  Multi-objective combinatorial optimization problems (MOCOPs), one type of
complex optimization problems, widely exist in various real applications.
Although meta-heuristics have been successfully applied to address MOCOPs, the
calculation time is often much longer. Recently, a number of deep reinforcement
learning (DRL) methods have been proposed to generate approximate optimal
solutions to the combinatorial optimization problems. However, the existing
studies on DRL have seldom focused on MOCOPs. This study proposes a
single-model deep reinforcement learning framework, called multi-objective
Pointer Network (MOPN), where the input structure of PN is effectively improved
so that the single PN is capable of solving MOCOPs. In addition, two training
strategies, based on representative model and transfer learning, respectively,
are proposed to further enhance the performance of MOPN in different
application scenarios. Moreover, compared to classical meta-heuristics, MOPN
only consumes much less time on forward propagation to obtain the Pareto front.
Meanwhile, MOPN is insensitive to problem scale, meaning that a trained MOPN is
able to address MOCOPs with different scales. To verify the performance of
MOPN, extensive experiments are conducted on three multi-objective traveling
salesman problems, in comparison with one state-of-the-art model DRL-MOA and
three classical multi-objective meta-heuristics. Experimental results
demonstrate that the proposed model outperforms all the comparative methods
with only 20\% to 40\% training time of DRL-MOA.
