Distributionally Robust Optimization for Multi-Period Multi-Item Lot-Sizing Problems under Yield Uncertainty

Academic Background

In modern manufacturing, yield uncertainty is a prevalent issue, particularly in industries such as agriculture, food processing, and textiles. The production processes in these industries rely on uncontrollable external factors (e.g., climate, raw material quality), making production output difficult to predict. Yield uncertainty not only increases production costs but may also lead to insufficient or excessive inventory, thereby affecting a company’s profitability and market competitiveness.

To address this issue, the academic community has proposed various optimization methods, including stochastic programming (SP) and robust optimization (RO). However, these methods have their limitations: stochastic programming relies on accurate estimates of the probability distribution of uncertain parameters, which is often difficult to achieve in real-world production; robust optimization tends to be overly conservative, potentially leading to high-cost production plans.

Distributionally robust optimization (DRO) combines the strengths of stochastic programming and robust optimization by optimizing the expected cost against a set of possible probability distributions, offering a more flexible and efficient solution. This research is based on this context and aims to address the production planning optimization problem for multi-period multi-item lot-sizing problems (LSP) under yield uncertainty by constructing a new data-driven DRO framework.

Source of the Paper

This paper is a collaborative effort by scholars from various academic institutions, including Paula Metzker, Simon Thevenin, Yossiri Adulyasak, and IEEE member Alexandre Dolgui. The paper was published in IEEE Transactions on Automation Science and Engineering in 2025, titled “Distributionally Robust Optimization for the Multi-Period Multi-Item Lot-Sizing Problems under Yield Uncertainty.”

Research Process and Detailed Content

Research Process

1. Problem Definition and Model Construction
   The core of this study is the multi-period multi-item lot-sizing problem (LSP), which involves determining optimal production setups and lot sizes for multiple items across several production periods to minimize total costs. The innovation lies in introducing yield uncertainty and constructing a mixed-integer DRO model. The model aims to minimize total costs, including setup costs, production costs, and inventory costs, while accounting for the impact of yield uncertainty on production planning.

2. Data-Driven Framework and Scenario Partitioning
   To handle yield uncertainty, the research team adopted a data-driven approach, partitioning historical data into scenarios to represent yield uncertainty as probability distributions under different scenarios. Specifically, the team used the K-means clustering algorithm to divide historical production data into multiple scenarios, each representing a possible production pattern. This scenario partitioning method captures different patterns of yield uncertainty, providing a foundation for the subsequent optimization model.

3. Construction and Solution of the DRO Model
   Based on scenario partitioning, the research team constructed two DRO models: the mean absolute DRO (MDRO) and the Wasserstein DRO (WDro). These models introduce a scenario-wise ambiguity set to represent uncertainty as probability distributions under different scenarios and solve them in the form of mixed-integer linear programming (MILP). The CPLEX solver was used to efficiently solve the models.

4. Experimental Design and Performance Evaluation
   To validate the effectiveness of the proposed models, the research team conducted large-scale Monte Carlo simulation experiments, generating 5,000 random scenarios to evaluate the performance of different models in production planning. The experiments compared DRO models with traditional stochastic programming and robust optimization models in terms of production costs, inventory management, and computational time.

Key Results

1. Model Performance Comparison
   The experimental results show that DRO models (particularly WDro and MDro) outperform traditional stochastic programming and robust optimization models in cost control. Specifically, the WDro model performs excellently in terms of average cost, 95th percentile cost, and 99th percentile cost. For example, in the in-sample simulation, the WDro model’s average cost is 1,432,243, which is 1.7% lower than the robust optimization model and 3.2% lower than the stochastic programming model.

2. Ability to Handle Uncertainty
   In the out-of-sample simulation (i.e., where the yield distribution in the simulation environment differs from that used in the optimization), the WDro model still demonstrates strong robustness. Although the average cost of the WDro model is slightly higher than that of the stochastic programming model (1,750,526 vs. 1,743,162), its performance in the 95th percentile and 99th percentile costs is significantly better, indicating its superior ability to handle extreme scenarios.

3. Inventory and Stockout Management
   The study also found that DRO models perform more balanced in inventory management and stockout control. Compared to the robust optimization model, the WDro model maintains lower inventory levels while reducing the risk of stockouts. This balanced strategy makes the WDro model more advantageous in complex production environments.

Conclusion and Significance

The DRO models proposed in this paper provide a new solution for production planning optimization in multi-period multi-item lot-sizing problems under yield uncertainty. By combining data-driven scenario partitioning and distributionally robust optimization techniques, the models offer more robust and efficient production plans in uncertain environments. This research not only fills a gap in the study of multi-item lot-sizing problems under yield uncertainty but also provides theoretical support and practical tools for decision-making in real-world production.

Research Highlights

1. Innovative Model: This paper is the first to apply DRO techniques to multi-item lot-sizing problems, introducing scenario-wise ambiguity sets that effectively capture the diversity of yield uncertainty.
2. Data-Driven Approach: The use of the K-means clustering algorithm to partition historical production data into scenarios provides reliable data support for the optimization model.
3. Practical Application Value: The proposed models demonstrate strong robustness and cost control capabilities in real-world production environments, offering practical decision-making tools for manufacturing enterprises.

Other Valuable Information

This paper also discusses the performance of the models under different parameter settings, particularly the impact of the Wasserstein radius and the number of scenarios on the model results. The research team found that using an ambiguity set with 3 scenarios achieves the best balance between computational time and model performance, while setting the Wasserstein radius to its maximum value (θ=1) provides stronger robustness without excessively increasing costs.

This research offers new ideas and methods for production planning optimization in manufacturing, demonstrating significant theoretical value and practical application.