The structure of the multi-graph has been considered given and fixed, representing the real-world connections between the nodes. So far, the research on the abovementioned optimisation problems focused mainly on the search algorithms for finding the best solutions using the multi-graph formulation of the problem.
![graph paper maker with numbers graph paper maker with numbers](https://i.pinimg.com/736x/91/b2/04/91b204b92a3c4cd5e371a59346c78598--weather-numbers.jpg)
Furthermore, there are often various constraints which have to be satisfied by the solutions in order to be feasible, for example a delivery vehicle must visit customers in specified time windows. The parallel edges can represent routes with different costs in the multi-objective vehicle routing problem 1, 2, 3 and hazardous material transportation 4, different modes of transport 5, 6 in the multimodal shortest path problem and tour planning 7, or different speed profiles in the trajectory based traffic management 8, 9, 10. The multiple parallel edges between pairs of nodes of the multi-graph offer a convenient way of modelling the real-world structure and inherent multi-objective nature of the problems including time, economic or environmental objectives. An example of problems include the vehicle routing problem, hazardous material transportation, multimodal shortest path problem and airport ground movement problem to name a few. Many optimisation problems in transportation, logistics or telecommunications can be formulated as search on a multi-graph. The findings pave the way to an informed approach to multi-graph creation for similar problems based on multi-graphs.
![graph paper maker with numbers graph paper maker with numbers](http://mimeticsoft.farvista.net/graphpapergenerator/pictures/graphpapergeneratorscreenshot.png)
Furthermore, we show that including edges with dominated costs in the multi-graph can also improve the results in the presence of time window constraints. An indicator is further proposed which can estimate when the multi-graph would benefit from a higher number of parallel edges. The results show that the number of parallel edges not only affects the computational complexity but also the number of trade-off solutions and the quality of the found solutions. Using the airport ground movement problem as an example, we analyse how the number of parallel edges and their costs in multi-graph structure influence the quality of obtained solutions found by the routing algorithm. In this study, we conduct case studies for a special type of constrained routing and scheduling problems.
![graph paper maker with numbers graph paper maker with numbers](https://www.printablee.com/postpic/2015/06/4-quadrant-graph-paper-20-x-20-large_405107.png)
The multi-graph structure is often based on infrastructure and available connections between nodes. Multi-graphs where several edges connect a pair of nodes are an important modelling approach for many real-world optimisation problems.