Learning Optimal Solutions via an LSTM-Optimization Framework

Abstract

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 production and inventory levels that must meet the periodic demand, 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 240000 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 ( , 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.

An expandable machine learning-optimization framework to sequential decision-making

“…In this study, we present an expandable framework based on a sequence-to-sequence neural machine translation system to solve sequentially dependent optimization problems with all feasible predictions, which are either optimal or very close to optimal. Yilmaz and Büyüktahtakın (2023b) present one of the pioneering studies that utilize an MLbased prediction methodology to reduce the solution times of repeatedly-solved combinatorial problems in a multi-period setting. Specifically, Yilmaz and Büyüktahtakın (2023b) harness bidirectional Long Short-Term Memory (LSTM) networks to predict binary decision variables that denote the production decision in the capacitated lot-sizing problem.…”

“…Yilmaz and Büyüktahtakın (2023b) present one of the pioneering studies that utilize an MLbased prediction methodology to reduce the solution times of repeatedly-solved combinatorial problems in a multi-period setting. Specifically, Yilmaz and Büyüktahtakın (2023b) harness bidirectional Long Short-Term Memory (LSTM) networks to predict binary decision variables that denote the production decision in the capacitated lot-sizing problem. Models are trained using the solutions of problems that are solved to optimality using CPLEX.…”

“…In Yilmaz and Büyüktahtakın (2023b), predictions for problems with longer planning horizons are generated by concatenating the predictions generated by models trained using problems with shorter planning horizons; e.g., predictions for 270-period problems are generated as three independent prediction sets first from 1 to 90, second from 91 to 180, and third from 181 to 270 using the model that trained 90 periods with. However, this approach disregards the dependence between consecutive sets; e.g., prediction sets 1 to 90 and 91 to 180 are generated independently, which may impact the quality of the predictions.…”

“…In this study, we integrate the attention-based encoder-decoder network presented in Luong et al (2015) into a prediction-optimization framework to overcome the sequential dependence problems, since the output sequence is allowed to be of variable size. Furthermore, in Yilmaz and Büyüktahtakın (2023b), the authors show that the percentage of predicted binary variables that are added to the problem can have a major impact on the quality of the solutions. In other words, the use of high-level predictions may lead to high infeasibility, since the predicted values are fixed in the problem.…”

“…In other words, the use of high-level predictions may lead to high infeasibility, since the predicted values are fixed in the problem. Yilmaz and Büyüktahtakın (2023b) states that fixing all predictions of binary variables may cause infeasibility in one of the two problems in some cases, which is highly undesirable. We propose an iterative algorithm to eliminate infeasible predictions to overcome this issue.…”