,d) cTx }. (18)

Second-stage feasibility has two separate sources. Because the service slack absorbs unserved demand, demand realizations cannot create second-stage infeasibility. Only the Service Row (2) sees the realization, becoming Σm Qml + δl ≥ dl + ξl d̂l; the Absorption Bound (3) enters the recourse problem unchanged as Σm Qml ≤ dl + d̂l, because it is the physical envelope of what market l can take rather than what it happens to demand. Every remaining constraint is independent of ξ, so recourse feasibility is a property of the first-stage design alone. Feasibility of the fixed return stream is separate, since Eq. (4) requires the observed tonnage to clear the network whatever the demand. We impose it on the first stage explicitly. Writing R for total returns, the design must satisfy .

∑i∈I cciYi ≥ R, min { ∑j∈J crjZj , ∑m∈M cemWm } ≥ max{0, R − ∑n∈N cdn}. (19)

The first inequality opens enough collection capacity to receive the returns; the second and third open enough recycling and redistribution capacity to absorb what disposal cannot take, since the Yield Ceiling (6) and Conservation (5) send the remainder to disposal, which is capacity-limited. With the data condition that total disposal capacity is at least sR, which holds here, these are sufficient for the return stream to clear, and with the first observation above they are sufficient for relatively complete recourse in the network structure considered. We claim no more: A network with additional echelons, return-side uncertainty, or constraints coupling realizations across regions would need its own argument. In the reported runs, they never bind, and imposing them changes no result; we state them so the guarantee is constructive rather than incidental. We solve the two-stage problem by column-and-constraint generation [29], an exact algorithm for two-stage robust programs. The master carries one replicated block of recourse variables per generated scenario and yields a lower bound; the separation problem returns the realization in UΓ maximizing recourse cost for the incumbent first-stage decision, and the first-stage cost plus that worst case gives an upper bound. The algorithm stops when the relative gap (UB − LB) / max(1, |UB|) falls to 10⁻⁶ or below. Because the master's incumbent supplies the lower bound, the master is solved at a relative MIP tolerance of 10⁻⁹, so its incumbent does not sit measurably above its own optimum and produce a spurious negative gap. The recourse value is convex in the right-hand side, so the inner maximum is attained at a vertex of UΓ; for integral Γ the vertices are the zero-one vectors with at most Γ ones, and we enumerate them exactly, which avoids a dualization step and is tractable at this size.

Section 5.4 reports the two-stage results. Their purpose is to test whether the invariance observed in the static formulation is an artefact of conflating the two stages.

Let x0 denote the optimal solution at Γ = 0 and xΓ the optimal solution at budget level Γ. We characterize the response of the model to the budget along four layers, each of which a decision maker would act on differently.

The strategic response is the Hamming distance between the facility-opening vectors, that is, the number of positions in which the vector (Y, Z, W) differs between the two solutions. This layer commits capital. The service response is the change in recovered volume delivered to the secondary market, reported both in total and as the number of market regions whose delivered quantity changes. Aggregate and regional service can move independently: A fixed total may be re-allocated across regions.

The operational response is the change in the flow variables. For each arc family, we report the relative L1 change, the number of arcs whose flow changes by more than a tolerance, and the tonnage re-routed, defined as one half of the L1 change because a tonne that leaves one arc joins another.

The penalty response is the change in shortfall cost, reported together with the residual change in all non-penalty cost terms, so that the two components of the objective change are visible separately.

For the operational layer, we use the normalized statistic:

‖xΓ − x0‖1

max(1,‖x ‖1). (20)

Computed per arc family and in aggregate. A budget is termed strategically inactive on an instance when the strategic response is zero across the admissible range of Γ, and upstream-inactive when, in addition, the flow change on every arc family other than the terminal one is zero. These are weaker and more precise statements than decision invariance, which our results do not support and which we are careful not to claim.

The purpose of the decomposition is to distinguish budgets that induce decision adaptation from budgets whose additional cost is primarily absorbed through shortage penalties. We do not call the latter ineffective. A budget that leaves the design unchanged while quantifying worst-case shortage exposure is doing something a planner may legitimately want, namely pricing insufficient capacity; what it is not doing is changing what gets built or how material moves, and the decomposition exists so that a reader can tell the two apart rather than infer either from a cost curve. It requires no computation beyond recording the solution vectors, which any solver returns, and we recommend it be reported alongside, or in place of, the price-of-robustness curve.

Two elements of the instance are taken directly from the waste balance. The return quantity rk at each of the 86 sources is the reported annual tonnage of collected electrical and electronic equipment for that body, and the count of civic amenity sites it operates is used to select and size candidate consolidation hubs. Nothing else in the instance is observed.

The remaining elements are constructed. We state the full procedure and every value, so that the instance can be regenerated exactly; our implementation performs these steps in the order given. All pseudo-random draws use a single NumPy PCG64 generator seeded with 7, and the draws are taken in the sequence listed, since the sequence determines the realization.

Coordinates. The balance reports site counts but not locations, so each body is assigned a coordinate drawn independently and uniformly from the Bavarian bounding box, latitude in [47.3, 50.5] and longitude in [9.0, 13.8]. Distances are great-circle distances multiplied by a road-detour factor of 1.25, and are therefore indicative rather than routed.

Candidate sites. The candidate hubs are the bodies reporting the most civic amenity sites, each inheriting that body's coordinate. Recycling facilities, redistribution centers, disposal sites, and market regions take coordinates drawn without replacement from the 86, in that order.

Market-demand allocation. Total demand is ρ times recoverable supply, (1 − s) times total returns, distributed across regions by weights drawn uniformly from [0.6, 1.4] and normalized to sum to one, giving a largest-to-smallest ratio of 2.19 in the realization used. Deviations are proportional, d̂l = α dl. One coincidence should be flagged: because demand is ρ times recoverable supply, at ρ = 1 total nominal demand and recoverable supply are the same (89,877 tonnes), and below that figure denotes recoverable supply unless demand is named because the allocation shapes the uncertainty set. Section 5.5 tests three alternatives.

Hub capacities. Hub capacity scales the base figure by the body's site count relative to the mean across candidates, clipped to [0.35, 3.0], giving 168,000 tonnes in total against 105,738 tonnes of returns. All other capacities are uniform across sites.

Table 4 lists every parameter of the instance, marking which values are observed and which are constructed. Four of these values carry the analysis, and none is observed: the residual fraction, the deviation ratio, the shortfall penalty, and the redistribution fixed cost. Section 5.5 perturbs each of them across a range and reports the effect, so their influence is measured rather than assumed away. The recovery-demand ratio is not treated as a parameter at all but as the experimental variable, precisely because secondary-market demand for recovered material is not a municipal statistic and we decline to fix it at an invented level.

Column-and-constraint generation replicates the recourse block once per generated scenario, so the two-stage experiment of Section 5.4 uses a reduced instance. The 86 sources are grouped into 8 collection regions by a longest-processing-time rule, assigning bodies in descending order of return quantity to the region with the smallest accumulated total, which balances volume across regions; the return quantity of a region is the sum of its members, and its arc costs are the member means. The candidate sites are reduced to 6 hubs, 4 recycling facilities, 3 redistribution centers, 3 disposal sites, and 8 market regions. Each echelon keeps its total capacity and its fixed cost per unit of installed capacity: both are scaled by the ratio of the original to the reduced site count. The reduced instance therefore represents the same physical system over fewer, larger sites rather than a capacity-starved one. Total demand is rescaled to hold ρ at its intended value.

All models were built with the Gurobi Python interface and solved with Gurobi 13.0.2 under Python 3.12 on Linux (x86-64). Unless stated otherwise, the relative MIP tolerance was 1e-4, the time limit 3600 seconds, and the thread count the solver default; no other parameter was altered, and no tuning, warm start, or callback was used. Instance data are seeded through a fixed FNV-1a hash rather than the interpreter hash, so a given name and seed reproduce identical data across machines.

Every model reached proven optimality within these settings, except the column-and-constraint generation runs, which terminate on the bound criterion of Section 3.4. Instances contain 1,400 to 1,800 variables and solve in well under a second, so no heuristic was required, and none is reported; we draw no wider conclusion, since establishing when metaheuristics are warranted would need a purpose-built benchmark across instance sizes and structures. Every number below comes from scripts that read the extracted waste balance and write the result tables directly.

Every number reported below is produced by scripts that read the extracted waste balance and write the result tables directly; no figure or table is drawn from any other source.

We first fix the recovery-demand ratio at unity, so that secondary-market demand equals recoverable supply, and sweep the budget across its full admissible range under each service policy. Table 5 reports all eighteen.

The four layers behave very differently, and the aggregate cost curve hides that fact.

The strategic layer is invariant. Across all eighteen solves, nine per service policy, the Hamming distance between opening vectors is zero: the same eight consolidation hubs, three recycling facilities, and four redistribution centers are selected at every budget level under both policies. No capital decision responds to the budget.

Aggregate service is invariant, at 88,000 tonnes throughout, but regional service is not. Under the regional policy, all twelve market regions receive a different quantity at positive budget; under the aggregate policy, the number of re-served regions grows from two to five as the budget increases. A fixed quantity of recovered material is being re-allocated toward the regions whose protected demand has risen fastest.

The operational layer separates sharply by position in the chain. Table 6 gives the arc-family detail at the maximum budget. Flows on the collection, recycling, redistribution, disposal, and forward arcs are invariant to the tonne, with an L1 change of exactly zero. The entire operational response is carried by the terminal redistribution-to-market arcs, where the relative L1 change reaches 0.50 under the regional policy and 0.81 under the aggregate policy. In other words, the recovery chain up to the point of delivery is frozen, and only the final allocation moves.

Two conclusions follow, and their wording matters. The budget here is strategically inactive and upstream-inactive: it changes no capital decision and no movement in the recovery chain. It is not decision-invariant, since terminal allocation does change, and a study reporting only openings and total delivered volume would have overstated the invariance. Moving from the regional to the aggregate policy alters the shape of the response without activating the strategic layer: at full protection the two coincide exactly, and they differ only at intermediate protection, as Section 3.3.3 shows. Because the two express different guarantees rather than two counterparts of one problem, this is not a repaired budget failing but the ceiling binding under both. The mechanism is the admissibility limitation of [5]: Protected demand cannot be served, because installed capacity binds and the physical return stream lies only 2.1 percent beyond it. Section 5.5 isolates both ceilings and shows that neither responds to the budget.

This section explains the service layer analytically: it accounts for when aggregate delivery stops responding from above, and says nothing about the strategic layer, which Section 5.5 shows is governed differently. The coarse grid brackets a regime change between ρ = 0.9 and 1.0, and the layers do not switch together. Table 7 shows delivery and upstream flow responding at all six ratios below the boundary and at none above, while the opening vector changes at five of the six under the static formulation and four of five under the two-stage formulation of Section 5.4. The boundary is sharp for aggregate delivery and is not a boundary for strategic response at all. We treat it as a threshold for the delivery layer alone.

Table 7 reports the sweep over ρ under the aggregate-service policy, comparing the zero-budget and maximum-budget solutions at each ratio.

Let R denote total returns and s the residual fraction. For a given installed design, the maximum deliverable recovered volume is

S ̄(Y, Z, W) = min { (1 − s) ∑k∈K rk , ∑j∈J crjZj , ∑m∈M cemWm }. (21)

Total demand is ρ(1 − s)R by construction. Delivered volume need not equal protected demand everywhere below the ceiling, because the absorption bound (3) leaves room to place material that is cheaper to deliver than to dispose of, as Section 3.2 describes and as the ρ = 0.2 row of Table 7 shows. Near the upper boundary, however, once the protected-demand requirement is the active lower bound on delivery, delivered volume increases with protection until the installed service ceiling is reached. The boundary is therefore the ratio at which nominal demand itself reaches that ceiling,

Ρ∗(Y, Z, W) = (1 − s) ∑ r . (22)

k∈K k

When nominal demand reaches this boundary, additional protected demand cannot increase aggregate service without changing the installed design. The statement is conditional on (Y, Z, W); it concerns the upper boundary only, and it is not a sufficient condition for a strategic response.

Two thresholds must be separated, and only one of them is available in advance.

The physical threshold is ρ*phys = 1. Since the ceiling can never exceed (1 − s)R, whatever is installed, no design can raise aggregate delivery once nominal demand reaches recoverable supply. This quantity is genuinely pre-optimization: it needs only the return volumes and the residual fraction.

The installed-design threshold is ρ*(Y, Z, W), which is conditional on a facility configuration and can be evaluated once that configuration is known. It is generally unavailable before optimization because Z and W are decisions; Section 5.5 shows directly that raising the shortfall penalty or halving the redistribution fixed cost changes which capacity is installed and therefore moves the threshold.

We deliberately do not offer a closed-form threshold for strategic response. Whether the budget causes another facility to open depends on discrete economics, the fixed cost of the increment against the shortfall penalty avoided, the transport cost of reaching the additional demand, the size of the capacity increment and the geography of the demand it would serve. Section 5.5 shows each of these changing the outcome at a fixed ρ. No volume ratio can capture that, and presenting one would overstate what the identity supports.

The boundary is confirmed numerically. At baseline, the ceiling is 88,000 tonnes against a recoverable supply of 89,877, so ρ*(Y, Z, W) = 0.9791. The mechanism is visible in the zero-budget solutions approaching it, where delivery equals nominal demand to the tonne: 80,890 tonnes at ρ = 0.90 and 87,181 at ρ = 0.97, each equal to ρ(1 − s)R, then 88,000 from ρ = 0.9792 onward, where nominal demand has met the ceiling. A grid at spacing 0.01 shows aggregate delivery responding at every ratio up to 0.97 and saturated from 0.98 onward, and a six-step bisection brackets the switch in [0.97906, 0.97922], whose midpoint differs from the prediction by 3 x 10⁻⁵.

The numerical value belongs to this instance and this configuration. What transfers is the pair of thresholds and their different status: ρ*phys = 1 as a design-independent bound on aggregate service, and ρ*(Y, Z, W) as a conditional statement that becomes checkable once a candidate design is on the table, for instance when assessing whether an existing network will deliver more under demand-side protection. Neither speaks to the strategic layer, as Section 5.5 shows.

The price of robustness is not monotone in ρ. It rises to 12.21 percent at a ratio of 0.6, falls back to 6.65 percent at 0.8, rises again to 27.89 percent at unity, and then declines slowly to 25.69 percent at 1.5. The non-monotonicity is a further reason to distrust the curve as a summary statistic: its magnitude conflates the cost of adaptation, which occurs below saturation, with the cost of unavoidable shortfall, which dominates above it. The largest values of the curve occur precisely where the model is doing the least.

The results so far were obtained from the static formulation, in which openings and flows are chosen together. If the strategic invariance were an artefact of that conflation, it should disappear once openings are committed before the realization and flows become recourse. We test this directly.

The column-and-constraint generation algorithm converged to a relative upper-lower bound gap below 10⁻⁶ at every tested budget level, requiring at most two iterations. The largest gap observed was 8 x 10⁻¹⁶ across the budget sweep and 4 x 10⁻¹⁴ across the ratio sweep, with a single value of -2 x 10⁻¹⁵ that reflects machine precision rather than a bound crossing. We therefore treat the reported solutions as optimal to numerical tolerance. Table 8 reports the first-stage response at recovery-demand ratio 1.0. The aggregation used for this experiment is specified in Section 4.3.

The invariance survives. The first-stage design is identical at every budget level: three hubs, two recycling facilities, and three redistribution centers, with a Hamming distance of zero throughout, while the objective rises by 23.87 percent. Committing the openings before the realization and optimizing the flows afterwards does not cause the first stage to hedge in this instance, which is consistent with the fact that at this ratio no configuration can recover more than the return stream yields, so hedging cannot buy additional service. What this establishes is narrower than a general claim: the strategic invariance reported in Section 5.2 is reproduced when the two stages are separated, so it is not attributable to the static timing convention. It remains a finding about this instance at this ratio, on the aggregation of Section 4.3.

The two-stage model does respond at some ratios below saturation, which both complements that finding and bounds it. Table 9 compares the zero-budget and maximum-budget first-stage designs across ρ.

In the two-stage baseline experiment, no first-stage response is observed above service saturation. Below saturation, strategic response remains intermittent: the first stage reconfigures at ratios of 0.4, 0.6, 0.8, and 0.9 but not at 0.2, and the moves differ in kind, opening an additional redistribution center at 0.4 and 0.8 and relocating sites at unchanged counts at 0.6 and 0.9. These results demonstrate association within the tested instance rather than a general relation. Section 5.5 supplies one direct counterexample to necessity: a capacity perturbation at fixed ratio that moves two facilities with aggregate service unchanged. The service-saturation threshold itself governs aggregate-service response, not strategic response, and the two layers decouple in both directions.

Sections 5.2 and 5.3 established that service freezes, but not why. Several parameters could produce the same appearance, and one observation shows that the physical return stream is not the sole explanation: The delivered volume settles at 88,000 tonnes while recoverable supply, at a residual fraction of 0.15, is 89,877 tonnes. The freeze occurs 1,877 tonnes short of the physical bound, so something else binds first. We therefore perturb one parameter at a time and record which constraint is tight at the optimum, whether the delivered volume moves, and whether the strategic layer reactivates. Table 10 summarizes the results.

All 33 perturbations are shown; each varies one parameter with the others held at baseline. Delivery Δ is the change in aggregate delivered volume between Γ = 0 and Γmax, so a row with a non-zero openings change and a zero delivery Δ is one in which the design moves without delivering more. The tests separate three ceilings, which are nested rather than alternative.

The immediate ceiling at baseline is installed redistribution capacity. The four opened redistribution centers provide exactly 88,000 tonnes of capacity and are fully utilized at every budget level; the fifth candidate remains closed. This ceiling is endogenous: the model declines to open the fifth center because its fixed cost exceeds the value of the 1,877 tonnes of additional service it would permit. Halving the per-site capacity moves the freeze to 55,000 tonnes, and halving the fixed cost removes it by making the fifth center worth opening, both of which confirm the attribution.

Behind it lies the physical supply ceiling at (1 − s) times total returns. No perturbation of penalty, fixed cost, capacity, deviation level, or demand allocation ever produces a served volume above 89,877 tonnes. The only parameter that moves it is the residual fraction itself, and when s is lowered, the delivered volume tracks the new bound exactly, reaching 100,451 tonnes at s = 0.05 and 95,164 tonnes at s = 0.10, in each case equal to (1 − s) times 105,738 tonnes. The resulting bound is the ceiling that no design decision can cross.

Between the two lies an economic band. Raising the shortfall penalty to 360 euro per tonne, or halving the redistribution fixed cost, makes the last capacity increment worth installing and lifts service from 88,000 to 89,877 tonnes. Within that band, the choice not to serve is economic, and the band is narrow here, 1,877 tonnes or 2.1 percent of recoverable supply.

Two consequences follow, and both qualify claims made earlier in this paper.

First, the attribution in Section 5.2 must be stated as a composite. The budget is strategically inactive at baseline because the binding constraint is installed capacity, and raising protection cannot justify installing more, since the additional volume that could then be served is bounded by physical supply only 2.1 percent away. It is not correct to attribute the freeze to physical supply alone.

Second, and more consequentially, the perturbations separate what governs each layer. A budget can increase aggregate service only when additional protected demand remains physically and operationally servable. Strategic response is governed separately: it may arise through capacity expansion when additional service is economically worthwhile, or through spatial reconfiguration when the geographic pattern of protected demand changes. Consequently, no volume ratio alone determines the strategic layer.

The evidence is thinner than the mechanism. Only three of the thirty-three perturbations change the opening vector: lowering the residual fraction to 0.05 or 0.10, which raises the physical ceiling and moves four and three facilities while delivery rises with it, and tripling redistribution capacity, which moves two facilities while delivery stays at 89,877 tonnes. Only that third case separates the layers, and the two delivery columns of Table 10 make it checkable; we report it as a single demonstration rather than a pattern. Several perturbations change what is installed without changing the opening count: raising the penalty to 360 euros per tonne, halving the redistribution fixed cost, or increasing capacity each lifts installed redistribution throughput from 88,000 to 110,000 or 132,000 tonnes and moves delivery to the physical ceiling.

Two further findings. The deviation level α moves the price of robustness from 6.90 percent at 0.05 to 56.63 percent at 0.40 without changing delivered volume, the binding constraint, or the opened facilities at any level. The curve's height therefore carries no reliable information about whether the strategic or aggregate-service layers respond. Only the terminal operational component moves with it, re-allocation growing from 9,858 to 55,099 tonnes while upstream L1 stays exactly zero. A curve that moves eightfold tracks one layer while three stay fixed, which is what the decomposition exists to expose. Demand allocation behaves differently again: it changes which constraint binds and how much is delivered, a skewed allocation reaching the physical ceiling of 89,877 tonnes rather than 88,000, yet none of the three allocations produces any response to the budget, the opening vector and delivered volume being identical at Γ = 0 and Γmax. The allocation shapes the plan; the budget does not.

The results answer the question posed in Section 1: Of the four responses a budget can produce, only two moved in this network, and neither was the one that commits capital. We now interpret that for practice and for method.

For a network planner, the finding is a screening rule whose scope depends on purpose. For capital expansion intended to increase aggregate delivery, two conditions must hold together: additional protected demand must remain servable, and the required capacity increment must be worth installing. The saturation boundary of Section 5.3 governs the first and can be checked from volumes; the second turns on the fixed cost of the increment against the penalty it avoids, and cannot be checked. Strategic redesign can also occur without higher aggregate delivery, when protection changes the spatial allocation of demand. Section 5.5 supplies the case: Tripling redistribution capacity moves two facilities while delivery stays at 89,877 tonnes, so a planner reading delivery volumes alone would conclude the design had not moved when it had. Section 3.3.3 supplies a larger one, where matched policies deliver the same total to a different mix of regions.

The response decomposition is inexpensive, and we suggest it become routine. A robust network design study that reports a price-of-robustness curve alone leaves the reader unable to tell which layer moved, and our results show that the layers can move independently: in the same instance, the strategic layer is completely inactive while the terminal operational layer changes by up to 81 percent in relative L1. We also caution against the shortcut we ourselves first took, of characterizing invariance through opened facilities and aggregate delivered volume only. Both were constant here, yet 22,043 tonnes were re-routed. Reporting a flow statistic is necessary; the opening vector alone is not sufficient.

The results also separate two previously independent limitations. The cardinality degeneracy of [3] and the admissibility limitation of [5] are distinct: the aggregate-service policy removes the former while the latter binds under both policies. A modeller who addresses only cardinality sees a changed response profile and may conclude the budget has become meaningful while the strategic layer remains inactive. A second trap sits alongside it: pooling regional constraints into an aggregate one gives the budget something to bind on, and may be the right choice, but it silently replaces a per-region requirement with a system-wide one.

Seven limitations bound the claims, and two of them concern the formulations themselves rather than the data.

Static formulation as the primary vehicle. The layered results for the full 86-source instance come from the static formulation, in which openings and flows are chosen together. That formulation is not a model of how the decisions are actually taken; we retain it because it is what the applied literature uses, and the two-stage check of Section 5.4 corroborates its central finding on a reduced instance. It does not follow that every static result would survive at full scale under recourse, and the affinely adjustable formulation of [27] and the adaptive network-flow formulations of [28] are not tested here at all. Claims resting on the static model should be read as properties of that model.

Aggregate-service formulation and its fungibility assumption. Results obtained under the aggregate-service policy describe a system-wide requirement and rest on an assumption our data cannot test, namely that recovered material is fungible across buyers. The waste balance covers collection, not downstream marketing, so we adopt fungibility as a modeling assumption for the case instance rather than as an established property of this waste stream. Those results establish nothing about how protection is allocated across regions, and Section 3.3.3 quantifies the difference: at matched aggregate protection, the aggregate-service solution delivers less recovered material to as many as 10 of 12 regions than the regional-service solution does. Where fungibility fails, the aggregate policy is inadmissible and only the regional results apply, so readers should not transfer the aggregate findings to settings with binding regional service obligations.

Mechanism tests. These perturb one parameter at a time from a single baseline, so interactions among them are not mapped; the nested structure is robust across the range tested, but the numerical width of the economic band, 2.1 percent here, is instance-specific.

Two-stage instance size. Column-and-constraint generation replicates the recourse block per scenario, so the two-stage experiment uses an aggregated instance. Total capacity per echelon is preserved, but the spatial detail of the 86 bodies is not, so those results corroborate the static findings rather than standing as a case study in their own right.

Saturation boundary. The value 0.979 belongs to this instance and this installed configuration. The boundary (22) is conditional on (Y, Z, W) and is therefore not in general a pre-optimization quantity; only the physical bound at unity is, and it is the weaker of the two. Both concern the upper boundary only. In this instance, service responds at every feasible ratio below it, but the instance is feasible only above approximately ρ = 0.053, so we have not exercised the low-demand regime in which a standing delivery might already cover protected demand; a network with more disposal capacity could exhibit one. Neither boundary predicts strategic response, and we make no claim that they do.

Unobserved demand side. Secondary-market demand is not measured, and although we sweep it rather than fix it, a case study with observed demand for recovered material would be stronger. Coordinates are likewise not published in the waste balance, so distances are indicative; every layered comparison was made under a fixed distance realization across all budget levels, so the comparisons are internally consistent even though absolute costs are not.

Flow statistics. The decomposition reports flow change through L1 statistics and arc counts, which capture magnitude but not structure; a study concerned with which markets gain and which lose would need a distributional or equity measure in addition.

A body-level multi-year panel would allow the deviation ratio to be estimated from observed year-to-year variation rather than assumed. The per-body files published by the state office cover a single year, and we chose not to construct a panel we could not verify against the primary source.