Current literature addressing the interpretability of solutions to optimization models involves formulating the solution as a decision tree. Bertsimas et al. [2] trained decision trees on data generated from numerous solved instances of the optimization problem; however, this approach is computationally expensive and has been validated only on small-scale problems. In contrast, Goerigk et al. [7] integrated decision trees directly into the optimization model, deriving the tree structure as part of the solution process. However, this approach is also computationally expensive, and in their work, they propose a greedy heuristic approach to counter the computational expense. Both methods attempt to construct a decision tree that maps a particular scenario to its nearest optimal solution. However, the number of optimal solutions that can be considered is restricted by the depth of the tree, which poses a potential limitation to both approaches. Furthermore, Li et al. [8] introduced a chatbot framework, Optiguide, which translates user queries into counterfactual questions for a supply chain model and provides natural language explanations using large language models. While innovative, this method requires highly precise queries to avoid confusion, can occasionally produce incorrect code, and struggles with new or unseen queries.
Complementing these decision tree-based methods, recent contributions include argumentation-based approaches, fuzzy inference systems, and scenario-driven techniques. Argumentation-based methods utilize abstract argumentation to model scheduling problems, producing structured natural language justifications that elucidate exactly why specific schedules are optimal or feasible by linking optimality conditions to argumentation constructs [3]. Fuzzy inference methods generate qualitative mappings between model parameters and optimal solutions using fuzzy clustering and rule induction, offering intuitive explanations regarding how sensitive optimal decisions are to operational factors [4]. Furthermore, scenario aggregation and recourse simplification methodologies have been developed to enhance the interpretability of stochastic optimization solutions. These approaches cluster scenarios based on similarities in recourse actions and identify representative scenarios, effectively reducing model complexity and clarifying the essential drivers of decision-making under uncertainty [9]. Nonetheless, these techniques typically remain specialized to specific application domains, face computational challenges at larger scales, and often yield explanations that are limited to localized interpretations. To the best of our knowledge, there remains a gap in the literature for approaches to develop interpretable reordering policies for inventory management in multi-echelon supply chain networks that not only outperform traditional (s, S) and (r, Q) policies but are also computationally efficient to derive.
In our work, we propose the use of Linear Decision Rules (LDR) to derive interpretable reordering policies. LDR approximates decision variables (e.g., reordering quantities) as affine functions of uncertain parameters, such as demand and initial inventory. Our preliminary analysis demonstrates that LDR-based reordering policies are highly interpretable, as the affine functional form explicitly shows how decisions depend on uncertain parameters. Performance evaluations reveal that, under easy-to-satisfy demand scenarios with short lead times, LDR-based policies yield optimality gaps between 5% and 15% when the variance in uncertain parameters is between 10% and 30%. In contrast, the (r, Q) policy exhibits gaps between 25% and 30%, while the (s, S) policy shows gaps between 45% and 60% under similar conditions. Furthermore, under more challenging demand scenarios and extended lead times, LDR-based policies remain robust, maintaining gaps within 10% to 15%, whereas heuristic policies deteriorate significantly, with gaps exceeding 170% for a 10% to 30% variance in uncertain parameters. However, as the variance increases to 50%, the optimality gap for LDR-based reordering policies rises to approximately 40%, indicating that increased uncertainty degrades their performance.
The LDR approach encounters scalability issues as the dimensionality of the underlying optimization problem expands with increasing network size or planning horizon. In our analysis, we observed significant sparsity in the decision rules derived from LDR, with each decision variable typically depending only on a small subset of uncertain parameters. Exploiting this sparsity, we proposed restricted linear decision rules that constrain each decision variable to depend only on a preselected subset of influential uncertain parameters. This substantially reduces the size of the model and the computational effort, thereby effectively addressing scalability concerns. Nevertheless, this strategy necessitates the careful selection of parameter subsets based on domain-specific expertise. Future research will focus on developing methodologies to automatically identify these influential parameters, thereby enhancing the scalability and applicability to broader supply chain networks.
References:
[1] Manuel P Baganha and Morris A Cohen. The stabilizing effect of inventory in supply chains. Operations Research, 46(3-supplement-3):S72–S83, 1998.
[2] Dimitris Bertsimas and Bartolomeo Stellato. The voice of optimization. Machine Learning, 110(2):249–277, 2021.
[3] Kristijonas Cyras, Dimitrios Letsios, Ruth Misener, and Francesca Toni. Argumentation for explainable scheduling. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 2752–2759, 2019.
[4] Tewodros L Deneke, Ricardo H Dunia, and Michael Baldea. Explainable optimal solutions using fuzzy inference. In 2024 American Control Conference (ACC), pages 51–55. IEEE, 2024.
[5] Richard Ehrhardt. (s, s) policies for a dynamic inventory model with stochastic lead times. Operations Research, 32(1):121–132, 1984.
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[7] Marc Goerigk and Michael Hartisch. A framework for inherently interpretable optimization models. European Journal of Operational Research, 310(3):1312–1324, 2023.
[8] Beibin Li, Konstantina Mellou, Bo Zhang, Jeevan Pathuri, and Ishai Menache. Large language models for supply chain optimization. arXiv preprint arXiv:2307.03875, 2023.
[9] Tushar Rathi, Rishabh Gupta, Jose M Pinto, and Qi Zhang. Enhancing explainability of stochastic programming solutions via scenario and recourse reduction. Optimization and Engineering, 25(2):795–820, 2024.