[Paper Review] Predicting Tactical Solutions to Operational Planning Problems under Imperfect Information

Eric Larsen, Sébastien Lachapelle+4 more|arXiv (Cornell University)|Jan 1, 2022

Transportation Planning and Optimization36 references31 citations

TL;DR

This paper proposes a supervised machine learning approach to rapidly predict expected tactical descriptions of operational solutions (TDOS) in two-stage stochastic programming under imperfect information. By training deep learning models on synthetically generated deterministic problem instances with controlled uncertainty, the method achieves millisecond-scale predictions with accuracy close to theoretical lower bounds, enabling faster tactical decision-making in rail load planning and similar applications.

ABSTRACT

This paper offers a methodological contribution at the intersection of machine learning and operations research. Namely, we propose a methodology to quickly predict expected tactical descriptions of operational solutions (TDOSs). The problem we address occurs in the context of two-stage stochastic programming, where the second stage is demanding computationally. We aim to predict at a high speed the expected TDOS associated with the second-stage problem, conditionally on the first-stage variables. This may be used in support of the solution to the overall two-stage problem by avoiding the online generation of multiple second-stage scenarios and solutions. We formulate the tactical prediction problem as a stochastic optimal prediction program, whose solution we approximate with supervised machine learning. The training data set consists of a large number of deterministic operational problems generated by controlled probabilistic sampling. The labels are computed based on solutions to these problems (solved independently and offline), employing appropriate aggregation and subselection methods to address uncertainty. Results on our motivating application on load planning for rail transportation show that deep learning models produce accurate predictions in very short computing time (milliseconds or less). The predictive accuracy is close to the lower bounds calculated based on sample average approximation of the stochastic prediction programs.

Motivation & Objective

• To address the computational bottleneck of solving second-stage operational problems online in two-stage stochastic programming under imperfect information.
• To enable fast, high-accuracy prediction of expected tactical descriptions of operational solutions (TDOS) for use in tactical decision-making.
• To reduce reliance on online scenario generation and repeated solution of computationally expensive second-stage problems.
• To bridge operations research and machine learning by using ML to approximate expected solutions of NP-hard operational problems.

Proposed method

• Generate large-scale training data by sampling deterministic operational problem instances under controlled probabilistic distributions of uncertain inputs (e.g., container weights).
• Solve each sampled instance offline using an integer linear programming solver to obtain detailed operational solutions.
• Aggregate and subselect these solutions into a compact TDOS representation (e.g., number of containers of each type loaded, railcars used).
• Train supervised machine learning models—particularly deep feedforward neural networks—on the input (first-stage variables and uncertain features) and output (TDOS) pairs.
• Account for missing or uncertain input features through appropriate sampling, aggregation, and subselection techniques during data generation.
• Use the trained model to predict expected TDOS in real time, with low latency and high accuracy, under imperfect information at decision time.

Experimental results

Research questions

• RQ1 Can machine learning models accurately predict the expected tactical description of operational solutions under imperfect information in two-stage stochastic programs?
• RQ2 How does the predictive performance of different ML models (e.g., MLP, logistic regression, linear regression) compare to theoretical lower bounds based on sample average approximation?
• RQ3 To what extent can a model trained on simpler problem instances generalize to harder, more complex instances without retraining?
• RQ4 How does the inclusion of uncertainty handling in data generation affect the robustness and accuracy of the predictions?
• RQ5 Can the proposed method significantly reduce online computation time compared to solving the second-stage problem from scratch?

Key findings

• A regression feedforward neural network (RegMLP) achieved the best overall performance, with a mean absolute error (MAE) of 2.372 on test data, close to the theoretical lower bound.
• Predictions were generated in less than one millisecond per instance, significantly faster than solving a single deterministic ILP instance.
• The model trained on simpler instances generalized well to harder instances without retraining, demonstrating strong generalization capability.
• The method’s predictive accuracy was within 0.05 of the lower bound calculated via sample average approximation, indicating near-optimality.
• The approach reduced the need for online scenario generation and second-stage solution computation, enabling faster tactical decisions in rail load planning.
• The results suggest that the methodology can be extended to other two-stage stochastic problems, such as vehicle routing with stochastic demands.