## 6 Results
### Pareto Set Quality Results  
We now turn our attention to the quality of the Pareto sets generated by ML-TZSA and for completeness include results for those of RAPTOR solution methodologies. Our Pareto set metrics include dominance ratio, non-dominance ratio, reference set gap average and reference set gap standard deviation. Dominance ratio and non-dominance ratio are Pareto set metrics based on comparing the Pareto sets generated by two different methodologies for the same query directly against each other. The dominance ratio of one Pareto set is defined as the portion of its solutions that dominate at least one solution from the Pareto set it is being compared with. The non-dominance ratio of one Pareto set is defined as the portion of its solutions that are not dominated by any solution from the Pareto set it is being compared with. The reference set gap average of a Pareto set is defined as the average distance (in normalised objective value space) between each of a reference Pareto set's solutions and the nearest solutions in the Pareto set being measured. This measure is zero when a solution method can generate the same Pareto set as the reference set. Regarding the reference set, ideally this would be the optimal Pareto set; however, currently no algorithm scalable enough exists to calculate it within a reasonable amount of time for the large sized instances considered here. Instead, we generate a reference Pareto set for each query composed of the combined set of mutually non-dominated solutions from all available solution methodologies. The available solution methodologies in this case are ML-TZSA for all transfer-zone sample sizes considered, as well as those generated by RAPTOR solution methodologies. Wherever an average can be calculated, a standard deviation can also be calculated; the reference set gap standard deviation measures the variance in the average distances between each of a reference set's solutions and the nearest ones in the Pareto set being evaluated. Low values of this metric indicate a Pareto set consisting of a diverse set of solutions, which is a desirable feature of the solutions we would like to provide to a commuter. In order to ensure a meaningful comparison, we restrict our attention to the set of queries of the 2000 test queries for which all methodologies found solutions. This excludes cases where query origins and destinations lie in the same transfer zone. In such cases, queries are generally of a short total distance, as such multi-modal journeys are less appropriate, and other methodologies are sufficient for trip recommendation purposes. Note that to overcome this issue it is possible to prepare the algorithm for different transfer-zone structures and to then use ones where the query origin and destination are not in the same transfer zone. Tables 2 and 3 provide details regarding the numbers of solutions that each separate methodology contributed to the reference sets over all of the considered test queries, listed in decreasing total contribution order, for the cases of (i) walking and public transport being available and (ii) walking, public transport and e-scooters being available. Generally speaking, ML-TZSA with high transfer-zone sample sizes contributes most to the reference sets, and RAPTOR the least, as expressed by columns 2 and 3. Columns 4 and 5 state the total number of non-dominated solutions generated by each methodology and indicate that (very) roughly, a third of the non-dominated solutions generated by each method form part of the reference sets. Reference set contributions are proportional to the relative numbers of non-dominated solutions generated by each methodology. This indicates that all methodologies are generating consistently good quality solutions, but different numbers of them. Figures 11 and 12 show firstly that, generally speaking, ML-TZSA outperforms RAPTOR in terms of all Pareto set metrics by a significant amount, which is what ML-TZSA is designed for.