Pareto surface for three objectives

When focusing on all three objectives the economic, the environmental and the social one the resulting set of non-dominated solutions forms a Pareto surface. The approx-imated Pareto-set of solutions for instance vienna obtained by LNS embedded in the heuristic cuboid splitting described in Subsection 5.3 is shown in Figure 12.

Figure 12: Pareto surface of instance vienna with economic, environmental and social objec-tives.

When considering all three objectives, the non-dominated solutions found for instance vienna (see Figure 12) underline the impact of considering all three objectives at once.

The Pareto-set forms a kind of a 'bowl', such that solutions which are very cost-ecient cause a rather high amount of emissions but are nevertheless good regarding disturbance.

On the other side of the 'bowl' (upper right point in Figure 12), the solution found is very good with respect to emissions, but costly and causing a high amount of distur-bance. Solutions with lowest disturbance show an average performance regarding cost and emissions. Hence, this Pareto surface provides decision support in all three aspects of sustainability.

7 Conclusion

In this paper, we deal with the MO2eVRPSynGZ. The assignment to echelons of cus-tomers located in the 'grey zone', which is the zone between the inner-city center and the area around it, is part of the model and is done in the solution procedure of the optimization problem.

In the model we focus on the economic objective expressed in costs as rst objective.

Additional objectives of citizens and municipal authorities are included by considering negative external eects of freight transport such as GHG emissions as environmental objective and disturbance of citizens, caused by noise and congestion, as social objec-tive.

Solutions for the formulated multi-objective optimization problem are found by LNS embedded in an - constraint method the heuristic rectangle splitting. The latter method is extended in this paper to a heuristic cuboid splitting which is able to deal with three objectives.

Computational results for articial test instances and a realistic instance of Vienna show that the quality of the provided LNS can improve results signicantly compared to GRASP+PR, the application of the 'grey zone' concept can improve the solution quality of the single-objective problem and also of the approximated Pareto-set of solutions in the multi-objective case, especially if customers are clustered. Furthermore, our results emphasize that the city layout has to be considered in the decision process if the pro-vided freight distribution scheme should be established, because especially for cities in which the city center is located in the middle of the city this scheme turned out to be advantageous. Finally, the economic, the environmental and the social objectives can be considered at the same time by the provided heuristic cuboid splitting, which yields a Pareto surface and underlines the trade-o between all three objectives. Because of lacking data, disturbance as additional objective to economic costs and GHG emissions was considered only for the realistic instance of Vienna. Therefore, future work has to focus on additional tests with all three objectives and, hence, realistic test instances for other cities with data about population density as well as points of interest are required to further conrm the ndings of this paper.

Acknowledgements

Part of this work was funded by the Austrian Research Promotion Agency as part of the JPI CONCOORD project (FFG project No. 839739). Furthermore, we gratefully acknowledge the travel time data provided by Taxi 31300, as well as the support of the

Natural Sciences and Engineering Research Council of Canada (NSERC) through its Discovery Grants program.

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Appendix

Table 4: Data for the calculation of GHG emissions

First echelon (Light duty vehicle [3.5t]) Second echelon (eBullitt)

Average fuel 0.085 l/km 10.45 Wh/km

(diesel)/ power Umweltbundesamt (2017) Larry vs. Harry (2017)

consumption

CO2-factor 0.52 (status 2017)

Umweltbundesamt (n. d.)^∗

CO2-equivalent 279.4 g/km (status 2017) 5 g/km

Umweltbundesamt (n. d.)^∗ based on above numbers

∗Provided numbers are subject to regular updates, previous data is available on request

Table 5: Instance characteristics

Table 6: Performance of LNS operators

Operator New best Average computational

Figure 13: Population density of city districts provided by the municipality of Vienna, location of POIs considered (schools and hospitals) and nodes of instance vienna.

Figure 14: Basis for calculation of disturbance of a certain path of instance vienna; path is divided into segments based on city districts.

Figure 15: Settings for test instances. On the left: setting A with city center in the middle of the city area and rst-echelon depot on the outskirts of the city. In the middle: setting B with city center located at the border of the city and rst-echelon depot opposite, on the other side of the city. On the right: setting C with city center located at the border of the city and rst-echelon depot diagonally opposite.

Figure 16: Parameter tests for LNS; total cost vs. cpu time as normalized values.