  The facility location problems (FLPs) are a typical class of NP-hard
combinatorial optimization problems, which are widely seen in the supply chain
and logistics. Many mathematical and heuristic algorithms have been developed
for optimizing the FLP. In addition to the transportation cost, there are
usually multiple conflicting objectives in realistic applications. It is
therefore desirable to design algorithms that find a set of Pareto solutions
efficiently without enormous search cost. In this paper, we consider the
multi-objective facility location problem (MO-FLP) that simultaneously
minimizes the overall cost and maximizes the system reliability. We develop a
learning-based approach to predicting the distribution probability of the
entire Pareto set for a given problem. To this end, the MO-FLP is modeled as a
bipartite graph optimization problem and two graph neural networks are
constructed to learn the implicit graph representation on nodes and edges. The
network outputs are then converted into the probability distribution of the
Pareto set, from which a set of non-dominated solutions can be sampled
non-autoregressively. Experimental results on MO-FLP instances of different
scales show that the proposed approach achieves a comparable performance to a
widely used multi-objective evolutionary algorithm in terms of the solution
quality while significantly reducing the computational cost for search.
