**5. Conclusions**

The opening of Arctic routes will be no longer a dream in the coming future with climate change; route planning is necessary for vessels to navigation on the Arctic region from different points of view (safe, economic cost, time etc.). This paper presents a modified A\* algorithm where the hesitant fuzzy set theory is incorporated for the purpose of solving the MCDM problem in Arctic route planning with large uncertainties originating from multi-climate models and experts' knowledge. Compared to the traditional A\* algorithm, the navigability of the Arctic route is firstly analyzed as a measure to determine the obstacle nodes, and three key factors to vessel navigation, including sailing time, economic cost and risk are overall considered in the HFS-A\* algorithm.

A numerical experiment, which is to find the optimal route between Bergen port and Shanghai port on the NSR, is presented to test the performance of the proposed algorithm. Multi-model ensemble forecast displays that the IB-class 3800 TEU container vessels can navigate on the NSR lasting for 3 to 5 months in the year of 2050. Most model outputs show the navigable time starts from August to October, while merely 2 to 4 models extend the navigable time (from July to November). The sensitivity analysis for the aggregation operators examines that the GHFHG1 operator has an advantage over other aggregation operators in route optimization, and its performance of integrating the three key factors in route planning is better than the performance of any other single factor.

In this paper, the improvement effects for this new approach have been evaluated theoretically and practically. Theoretically speaking, the simple A\* algorithm cannot handle the Arctic path planning problem which has multi-criteria attribution with large uncertainties. Even if we can synthesize the time, economic and uncertainty factors by addition and multiplication, the uncertainties existing in climate model prediction and expert knowledge cannot be portrayed by a simple A\* algorithm. Practically speaking, we compared the route planning result of HFS-A\* algorithm and single factor route planning result (see Figure 4). It can be found that there is a more realistic performance of the HFS-A\* route planning algorithm than compared with the simple A\* route planning algorithm. Overall, this new HFS-A\* algorithm can be well-applied to the Arctic region and to evaluate the strategic prospects for future Arctic routes.

**Author Contributions:** Conceptualization, Y.W..; methodology, Y.W.; software, Y.W.; validation, Y.W.; formal analysis, Y.W.; investigation, Y.W.; resources, L.Q.; data curation, L.Q.; writing—original draft preparation, Y.W.; writing—review and editing, R.Z.; supervision, R.Z.; funding acquisition, R.Z., L.Q.

**Funding:** This work has received funding from the National Natural Science Foundation of China under gran<sup>t</sup> agreemen<sup>t</sup> number 51609254 and number 41375002.

**Conflicts of Interest:** The authors declare no conflicts of interest.
