Resource Allocation under Stochastic Demands using Shrinking Horizon Optimization
Alexandros E. Tzikas, Nazim Kemal Ure, Mansur Arief, Mykel J. Kochenderfer, Stephen P. Boyd
TL;DR
The paper tackles multi-period resource allocation under stochastic demands by contrasting a static open-loop solution with a shrinking-horizon, model-predictive-control–style policy that updates future allocations as demands are realized. The static problem is solved via a duality-based approach and bisection on the dual variable $\nu^*$, yielding $a_t^* = F_t^{-1}(1-\nu^*/p_t)$ when feasible, with an exact integral form $\mathbb{E}[\min(d_t,a_t)]=\int_0^{a_t}\mathbb{P}(d_t\ge x)dx$. The sequential policy re-solves a reduced-horizon problem at each step conditioned on observed demands, using conditional quantile functions to adapt to temporal correlations; in the independent-demands case, it reduces to the static solution. Empirical results on jointly log-normal demands show the shrinking-horizon method achieves revenues near a prescient oracle and outperforms the static and roll-forward baselines, validating its effectiveness in leveraging information revealed over time. The work provides a scalable, distribution-driven approach to dynamic revenue management across domains like supply chain, pricing, healthcare, and energy, with future directions including learning demand distributions in closed loop.
Abstract
We consider the problem of optimally allocating a limited number of resources across time to maximize revenue under stochastic demands. This formulation is relevant in various areas of control, such as supply chain, ticket revenue maximization, healthcare operations, and energy allocation in power grids. We propose a bisection method to solve the static optimization problem and extend our approach to a shrinking horizon algorithm for the sequential problem. The shrinking horizon algorithm computes future allocations after updating the distribution of future demands by conditioning on the observed values of demand. We illustrate the method on a simple synthetic example with jointly log-normal demands, showing that it achieves performance close to a bound obtained by solving the prescient problem.