  In this study, we present a deep learning-optimization framework to tackle
dynamic mixed-integer programs. Specifically, we develop a bidirectional Long
Short Term Memory (LSTM) framework that can process information forward and
backward in time to learn optimal solutions to sequential decision-making
problems. We demonstrate our approach in predicting the optimal decisions for
the single-item capacitated lot-sizing problem (CLSP), where a binary variable
denotes whether to produce in a period or not. Due to the dynamic nature of the
problem, the CLSP can be treated as a sequence labeling task where a recurrent
neural network can capture the problem's temporal dynamics. Computational
results show that our LSTM-Optimization (LSTM-Opt) framework significantly
reduces the solution time of benchmark CLSP problems without much loss in
feasibility and optimality. For example, the predictions at the 85\% level
reduce the CPLEX solution time by a factor of 9 on average for over 240,000
test instances with an optimality gap of less than 0.05\% and 0.4\%
infeasibility in the test set. Also, models trained using shorter planning
horizons can successfully predict the optimal solution of the instances with
longer planning horizons. For the hardest data set, the LSTM predictions at the
25\% level reduce the solution time of 70 CPU hours to less than 2 CPU minutes
with an optimality gap of 0.8\% and without any infeasibility. The LSTM-Opt
framework outperforms classical ML algorithms, such as the logistic regression
and random forest, in terms of the solution quality, and exact approaches, such
as the ($\ell$, S) and dynamic programming-based inequalities, with respect to
the solution time improvement. Our machine learning approach could be
beneficial in tackling sequential decision-making problems similar to CLSP,
which need to be solved repetitively, frequently, and in a fast manner.
