A Dynamic Inventory Rationing Model for Last-Mile Delivery of Medication in Liberia's Community Health Program

Baris Ata; Yufan Chen; Robert Montgomery

math.OCOct 23, 2026

A Dynamic Inventory Rationing Model for Last-Mile Delivery of Medication in Liberia's Community Health Program

Authors: Baris Ata, Yufan Chen, Robert Montgomery

Organizations: Booth School of Business, University of Chicago · Rady School of Management, University of California San Diego

Abstract

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

\theta(n)

for malaria demand among children under five in the Fangonda cluster.

Parameter	Interpretation	Baseline Value
$T$	Planning horizon	$22$
$K$	Number of CHAs	$7$
$\theta(n)$	Weekly incidence probability	See Equation ( 6.2 ) and Figure 2
$\mu_{k}(n)$ , $\Gamma_{kk}(n)$	Weekly demand mean and variance	See Equations ( 6.3 )–( 6.4 )
$\Gamma_{k\ell}(n)$ , $k\neq\ell$	Cross-location covariance	$0$
$p_{k}$	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

\lambda=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

\lambda\in\{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.

$\kappa$	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

\lambda=0.25

Variance	Objective gap vs. fairness-only	Operating cost	Fairness utility	Coverage	Mismatch frequency	$\bar{S}$	$S_{+}$
1 $\times$	-2.794	71.420	335.799	0.676	0.852	1.276	1.497
2 $\times$	-4.698	73.660	334.153	0.668	0.725	1.458	2.012
5 $\times$	-7.470	79.714	332.506	0.647	0.536	1.861	3.474

Table 4: Demand-overdispersion response when

\lambda=0.25

$\rho$	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

\lambda=0.25

$N_{h}$	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

\lambda=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

\lambda=0.25

Figure 6: Operating-cost and raw-fairness trade-offs across fairness weights and scenarios. Circles show our policy for

\lambda=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

N_{h}=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,\dots,10j+9

Parameter	Coarse stage ( $M_{c}=110$ )	Fine stage ( $M=1{,}100$ )
Panel A: Network architecture.
Input dimension	$9$
$H_{m}$ / $G_{m}$ 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.

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Online Allocation with Unknown Shared Supply

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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

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