Uncovering expert objectives in production planning via inverse optimization: An industrial case study

Production planning in the manufacturing industry often relies on the use of optimization models, but defining an appropriate objective function can be a challenge. In practice, planners must balance competing goals, manage uncertainty, and account for qualitative business preferences that are difficult to quantify. As a result, many optimization models fail to match expert behavior, limiting trust and adoption. In this work, we propose a data-driven inverse optimization framework to infer the objective function implicitly captured in expert planners' decisions. We formulate the production planning problem as a mixed-integer linear program, where the unknown objective function is represented as a weighted sum of hypothesized cost terms. A suboptimality-loss-based inverse optimization method is then applied to learn the objective weights from historical production plans. The proposed approach is applied to a real industrial case provided by Dow, where the inferred weights reveal that avoiding inventory shortages and maintaining consistent cycle lengths dominate the planners' decision-making. Time- and product-dependent extensions further improve predictive accuracy and uncover evolving priorities. Expert interviews confirm the practical validity of these insights. Overall, this study shows that inverse optimization can transform tacit human expertise into interpretable models, enabling more accurate and trusted decision-support tools for complex industrial systems.

Hard Constraints, Smooth Gradients: Learning Feasible Inventory Policies via Differentiable Projection

Many operational problems are constrained sequential decision processes with large, combinatorial action spaces and interdependent feasibility constraints. Mixed-integer linear programs (MILPs) handle such constraints flexibly but scale poorly in stochastic environments. Deep reinforcement learning (DRL) promises scalable decision rules, but existing methods either penalize constraints rather than enforce them, or rely on feasibility mechanisms that break down once constraints interact. We bridge this gap by embedding a differentiable convex optimization module inside the policy: a neural network proposes continuous action targets, a quadratic program projects them onto the relaxed feasible set, and a dual-informed integer mapping restores integrality while preserving feasibility. Given a differentiable simulator, the policy trains end to end from sampled trajectories using pathwise gradients, while handling hard constraints with similar flexibility to MILPs. We show that our feasibility enforcement has bounded error relative to an exact integer projection and ensures the entire feasible action space is reachable. We apply the method to multi-echelon production-inventory planning under shared resource and material constraints. Our policy attains an average optimality gap below 1% on small instances. It further outperforms state-of-the-art echelon base-stock policies by up to 9.75% and a rolling-horizon multi-stage stochastic program by at least 7.7% in larger networks. On an industry-scale case study from ASML, it reduces average cost by up to 3.22% relative to the best-known benchmark policy. The savings are largest where planning is hardest: in tightly capacitated systems with high demand variability. More broadly, our work shows that DRL can deliver economically significant savings in sequential decision problems with interdependent hard constraints, which are widespread in practice.

Discrete-Time MDP Modeling for Multi-Item Capacitated Lot Sizing with Stochastic Demand Timing

This paper studies a finite-horizon multi-item capacitated lot-sizing problem in which demand quantities are deterministic, while demand-arrival periods are stochastic. Each demand occurs once within a known time window and must be satisfied no later than its deadline. The proposed model makes production and allocation decisions at the demand level, allowing it to represent capacity competition, demand-specific backlog, and allocation-dependent inventory dynamics. The stochastic problem is formulated as a discrete-time Markov decision process (DTMDP), including the state space, feasible actions, transition kernel, and one-period cost function. To isolate the computational effect of stochastic timing, each stochastic instance is first compared with a deterministic counterpart in which each arrival distribution is replaced by its most likely arrival period. This comparison shows that stochastic timing substantially increases the number of states, the number of transitions, solution time, and memory pressure. A genetic algorithm (GA) is then proposed for the stochastic-timing problem. The GA searches over feasible state-feedback policies and evaluates each policy exactly under the DTMDP transition model. Computational experiments on 330 benchmark instances show that the GA remains close to the exact stochastic solution whenever the latter is available, with an average optimality gap of about $3.44\%$. On the difficult benchmark instances, comprising 90 test cases, the GA remains below the $5\%$ optimality-gap threshold and achieves an average optimization speedup of $6.89 \pm 1.41$ at the $95\%$ confidence level. For instances that cannot be solved exactly on the available hardware, an empirical Bellman-time regression is used to estimate the missing exact resolution time and extrapolate the expected GA speedup.

Generative Robust Optimisation

Classical uncertainty sets for robust optimisation impose fixed geometric shapes that cannot represent the complex dependencies present in real-world data. We propose Generative Robust Optimisation (GRO), a framework in which a deep generative model defines the uncertainty set as the image of a neural network decoder over a calibrated latent set, naturally accommodating nonlinear correlations, asymmetry, and multimodality. A five-point evaluation framework (reconstruction fidelity, distribution matching, latent regularity, robust relevance, and computational tractability) provides systematic, model-agnostic criteria for assessing any neural network-based uncertainty set. We instantiate this framework with a Wasserstein Adversarial Autoencoder employing Gaussian mixture model-guided training for latent regularity and constraint-consistency regularisation for robust relevance. Restricting the decoder to ReLU activations enables exact worst-case verification through mixed-integer programming embedding. Extensive experiments on a production planning problem across six uncertainty distributions and six generative architectures, together with a multi-period facility location study, validate the framework and demonstrate that systematic attention to all five criteria yields uncertainty sets that are simultaneously expressive, well-calibrated, and optimisation-tractable.

SkillChain-Gym: A Benchmark for Reskilling-Aware Production-Inventory Control under Disruptions

Production planning increasingly has to treat workforce capability as a decision variable: certifications lapse when skills are not maintained, new products require skills the current workforce does not hold, and reskilling competes for the same worker hours needed for production. Existing operations benchmarks usually treat labor as exogenous, while workforce-planning models with skills and learning are rarely released as reusable testbeds. We introduce SkillChain-Gym, a benchmark specification for reskilling-aware production-inventory control: a single-site environment with stylized worker skill-state dynamics, hard threshold certification, forgetting, and capacity-consuming training actions constrained by the same per-worker time budget as production. The benchmark includes seed-controlled disruption scenarios, three feasibility modes with projection diagnostics, deterministic replay, and metrics covering operations, resilience, capability growth, and training-access distribution. We evaluate production-only, reactive adaptive, water-filling adaptive, and static-insurance policies with budget variants over 60-shift horizons with paired statistical tests. The results are regime-dependent rather than a ranking. Training-capable policies dominate the production-only baseline, and maintenance training is necessary under forgetting even without disruptions. Among training-capable classes, adaptive training helps when bottlenecks are visible in the forecast, while a lean static cross-training plan, a deliberately favorable comparator whose structure encodes relevant skill contingencies, acts as strong insurance under surprise shocks and absenteeism. Capacity slack and the forgetting rate govern the boundary between these regimes. No policy class dominates across regimes, motivating forecast-driven controllers that decide when to buy skill insurance and when to react.