(15c) Inverse Mixed-Integer Optimization for Learning Interpretable Decision Rules

Humans rely on decision rules as a means of coping with the complexity and ambiguity of real-world decision-making. Decision rules are simplified models or mental shortcuts that enable individuals to make decisions based on limited information and cognitive resources. While decision rules can be useful, they are also prone to biases and errors, as they are based on limited information and assumptions. Therefore, learning these rules can be beneficial as it can help individuals identify and correct biases in their decision-making. Moreover, learning an expert's decision rules can be a valuable tool for training novices in a particular domain. By learning how experts simplify complex interplay between different decisions, novices can gain insights into the strategies that have been developed and refined over years of experience. This motivates us to consider the development of a framework to learn and interpret decision rules from prior decision data.

To accomplish this, we apply the concept of data-driven inverse optimization (IO) (Ahuja & Orlin, 2001; Aswani, Shen, & Siddiq, 2018; Chan, Mahmood, & Zhu, 2021). Mathematical optimization is a natural model for decision-making, assuming that the decision maker aims to make the best possible decision given a set of available options. This makes IO an ideal approach for learning unknown decision-making mechanisms as it finds an optimization model that can generate decisions close to the observed data. In our previous work (Gupta & Zhang, 2022, 2023), we have shown that IO is very data-efficient in learning interpretable models, making it an even more attractive learning paradigm.

Commonly used decision rules include ranking options, bounding values, and assigning causality, many of which involve logical relationships between variables. This naturally motivates the use of disjunctions and discrete variables (Raman & Grossmann, 1994) to model these logical relationships. Therefore, in this work, we propose an approach to learning human decision rules using inverse optimization with a mixed-integer linear program (MILP) as the forward problem.

Our approach formulates an MILP with constraints defining the hypothesis space of possible decision rules. Leveraging domain knowledge and the modeling flexibility of mixed-integer optimization, we include two types of constraints: 1) inherent physical limitations of the system faced by the decision maker, and 2) a superset of choices that the decision maker needs to consider while arriving at their decisions. We then use data-driven IO to find the weights that the decision maker assigns to each of the decisions in the objective function. This approach makes our model inherently interpretable as the values of these parameters being estimated have a direct meaning that reflects the decision policy. We apply the proposed approach to a supply chain planning problem to demonstrate the accuracy and interpretability of the resulting decision-making models.

References:

Ahuja, R. K., & Orlin, J. B. (2001). Inverse optimization. Operations Research, 49(5), 771–783.

Aswani, A., Shen, Z. J. M., & Siddiq, A. (2018). Inverse optimization with noisy data. Operations Research, 66(3), 870–892.

Chan, T. C. Y., Mahmood, R., & Zhu, I. Y. (2021). Inverse Optimization: Theory and Applications. arXiv preprint arXiv:2109.03920 (2021).

Gupta, R., & Zhang, Q. (2022). Decomposition and Adaptive Sampling for Data-Driven Inverse Linear Optimization. INFORMS Journal on Computing, 34(5), 2720–2735.

Gupta, R., & Zhang, Q. (2023). Efficient learning of decision-making models: A penalty block coordinate descent algorithm for data-driven inverse optimization. Computers and Chemical Engineering, 170(December 2022).

Raman, R., & Grossmann, I. E. (1994). Modelling and computational techniques for logic based integer programming. Computers and Chemical Engineering, 18(7), 563–578.