Dividing splittable goods evenly and with limited fragmentation

A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the cases F= 1 and F= 2. For F= 1 , the max–min and min–max problems are solvable in linear time. The case F= 2 has neat formulations and structural characterizations in terms of weighted graphs. First we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in linear time if m≥ n- 1 , and a solution always exists in this case, in contrast to F= 1. Moreover, the problem is fixed-parameter tractable in the parameter 2 m- n. (Note that this parameter measures the number of agents above the trivial threshold m= n/ 2.) The structural results suggest another related problem where unsplittable items shall be assigned to subsets so as to balance the average sizes (rather than the total sizes) in these subsets. We give an approximation-preserving reduction from our original splitting problem with fragmentation F= 2 to this averaging problem, and some approximation results in cases when m is close to either n or n\ua0/\ua02

Dividing splittable goods evenly and with limited fragmentation

A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares of at most F pieces. We call F the fragmentation. For F=1 we can solve the max-min and min-max problems in linear time. The case F=2 has neat formulations and structural characterizations in terms of weighted graphs. Here we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in linear time if m>n-2, and a solution always exists in this case. Moreover, case F=2 is fixed-parameter tractable in the parameter 2m-n. The results also give rise to various open problems

Fuelling the zero-emissions road freight of the future: routing of mobile fuellers

The future of zero-emissions road freight is closely tied to the sufficient availability of new and clean fuel options such as electricity and Hydrogen. In goods distribution using Electric Commercial Vehicles (ECVs) and Hydrogen Fuel Cell Vehicles (HFCVs) a major challenge in the transition period would pertain to their limited autonomy and scarce and unevenly distributed refuelling stations. One viable solution to facilitate and speed up the adoption of ECVs/HFCVs by logistics, however, is to get the fuel to the point where it is needed (instead of diverting the route of delivery vehicles to refuelling stations) using "Mobile Fuellers (MFs)". These are mobile battery swapping/recharging vans or mobile Hydrogen fuellers that can travel to a running ECV/HFCV to provide the fuel they require to complete their delivery routes at a rendezvous time and space. In this presentation, new vehicle routing models will be presented for a third party company that provides MF services. In the proposed problem variant, the MF provider company receives routing plans of multiple customer companies and has to design routes for a fleet of capacitated MFs that have to synchronise their routes with the running vehicles to deliver the required amount of fuel on-the-fly. This presentation will discuss and compare several mathematical models based on different business models and collaborative logistics scenarios