[Paper Review] Online Resource Allocation for Reusable Resources

Xilin Zhang, Wang Chi Cheung|arXiv (Cornell University)|Dec 6, 2022

Risk and Portfolio Optimization4 citations

TL;DR

This paper proposes an online algorithm, iterated Multiplicative Weight Update (iMWU), for reusable resource allocation under model uncertainty, achieving a (1−ϵ) approximation of the optimal reward, where ϵ→0 as capacity and horizon grow. The method balances rewards, resource usage, and stochastic usage durations through iterative weight updates, enabling near-optimal performance in dynamic, heterogeneous customer settings with reusable resources.

ABSTRACT

We study a general model on reusable resource allocation under model uncertainty. A heterogeneous population of customers arrive at the decision maker's (DM's) platform sequentially. Upon observing a customer's type, the DM selects an allocation decision, which leads to rewards earned and resources occupied. Each resource unit is occupied for a random duration, and the unit is available for another allocation after the usage duration. Our model captures numerous applications involving admission control and assortment planning. The DM aims to simultaneously maximize multiple types of rewards, while satisfying the resource constraints and being uncertain about the customers' arrival process. We develop a near-optimal algorithm that achieves $(1-ε)$ fraction of the optimal expected rewards, where the error parameter $ε$ decays to zero as the resource capacity units and the length of the horizon grow. The algorithm iteratively applies the Multiplicative Weight Update algorithm in a novel manner, which balances the trade-off among the amounts of rewards earned, resources occupied and usage durations.

Motivation & Objective

• To address the gap in online algorithms for reusable resource allocation under model uncertainty.
• To develop a general framework that captures multiple objectives, customer heterogeneity, and finite usage durations in online settings.
• To achieve near-optimal performance guarantees in terms of reward maximization despite unknown customer arrival processes.
• To extend the theoretical foundations of online non-reusable resource allocation to the more complex reusable resource setting.
• To validate the algorithm’s performance through numerical experiments across varying problem scales.

Proposed method

• The iMWU algorithm applies the Multiplicative Weight Update method in an iterative fashion to balance rewards, resource consumption, and usage durations.
• It uses a dual-based approach to maintain feasibility and optimize over time, updating weights based on observed outcomes and predicted performance.
• The algorithm operates in an online fashion, making decisions upon each customer’s arrival type without prior knowledge of the arrival process.
• It incorporates a discount parameter to buffer against inventory shortages, inspired by prior work on non-reusable settings.
• The method leverages a linear programming relaxation (LP-S) to derive baseline performance, which is used for comparison and theoretical analysis.
• Theoretical analysis shows that the optimality gap decays as O(√(ξ log(|I_c|/ξ))), where ξ is the inverse of relative capacity, approaching zero as capacity increases.

Experimental results

Research questions

• RQ1 Can near-optimal performance be achieved in online reusable resource allocation under model uncertainty?
• RQ2 How can an online algorithm effectively balance multiple objectives, resource consumption, and stochastic usage durations?
• RQ3 Does the iMWU algorithm maintain strong performance guarantees when the customer arrival process is unknown?
• RQ4 How does the algorithm scale with respect to the number of customer types and resource constraints?
• RQ5 Can the theoretical approximation factor be empirically validated across varying problem instances?

Key findings

• The iMWU algorithm achieves a (1−ϵ) approximation of the optimal expected reward, where ϵ→0 as resource capacity and planning horizon grow.
• The algorithm converges to the performance of an offline static algorithm (OSA) with full model knowledge, especially as the horizon increases.
• The optimality gap is bounded by O(√(ξ log(|I_c|/ξ))), which improves as relative capacity increases (i.e., smaller ξ).
• Numerical experiments show that the iMWU algorithm’s performance is robust and nearly independent of the number of customer types (|J|), even when |J| varies from 100 to 1000.
• The normalized reward of iMWU asymptotically approaches that of the OSA, confirming the theoretical convergence to near-optimality.
• The algorithm maintains strong performance even under model uncertainty, with minimal degradation compared to the offline benchmark.