Small is Beautiful: A Framework for Evaluating and Optimizing Live-Cube Compact Storage Systems

Warehouses occupy much space and land, which has become increasingly scarce in many parts of Europe, Asia, and the United States, particularly close to areas where demand is generated, such as large cities. This paper studies live-cube compact storage systems that may solve this space shortage problem as they do not require travel aisles. Each stored unit load is accessible individually and can be moved in x and y directions by a shuttle as long as an empty location is available, comparable to the well-known 15-puzzle in which 15 numbered tiles slide within a 4 × 4 grid. When multiple empty locations are available on a level, the shuttles can cooperate to create a virtual aisle for fast retrieval of a desired unit load. A lift moves the unit loads across different levels in z direction. Such storage systems are increasingly used in different service sectors like car parking, warehousing, and container handling, but so far they have hardly been studied. For live-cube systems, many research questions still have to be answered, including cycle time calculations, cost comparisons, and energy requirements. In this paper, we first derive simple to use closed-form formulas for expected retrieval time of an arbitrary unit load and validate the quality of these formulas by comparing them with a real application. Second, we propose and solve a mixed-integer nonlinear model to optimize system dimensions by minimizing the retrieval time. We obtain closed-form expressions for minimum retrieval time that are simple to apply in practice. Third, we compare the investment, operational costs, and energy consumption of live-cube systems with traditional systems based on a real application.