**1. Introduction**

The dramatic variation of sea ice in the Arctic region, due to global warming, has attracted many researchers in science and engineering, where shipping in the Arctic water is one of the hottest issues. Compared to the traditional shipping routes, the Arctic routes have shorter distances linking Asia and North America, as well as linking Asia and Europe, and are a more open navigation environment, more access to the abundant oil and gas resources and lower piracy risk [1]. Therefore, navigation through Arctic routes is considered to be a money-making opportunity for shipping and oil gas companies.

Compared to the traditional ship path planning problem [2–6], sea ice condition becomes a key factor to the route planning in the Arctic region due to its impact on travel time and fuel consumption, as well as the risk of being stuck in the ice [7].

Reference [7] introduced a system for route optimization in ice-covered water, which consisted of an ice model, a ship transit model, and an end-user system. The system was operated on commercial vessels in the Baltic Sea, and its performance was tested. Reference [8] developed an ice navigation system combined with a sea ice model, transit model and a model for route planning which simulated the whole Arctic area. The system employed a modified transit model devised by [9], which integrated various parameters such as ice-breaking fee, port charge, capital cost, etc. Reference [10] developed another ice navigation system where the uncertainty of sea ice prediction and the extremely severe conditions were taken into consideration. The route optimization problem in ice-covered water was regarded as a dynamic stochastic path planning problem, and a heuristic route optimization model was proposed to solve it. Reference [11] devised an automatic ice navigation support system to find the safest- and- shortest routes in the Arctic area for different types of vessels with a geographic information system.

However, unlike the real-time path planning system, the model in this paper is mainly devised to evaluate the strategic prospects for future Arctic routes. Therefore, most data are incorporated for future prediction, where large uncertainties arise from the bias of current multi-climate models and the inconsistency of experts' cognition. Additionally, for commercial navigating in the Arctic routes, sailing safety, as well as economic benefits, should be guaranteed according to the harsh weather conditions in the Arctic area. Therefore, sailing time, economic cost, and navigation safety are all key factors to influence the route planning, which makes the problem a multi-criteria decision making (MCDM) problem.

Owing to the MCDM problem with large uncertainties in the route optimization model, information on each grid has variation so that a new path planning method is required for the model to handle this uncertain decision problem. Many studies have examined that the hesitant fuzzy sets theory is a powerful tool to solve the mentioned kind of problem [12–17]. Therefore, this paper develops a new ice navigation system with a modified A\* path planning algorithm called HFS-A\* algorithm, where the hesitant fuzzy set theory is incorporated to improve the traditional A\* algorithm. Three key factors, including sailing time, economic benefits, and navigation safety, are considered to the final decision-making in this system where multi-models of sea ice prediction and multiple experts' knowledge are used as input. More details related to hesitant fuzzy set theory and A\* algorithm can be seen in Section 2. Section 3 introduces the establishment of the HFS-A\* algorithm. A numerical experiment has been used to examine the proposed model in Section 4, and the conclusion can be seen in Section 5.
