## 4 Solving the Warehouse Delivery Problem
### Task Assignment and Sequencing
#### 4.3.2 Conflict Detection and Resolution
More pronounced, for the comparison of different robot diameters, increasing the size of the robots resulted in an increase in the number of memory failures and timeouts. The reason for this is that increasing the size of the robot means that the size of the clusters of conflicting vertices also needs to increase. However, increasing the cluster of conflicting vertices in turn leads to a combinatorial increase in the number of conflict constraints that need to be resolved. This both increases the problem size and the time needed to resolve these new constraints. Different mechanisms that can reduce the number of conflict constraints should be explored. One option would be to introduce additional primitives that group vertices together for the purpose of conflict resolution. This would separate the level of fine-grained detail needed to perform robot routing from the detail needed for conflict resolution. In principle, this could potentially increase the minimum makespan achievable for any problem instance, for example, robots could not follow each other quite so closely as would be possible otherwise. Nevertheless, such a trade-off may be acceptable if it leads to a significant increase in solver performance. This trade-off would need to be evaluated experimentally. Finally, industrial settings also require dynamic responses to events as new items are delivered to the warehouse. Because such events are unpredictable a working system requires re-planning, with new delivery tasks added to any waiting and currently executing tasks. While our encoding does allow for re-planning with partial assignments, nevertheless our evaluation infrastructure would need to be extended to cover this scenario. However, this is largely an engineering challenge, important from a practical perspective, but not something that requires the development of new ASP techniques and encodings. Conceptualization, methodology and data curation, D.R., T.S., P.W., K.C. and S.L. ; software and analysis, D.R., T.S. and P.W. ; investigation and resources, D.R., T.S., P.W., K.C., S.L. and T.C.S. ; writing, D.R., T.S., P.W. and T.C.S. ; review and editing, D.R., T.S. and P.W. All authors have read and agreed to the published version of the manuscript. This work was supported by Dorabot, China, and Potassco Solutions, Germany, as well as DFG grants SCHA 550/11 and 15, Germany. The ASP encodings and benchmark data used to generate the results is available at [[https://github.com/krr-up/robot-scheduling-encodings/releases/tag/v1.0](https://github.com/krr-up/robot-scheduling-encodings/releases/tag/v1.0)]([https://github.com/krr-up/robot-scheduling-encodings/releases/tag/v1.0](https://github.com/krr-up/robot-scheduling-encodings/releases/tag/v1.0)) (accessed on 1 February 2023). The authors declare no conflict of interest. ## References

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