## 3 Modeling an Automated Warehouse in the Steel Sector  
The shuttle-lift-crane automated storage and retrieval system is a prevalent storage solution in the steel industry, specifically designed to store bundles of long metal bars or tubes ranging from ten to twelve meters in length and weighing between 1000 and 5000 kg. Notably, this system does not involve picking individual bundles. Rather, each bundle enters and exits the warehouse without any alterations. As a result, the SLC-AS/RS does not employ loading units or boxes to store items since the bundles themselves serve as the unit loads. Typically, the overall storage area of an SLC-AS/RS is divided into several bearing metal structures known as racks. Each rack can reach heights of up to 20 to 25 m, widths of 12 m (depending on the stored tubes or bars' length), and lengths exceeding 100 m. To facilitate understanding this complex system, a schematic representation is provided in Figure 2, displaying a system comprising three racks and two input and two output locations.  
Figure 2: Schematic representation of an SLC-AS/RS.  
Unlike standard AS/RSs for pallets, each rack in the SLC-AS/RS is divided into perpendicular aisles, crossing the rack lengthwise (\(x\)-axis) and splitting it into multiple sections. Storage locations, composed of metal shelves, line both sides of these aisles, accommodating multiple bundles of varying types and lengths, as depicted in Figure 3. Consequently, these storage locations are not standardized, and previous allocations can affect the possibility of utilizing specific storage spaces. This unique aspect of the SLC-AS/RS could be formulated as a one-dimensional cutting stock problem to minimize the number of storage locations used.  
Resource-wise, an SLC-AS/RS encompasses four categories: (i) input/output (I/O) points; (ii) shuttles; (iii) lifts; and (iv) cranes. Each shuttle serves all racks but only one I/O point, while each lift serves a single crane and shuttle. Conversely, each crane serves every lift within its rack. A detailed breakdown of these machines is provided below:  
* I/O points: Chain conveyors serve as buffers between the system and external processes, such as truck loading or production lines. Input points, equipped with sensors for bundle alignment and weight control, are also depicted in Figure 4A.
* Shuttles: Vehicles handle horizontal movements, transporting bundles between I/O points and racks. They can transport one or two bundles, depending on single-depth or double-deep storage, as illustrated in Figure 4B.
* Lifts: Responsible for vertical movements and transporting bundles between shuttles and the top of racks or cranes. Figure 4C showcases an example of these lifts.
* Cranes: Essential for storage and retrieval operations, moving bundles between lifts and shelves or vice versa. Each rack houses one crane capable of three-axis movements, as detailed in Figure 4D.  
Figure 4: Main elements of a shuttle–lift–crane AS/RS.  
Figure 3: Example of bundle allocation in a single storage location.  
For a comprehensive understanding of the SLC-AS/RS operations and cycle time computation, refer to [PERSON] et al. (2018). In this work, we focus on the behavior and interactions of machines during input and output operations. The operation begins when the entering bundle reaches an input point. If the shuttle is not near the input point, it is recalled, the bundle is loaded upon arrival, and then proceeds to the assigned rack. Once the bundle is loaded onto the lift, it is transported to the top of the rack and requested by the crane for storage. The crane then moves the bundle to the designated storage location. Similarly, we will consider the process for retrieval or output operations. The operation is sent to the crane, which retrieves the required bundle from the storage location and transports it to the corresponding lift for output.