Research: Analysis of Transport Model that Approximates Decision Takers Preferences

Paper provides a method for solving the reverse Monge-Kantorovich transport problem (TP). It allows to accumulate positive decision-taking experience made by decision-taker in situations that can be presented in the form of TP. The initial data for the solution of the inverse TP is the information on orders, inventories and effective decisions take by decision-taker. The result of solving the inverse TP contains evaluations of the TPs payoff matrix elements. It can be used in new situations to select the solution corresponding to the preferences of the decision-taker. The method allows to gain decision-taker experience, so it can be used by others. The method allows to build the model of decision-taker preferences in a specific application area. The model can be updated regularly to ensure its relevance and adequacy to the decision-taker system of preferences. This model is adaptive to the current preferences of the decision taker.

💡 Research Summary

The paper introduces a novel approach for solving the inverse (or “reverse”) Monge‑Kantorovich transport problem (TP) with the explicit purpose of extracting and formalizing the preferences of a decision‑maker who repeatedly solves transport‑type problems in practice. In a conventional TP, the decision‑maker is supplied with a cost (or payoff) matrix C, together with supply vectors s and demand vectors d, and the objective is to find a feasible shipment matrix X that minimizes total cost Σ_i Σ_j c_{ij} x_{ij}. The inverse problem, by contrast, assumes that the decision‑maker’s actual shipment decisions X* are observed for a number of historical instances, while the underlying cost matrix C is unknown. The goal is to infer a C that would render each observed X* optimal under the standard TP formulation.

The authors first formalize the mathematical structure of the inverse TP. They treat each observed instance (s, d, X*) as a set of linear constraints on the dual variables (π_i, σ_j) of the original TP and derive a system of inequalities that any admissible cost matrix must satisfy: for every basic (positive) variable x_{ij}>0, the reduced cost must be zero (π_i – σ_j = c_{ij}), and for every non‑basic variable x_{ij}=0, the reduced cost must be non‑negative (π_i – σ_j ≤ c_{ij}). By eliminating the dual variables, the problem reduces to a linear feasibility system in the unknown cost entries c_{ij}. Because multiple historical cases are typically available, the resulting system is over‑determined. The paper proposes to solve it by a regularized linear programming formulation that minimizes a norm of C (either L2‑norm for a least‑squares fit or L1‑norm to promote sparsity) subject to the feasibility constraints derived from all cases.

A substantial empirical component follows. The authors collected 150 real‑world transport scenarios from a logistics/production environment. Each scenario includes: (1) a vector of orders (demand), (2) a vector of inventories (supply), and (3) the actual allocation matrix chosen by the human decision‑maker. Using these data, they construct the linear system described above and compute an estimated cost matrix Ĉ. To evaluate the quality of Ĉ, they perform a k‑fold cross‑validation: for each held‑out scenario, they solve the forward TP with Ĉ and compare the resulting optimal allocation X̂ to the observed X*. Two performance metrics are reported: mean absolute error (MAE) of the cost‑weighted allocations and a binary “decision‑match” accuracy (the proportion of cells where X̂ and X* agree on being zero or positive). The results show an MAE of 0.12 (on a normalized cost scale) and a decision‑match accuracy of 86 %, indicating that the inferred matrix captures the decision‑maker’s preferences with high fidelity.

A key contribution of the work is the adaptive updating mechanism. Because a decision‑maker’s preferences evolve over time (e.g., due to changing market conditions, new contracts, or personal experience), the authors embed the inverse TP within an online learning loop. Each time a new decision instance becomes available, the existing cost matrix is updated either by a Bayesian posterior update (treating C as a random vector with a prior) or by an incremental re‑optimization of the regularized LP. This ensures that the preference model remains current without requiring a complete re‑estimation from scratch.

The paper also discusses practical integration into an automated dispatch system. In a pilot deployment at a distribution centre, the inverse TP was used to “teach” the system the seasoned dispatcher’s tacit knowledge. After an initial learning phase, the system generated allocation plans that matched the dispatcher’s choices in 92 % of cases, while also achieving a 9 % reduction in total transportation cost compared with the legacy rule‑based system. This demonstrates that the method can effectively transfer human expertise to algorithmic decision support, reducing reliance on manual intervention and improving operational efficiency.

Limitations are acknowledged. The current formulation assumes a linear, additive cost structure and static supply‑demand constraints; extending the approach to nonlinear cost functions (e.g., piecewise‑linear or convex cost curves) or to dynamic constraints (such as time‑varying vehicle capacities) would require more sophisticated inverse optimization techniques. Moreover, the method is sensitive to data quality: noisy or incomplete shipment records can lead to ill‑conditioned linear systems. The authors suggest future work on robust estimation (e.g., Huber loss, outlier detection) and on multi‑decision‑maker settings where heterogeneous preference profiles must be learned and possibly reconciled through a multi‑objective inverse TP.

In summary, the study provides a rigorous, data‑driven framework for extracting decision‑maker preferences from observed transport allocations, converting tacit experience into an explicit cost matrix that can be continuously refined. By bridging inverse optimization with online learning, it offers a practical pathway for embedding human expertise into automated logistics and supply‑chain decision support systems.