## 5 Evaluation of the Solutions  
The solution generated by the proposed algorithm includes a set of operations \(j\in J\), encompassing both input and output operations aimed at fulfilling the requests. Each input operation corresponds to an empty storage location where the entry bundle will be stored. Similarly, every output operation is linked to a bundle in stock for retrieval. Sorting the operations by their scheduled execution time enables the computation of the makespan through a discrete event simulation. This simulation can accommodate stochastic processing times and is applicable irrespective of the system's complexity. Notably, the deterministic simulation considered here naturally extends to a stochastic one. During each simulation run, deterministic processing times can be substituted with randomly generated times using corresponding probability distributions. The specific probability distributions employed to model these random processing times are derived from fitted historical data. To elucidate the computation of the makespan analytically, a simple example is provided below, showcasing machines and operations:  
* \(r_{j}\), \(s_{j}\), and \(l_{j}\) represent the crane, shuttle, and lift necessary for operation \(j\), dependent on its associated operation and the bundle it involves. * \(j^{*}_{r,s,l}\) denotes the most recent operation involving machines \(r\), \(s\), and \(l\). Similarly, \(j^{*}_{r}\) signifies the most recent operation involving only \(r\), while \(j^{*}_{r,s}\) indicates the most recent operation involving crane \(r\) and shuttle \(s\). Key events occurring during the simulation of each operation are defined as follows:  
* \(AVALL_{m}(j)\) represents the moment when operation \(j\) becomes available for machine \(m\). * \(START_{m}(j)\) signifies the start time when machine \(m\) initiates work on operation \(j\). * \(END_{m}(j)\) indicates the completion time when machine \(m\) finishes work on operation \(j\). * \(P1_{j}\) and \(P2_{j}\) denote the first and second positions that crane \(r_{j}\) must visit to execute operation \(j\). If \(j\) is an input operation, \(P1_{j}\) represents the interchange point between lift \(l_{j}\) and crane \(r_{j}\), while \(P2_{j}\) denotes the storage location. Conversely, for an output operation, \(P1_{j}\) corresponds to the storage location, while \(P2_{j}\) indicates the interchange point between lift \(l_{j}\) and crane \(r_{j}\). An illustration of the SLC-AS/RS under consideration is depicted in Figure 5. The layout assumes a configuration with one crane and two lifts within the rack (with one lift designated for each I/O point). The warehouse comprises ten aisles, each spanning a length of one meter. For the input shuttle, traversal of the rack occurs through the fourth aisle, while the output shuttle navigates along the eighth aisle. Within each aisle, storage consists of three levels, each standing at a height of one meter. The crane's movement follows a uniform, linear trajectory, covering one meter per second in both length and height. The unidirectional travel time for the lifts is presumed to be 3 s, while the upload and download times are estimated at 5 s. Additionally, the one-way travel time for shuttles to reach the rack is set at 6 s. The operations to be executed include:  
* \(t_{1}=OUTPUT\); \(\rho_{1}=3\); \(P1_{1}=(5,1)\); \(P2_{1}=(8,0)\). * \(t_{2}=INPUT\); \(\rho_{2}=4\); \(P1_{2}=(4,0)\); \(P2_{2}=(7,2)\). * \(t_{3}=INPUT\); \(\rho_{3}=10\); \(P1_{3}=(4,0)\); \(P2_{3}=(1,3)\). * \(t_{4}=OUTPUT\); \(\rho_{4}=11\); \(P1_{4}=(7,2)\); \(P2_{4}=(8,0)\). * \(t_{5}=OUTPUT\); \(\rho_{5}=20\); \(P1_{5}=(7,1)\); \(P2_{5}=(8,0)\). In a properly executed simulation, the timing for each event (in seconds) is detailed in Table 1. Additionally, a Gantt chart depicting the sequence of operations is presented in Figure 6.