Optimal two-class-based storage in a live-cube compact storage system

Live-cube compact storage systems realize high storage space utilization and high throughput, due to full automation and independent movements of unit loads in three-dimensional space. Applying an optimal two-class-based storage policy where high-turnover products are stored at locations closer to the Input/Output point significantly reduces the response time. Live-cube systems are used in various sectors, such as warehouses and distribution centers, parking systems, and container yards. The system stores unit loads, such as pallets, cars, or containers, multi-deep at multiple levels of storage grids. Each unit load is located on its own shuttle. Shuttles move unit loads at each level in the x and y directions, with a lift taking care of the movement in the z-direction. Movement of a requested unit load to the lift location is comparable to solving a Sam Loyd’s puzzle game where 15 numbered tiles move in a 4 × 4 grid. However, with multiple empty locations, a virtual aisle can be created to shorten the retrieval time for a requested unit load. In this article, we optimize the dimensions and zone boundary of a two-class live-cube compact storage system leading to a minimum response time. We propose a mixed-integer nonlinear model that consists of 36 sub-cases, each representing a specific configuration and first zone boundary. Properties of the optimal system are used to simplify the model without losing any optimality. The overall optimal solutions are then obtained by solving the remaining sub-cases. Although the solution procedure is tedious, we eventually obtain two sets of closed-form expressions for the optimal system dimensions and first zone boundary for any desired system size. In addition, we propose an algorithm to obtain the optimal first zone boundary for situations where the optimal system dimensions cannot be achieved. To test the effectiveness of optimal system dimensions and first zone boundary on the performance of a two-class-based live-cube system, we perform a sensitivity analysis by varying the ABC curve, system size, first zone size, and shape factor. The results show that for most cases an optimal two-class-based storage outperforms random storage, with up to 45% shorter expected retrieval time.

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