We study the dynamic allocation of scarce medications in Liberia's community health program. A central warehouse supplies a health facility and the Community Health Assistants (CHAs) it supervises. When a CHA runs out of medication, patients are referred to the facility, incurring travel costs if they complete the referral and remain untreated otherwise. Allocating inventory to CHAs allows more patients to receive treatment without traveling, while retaining inventory centrally preserves flexibility to respond to uncertain demand. The allocation problem must balance these considerations while accounting for fairness across locations. We formulate this problem as a finite-horizon stochastic control problem whose objective combines the costs of referrals and untreated patients with a demand-weighted fairness utility. Because standard dynamic programming methods are computationally intractable in this setting, we develop a continuous-time approximation that involves controlling the drift rate of a reflected Brownian motion. We then adapt the deep splitting method, using deep neural networks to approximate the value function and its gradient and compute a dynamic allocation policy. We calibrate the model using data on malaria cases among children in the Fangonda region of Lofa County, Liberia, and evaluate the proposed policy in the original discrete-time model. In the baseline setting, our policy reduces operating costs by about 6% relative to the strongest benchmark, the fairness-only policy, while retaining approximately 98% of its fairness utility. Additional experiments show that higher referral costs shift inventory toward CHAs and reduce local unmet demand, while greater demand variability increases our policy's performance over the fairness-only benchmark. Our model and computational approach provide a framework for studying similar last-mile allocation problems.
Figures & tables
Figure 1: An illustration of the patient and medication flows within the model for a single Health Facility (HF) and K Community Health Assistants (CHAs).
Figure 2: Calibrated weekly incidence probability θ(n) for malaria demand among children under five in the Fangonda cluster.
Parameter
Interpretation
Baseline Value
T
Planning horizon
22
K
Number of CHAs
7
θ(n)
Weekly incidence probability
See Equation ( 6.2 ) and Figure 2
μk(n) , Γkk(n)
Weekly demand mean and variance
See Equations ( 6.3 )–( 6.4 )
Γkℓ(n) , k=ℓ
Cross-location covariance
0
pk
Referral compliance probability
0.69
Table 1: Model parameters in the test problem for the Fangonda cluster.
Policy
Total objective
Operating cost
Fairness utility
Coverage
Mean CHA local fill
HF shipment share
Our policy
-12.530
71.420
335.799
0.676
0.661
0.531
80–20 Rule
12.300
72.799
241.994
0.688
0.293
0.800
Static
-0.909
70.683
286.370
0.679
0.672
0.528
Fairness-only
-9.736
76.090
343.307
0.655
0.624
0.528
Myopic
19.827
77.857
232.118
0.652
0.495
0.635
Fluid-optimal
-2.744
81.899
338.569
0.618
0.781
0.365
Table 2: Baseline policy comparison when λ=0.25 .
Figure 3: Weekly shipments under our policy in the baseline case. Blue lines show means over 10,000 demand paths, and shaded bands span the pointwise 5th–95th percentiles across paths. Orange dashed lines show mean weekly demand. The Warehouse panel compares total shipments with total local demand across all locations. Shipments and demand in week n are displayed over (n−1,n] .
Figure 4: Inventory trajectories under our policy in the baseline case. Blue lines show means over 10,000 demand paths, and shaded bands span the pointwise 5th–95th percentiles across paths. Orange dashed lines show mean weekly demand on the right axis, using total local demand for Warehouse and excluding referrals for HF.
Figure 5: Cost–fairness trade-off over λ∈{0.10,0.25,0.50,1.00} under the baseline calibration. The fairness-only policy is shown separately as the benchmark policy with maximum fairness.
κ
CHA shipment share
CHA local unmet
Release fraction through week 8
Mismatch frequency
Mean absolute severity
0.05
0.405
42.160
0.510
0.884
1.399
0.15
0.469
33.720
0.450
0.852
1.276
0.25
0.531
25.090
0.430
0.779
1.033
Table 3: Travel-cost response under λ=0.25 .
Variance
Objective gap vs. fairness-only
Operating cost
Fairness utility
Coverage
Mismatch frequency
Sˉ
S+
1 ×
-2.794
71.420
335.799
0.676
0.852
1.276
1.497
2 ×
-4.698
73.660
334.153
0.668
0.725
1.458
2.012
5 ×
-7.470
79.714
332.506
0.647
0.536
1.861
3.474
Table 4: Demand-overdispersion response when λ=0.25 .
ρ
Objective gap vs. fairness-only
Operating cost
Fairness utility
Mean CHA local fill
CHA shipment share
Mismatch frequency
Mean absolute severity
0.00
-2.794
71.420
335.799
0.661
0.469
0.852
1.276
0.50
-2.395
71.080
330.293
0.697
0.480
0.547
0.591
0.70
-2.868
71.338
332.842
0.757
0.526
0.405
0.384
Table 5: Demand-correlation response when λ=0.25 .
Nh
Total expected demand
Objective gap vs. fairness-only
HF shipment share
Coverage
HF-load fulfillment
Mean CHA local fill
199
197.5
-3.724
0.500
0.709
0.597
0.706
219
207.5
-2.794
0.531
0.676
0.572
0.661
243
219.5
-0.827
0.550
0.644
0.539
0.645
Table 6: HF-demand response when λ=0.25 .
Q(0)
Objective gap vs. fairness-only
Week-8 release, units (%)
Coverage
HF-load fulfillment
HF shipment share
HF shipment
CHA shipment
124.5
1.873
63.851 (51.3%)
0.598
0.451
0.477
59.331
65.169
142
-2.794
63.884 (45.0%)
0.676
0.572
0.531
75.441
66.557
166
-4.653
73.706 (44.4%)
0.790
0.745
0.587
97.491
68.491
Table 7: Initial-inventory response when λ=0.25 .
Figure 6: Operating-cost and raw-fairness trade-offs across fairness weights and scenarios. Circles show our policy for λ=0.10,0.15,0.25,0.35,0.50,0.70,1.00 , and stars show fairness-only policy. The left panel varies HF-demand exposure with Q(0)=142 , and the right panel varies initial inventory with Nh=219 .
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 7: Componentwise state coverage in the baseline case.
Figure 8: An illustration of blockwise initialization on the coarse interval j and the corresponding ten fine intervals, 10j,10j+1,…,10j+9 .
Parameter
Coarse stage ( Mc=110 )
Fine stage ( M=1,100 )
Panel A: Network architecture.
Input dimension
9
Hm / Gm output dimension
1 , 8
Number of hidden layers
4
Number of neurons per layer
512
Hidden-layer activation
GELU
Appendix
Table 8: Network architecture and training parameters for the two-stage backward training routine.
Inventory control studies how to order, allocate, and position stock to balance service levels, operating costs, uncertainty, and capacity constraints.
Many real-world resource allocation systems, such as humanitarian logistics and vaccine distribution, must preposition limited supply across multiple locations before demand is realized while stockouts incur irreversible service losses. To study this, we introduce the Online Shared Supply Allocation (OSSA) problem, a stateful online model in which a central hub allocates a finite, unknown supply to multiple sites facing sequential demand under fixed-charge transportation costs and lost-sales penalties. Unlike classical make-to-stock or make-to-order inventory models, OSSA precludes backlogging and replenishment only hedges against future demand. To tackle OSSA, we propose a deterministic threshold-proportional policy GPA and prove that it achieves a 4/3-approximation to the offline optimum up to an additive term independent of the total supply. We complement this with matching lower bounds showing that the 4/3 ratio is tight and that the additive-error dependence is unavoidable, even for randomized algorithms that know the total supply upfront. Finally, we develop a learning-augmented extension to GPA that principally incorporates imperfect forecasts (e.g., from human experts or ML models) commonly available in practice, enabling us to exploit high-quality advice while being robust against arbitrary bad ones. Synthetic and real-world experiments show that GPA outperforms natural baselines with global supply is scarce.
Many social services assign scarce resources, such as housing assistance or hospital interventions, to people who arrive one at a time. Each arrival must receive a decision immediately, and the long-run usage of every resource must stay within its capacity. We study how to learn such a policy from logged observational data. Recent work (Tang et al., 2024) proposes a decision-blind pipeline that fits one outcome model per treatment by regression, prices each capacitated resource from those models, and assigns each arrival the treatment with the largest predicted outcome minus price. We instead train the outcome models end-to-end, differentiating an off-policy estimate of the deployed policy's value through the dual prices themselves. We study two formulations: an exact nonconvex one, and a convex relaxation whose optimum always satisfies the capacity constraints in expectation and whose suboptimality can be explicitly bounded. Every method is evaluated in a queueing simulation to capture the online decision structure. Across six datasets, our two end-to-end variants lead the delay-adjusted policy value wherever capacity is scarce, once a waiting period carries even a small cost. In particular, decision-blind regression predicts ground truth more accurately, but when resource capacities are stringent these baselines frequently violate them and incur much longer queueing delays. Thus, end-to-end training is best suited to settings where resources are genuinely scarce and feasibility matters.
Mohammadsaeed Haghi, Mahdi Salmani, Nima Kelidari +2
We consider online resource allocation under a typical non-profit setting, where limited or even scarce resources are administered by a not-for-profit organization like a government. We focus on the internal-equity by assuming that arriving requesters are homogeneous in terms of their external factors like demands but heterogeneous for their internal attributes like demographics. Specifically, we associate each arriving requester with one or several groups based on their demographics (i.e., race, gender, and age), and we aim to design an equitable distributing strategy such that every group of requesters can receive a fair share of resources proportional to a preset target ratio. We present two LP-based sampling algorithms and investigate them both theoretically (in terms of competitive-ratio analysis) and experimentally based on real COVID-19 vaccination data maintained by the Minnesota Department of Health. Both theoretical and numerical results show that our LP-based sampling strategies can effectively promote equity, especially when the arrival population is disproportionately represented, as observed in the early stage of the COVID-19 vaccine rollout.
Pan Xu, Yifan Xu
Department of Computer Science, New Jersey Institute of Technology, Newark, USA · Key Lab of CNII, MOE, Southeast University, Nanjing, China