## 6 Conclusions  
We presented formulations and experimental evaluations of three different approaches to the vehicle dispatching problem for a fleet of autonomous mobile robots, based on the Stable Marriage Problem (SMP) and the Linear Sum Assignment Problem (LSAP). These formulations make the approaches particularly well suited for a synchronous computation of dispatching decisions for the whole fleet instead of cascading asynchronous dispatching decisions for each individual vehicle. We compare the relative performance of the methods in a broad range of environment configurations, covering differences in terms of floorplan or factory layout, transportation volume (P/T ratio), and capacity of the material handling system (vehicle utilisation). The evaluation focuses specifically on Automated Material Handling Systems (AMHS) based on Autonomous Mobile Robots (AMR), contrasting systems that are highly-constrained by unidirectional guide/flow paths which are much more commonly investigated. For this, we use our own testbed simulation, which was designed and implemented based on an agent-based model (ABM) around a digital twin representation as a backend. The digital twin bridges the fleet management with the execution layer of the individual agents. Overall, the methods were adapted and applied to the AMR domain successfully. From our experimental evaluation, we conclude that the approach is suitable for many real-world applications. For our experimental evaluation, we tracked the absolute performance measures _throughput rate_ and _total travelled distance_. In addition, we proposed two metrics related to the efficiency of the transportation system, _Travel Efficiency_, which puts total travelled distance in relation to the requested transport distance, and _Throughput Effort_ which represents the average travel distance per throughput unit. Generally, the results are quite similar for a broad range of environment configurations. We found that consistently, for each method individually, a lower vehicle utilisation results in a larger throughput rate. That is, throughput can be significantly improved by not operating the vehicle fleet near full capacity or maximum utilisation. This comes at the cost of some decline in efficiency however, which we found to increase with higher vehicle utilisation. We showed that the methods facilitating the reassignment window can indeed improve the efficiency of the transport system, but not so much the actual throughput. Instead, while we found that for most settings the method without reassignment would generally perform worse in terms of total travelled distance, we noticed that often it would also score better at the throughput rate. Contrary to our initial assumption we observed that for most settings, there is no significant difference between the ISAP and SMP formulation with reassignment. In particular, the random environments show a clear tendency towards the above-mentioned observations. We identified some peculiarities with the custom layouts, where we were able to reproduce issues probably related to the employed dispatching rule (minimum distance to the pick-up location). While this rule optimises for the average load waiting time, it does not necessarily lead to a better system performance. Instead, we found that the structural layout of the environment can dictate implicit job prioritisation, which can be amplified by reassignment. That is, when the input storages are located in the vicinity of common drop-off locations (e.g., the output storage), the delivery of input parts would be prioritised. In contrast, when the vehicles would need to travel along the assembly stations on their way to the input storage, they would implicitly prioritise the delivery of output products. The method without reassignment would be less affected, as the time window for this effect to occur would be much smaller. All in all, we identify several potential angles of approach for further improvements.