Freight planners can skip per-scenario solves: perturbing single-scenario solutions cuts cost-estimate error to 1.33%
Synopsis
Two-stage stochastic service network design must fix how many vehicles run on each arc before demand is known, and the recourse cost of adding spot vehicles makes the problem hard to solve. A feedforward neural network replaces that recourse subproblem and is embedded in a mixed-integer program; because first-stage variables are general integers, training samples come from perturbing solutions of single-scenario problems. Across 18 instances, designs from the perturbation approach deviate 1.33% on average from their full-scenario evaluated cost, versus 10.81% for random sampling.
Interpretation
A freight carrier planning next season's capacity under unknown volumes can now solve a single mixed-integer program in which the recourse cost is replaced by a neural network prediction, instead of modeling every demand scenario. Earlier surrogate approaches of this kind assumed bounded binary first-stage variables, whereas here the capacity decision is a general integer count of vehicles per arc, whose space is in principle unbounded. Across 18 instances, designs from the perturbation-trained network had estimated costs averaging 1.33% away from actual costs evaluated on all 100 scenarios, never exceeding 3.3% on an instance; the random approach averaged 10.81% and reached 22.3%.
Where the training data comes from determines whether the learned network yields usable capacity plans, and sampling near single-scenario optima beats purely random sampling. The random approach draws integer vehicle counts independently per arc from degree-based upper bounds, while the perturbation approach first solves several single-scenario deterministic problems and collects the integer-feasible solutions encountered as base solutions. On held-out test data, perturbation reached mean absolute error 9.11 and mean squared error 161.95, below the random approach's 9.88 and 187.89; on mean absolute percentage error the ranking reverses, with random at 2.02% against 4.99% for perturbation, because random datasets contain many sparse designs whose recourse-cost labels are higher.
Perturbing base solutions to enlarge the sample set, then mixing them with random samples, trades solution-space coverage against plan quality. The hybrid method assigns a fraction α of samples to perturbation and the rest to random generation; raising α from 0.25 to 0.75 lowered average mean absolute error from 11.66 to 9.35 and average actual cost from 1211.40 to 1185.71. α=0.75 attained the lowest or tied-lowest cost on 11 of 18 instances, against 8 for α=0.5 and 3 for α=0.25; on the larger instances α=0.75 was best on 8 of 9.
Against the commonly used sample average approximation, the surrogate yields comparable designs while shifting where the computational effort goes. Sample average approximation solves M mixed-integer programs each with N scenarios, whereas the surrogate solves one model and pays most of its cost in generating labeled training data. Perturbation and hybrid averaged gaps to the best design of 0.94% and 1.02%, against 0.07% for sample average approximation; on the larger 5-12 and 6-16 instances, sample average approximation consistently found better designs.
Perspective
The demonstrated setting is 18 randomly generated consolidation networks with 4 to 6 hubs, 8 to 16 satellite terminals and vehicle capacities of 90, 140 and 200 units, each with a fixed universe of 100 demand scenarios on which all second-stage evaluations are performed. The intended readers are researchers in network design and stochastic programming and carrier planners.
The surrogate is trained per instance, so a change in network configuration requires retraining, and generating training data consumed most of the computational time in the experiments. A single model spanning instances, graph neural networks, and direct learning-based heuristics are proposed as future directions.