A stochastic analysis of the dynamic space utilization in warehouses

Improving the volume space utilization of modern warehouses is a critical objective in warehouse layout design, making the accurate estimation of space utilization across different layouts essential. However, the dynamic demand and replenishment processes of heterogeneous stock keeping units (SKUs) in modern warehouses present significant challenges in accurately assessing space utilization. Traditional models have relied on deterministic frameworks that assume constant demand and production processes, which do not fully capture the heterogeneity and stochasticity in demand arrivals, sizes, and replenishment times encountered in practical scenarios. This study aims to develop a framework for computing volume space utilization in modern block stacking warehouses, incorporating both heterogeneity and stochasticity. We investigate two models in warehouse scenarios: the lost sale model, where excess demand is lost, and the backorder model, where unmet demand is backlogged. For each model, we derive the closed-form expressions of three key types of space wastes that characterize the space utilization of warehouses: honeycombing, aisle, and top-of-lane space wastes, using a continuous-time Markov chain analytical framework. Utilizing these expressions, we analyze the tradeoffs among these wastes and identify the optimal lane depth that maximizes space utilization. Our case studies show that, in a stochastic environment, the proposed computational framework allows for a more accurate estimation of space utilization, and the application of the obtained optimal lane depth can achieve significantly higher space utilization than deterministic methods found in the literature. This underscores the importance of incorporating stochasticity and heterogeneity of demand and replenishment in layout design. Additionally, by investigating the sensitivity of space utilization to varying demand processes and replenishment strategies, we provide managerial insights for adapting warehouse layout designs in response to changes in SKUs' demand and replenishment patterns.