A stochastic programming formulation to minimize the total traveling cost on the Northern Sea Route

We consider the problem minimizing the total traveling cost on a network under uncertainty in using edges due to some obstacles unable to pass the edges. We formulate this model as a two-stage stochastic program. Such a model can be applied to achieve the least-cost path crossing over the Northern Sea Route on the Arctic by considering ice floes as obstacles that incur some extra cost to break it or detour from the path. Lagrangian relaxation is employed to solve an instance of large-scale by relaxing the coupling constraints linking the two different types of decision variables with the associated Lagrangian multipliers. It turns out that the relaxed problem can be separate into the two well-known problems, the shortest-path problem and the 0-1 knapsack problem. In addition to establishing a lower bound from the Lagrangian relaxation, an upper bound can be obtained by finding a feasible solution via variable fixing.