## 3 Problem Description and Modeling
### Problem Description  
The traditional FJSP with AGVs (FJSP-AGV) is described as follows: In a flexible job shop, there are n jobs to be processed, m machines, and x AGVs. Each job consists of multiple operations, and each operation can be assigned to one machine selected from a set of eligible machines. The processing time for each operation is known and deterministic, and every operation can be processed by at least one machine. Initially, all jobs and AGVs are located at the loading/unloading (L/U) station. AGVs operate in three states: idle,unloaded running, and loaded running. For the first operation of a job, the AGV transports the job from the L/U station to the assigned machine in the loaded state. For subsequent operations, the AGV first travels in the unloaded state to the machine where the job was last processed, and then transports the job in the loaded state to the next assigned machine. Building upon this, this study incorporates AGV charging constraints into the problem formulation, resulting in the FjSPLSP-LTCC model. Specifically, the workshop is equipped with y AGV charger. AGVs, when in the unloaded state, may choose to proceed to a charger for battery replenishment. If the charger is occupied, the AGV must wait until it becomes available. To better reflect practical manufacturing scenarios, three types of AGV charging actions are defined: no charging (NC), opportunity charging (OC), and full charging (FC). OC refers to short-term charging during AGV idle periods to partially replenish battery levels, while FC involves longer charging durations that restore the battery to full capacity. Moreover, to prevent AGV shutdowns due to complete battery depletion, a mandatory maintenance battery change (MBC) policy is introduced, wherein maintenance personnel manually replace the battery for the AGV. Different strategies are suitable for different practical scenarios: the OC strategy is mainly applied when AGVs have short idle windows; the FC strategy is used when AGVs experience longer idle times or when future tasks are expected to have high energy demands; the MBC strategy serves as an emergency backup, automatically triggered by the system monitoring when AGV battery levels approach depletion. The effects of different charging strategies on AGV battery levels are illustrated in Figure 1. It should be noted that battery degradation effects are not considered in Figure 1; it only reflects the improvements in battery level under different charging strategies. In addition, the FjSPLSP-LTCC model incorporates the battery degradation effect of AGVs, characterized by a nonlinear decline in travel speed as the remaining battery level decreases. This study models the degradation through a four-phase velocity control function, described as follows: Saturation Zone (\(s\geq s_{\text{rated}}\)): When energy supply is sufficient, the AGV operates at its rated maximum speed. Decay Zone (\(s_{\text{critical}}<s<s_{\text{rated}}\)): The velocity is determined by a combination of a linear term \(\alpha\cdot s\) and an exponential decay term \(\beta\cdot e^{-\lambda(s_{\text{rated}}-s)}\), reflecting the effects of increased internal resistance and polarization that lead to compounded energy losses. Sustainment Zone (\(s_{\text{dead}}\leq s\leq s_{\text{critical}}\)): A smooth deceleration trend is modeled using \(v_{min}\cdot\text{tanh}\Big{(}\frac{s-s_{\text{dead}}}{s_{\text{critical}}-s _{\text{dead}}}\cdot\pi\Big{)}\). Failure Zone (\(s<s_{\text{dead}}\)): A deep discharge protection mechanism is triggered, forcing the AGV to shut down in order to prevent battery damage. The corresponding mathematical formulation is given in  
Figure 1: The different actions and states of AGVs during transport. Equation (1), and the velocity profile under different battery levels is illustrated in Figure 2. Here, \(\alpha=0.5\), \(\beta=0.8\), \(\lambda=2.5\). \[v(s)=