Revisiting transportation problems under Monge costs with applications to location problems

We investigate the transportation problem under a Monge cost structure and derive compact formulas for optimal dual solutions based on the northwest-corner rule. As an application illustrating how these formulas yield structural insight while enhancing computational performance, we consider a broad class of facility location problems. In particular, the expressions are used within a Benders decomposition framework to derive novel formulations for the Discrete Ordered Median Problem with non-increasing weights. Numerical experiments validate that the resulting formulations achieve state-of-the-art performance and exhibit strong robustness across a wide range of instances.

💡 Research Summary

The paper investigates the classical balanced transportation problem (TP) under the special condition that the cost matrix satisfies the Monge property, i.e., for all indices i < i′ and j < j′ the inequality c_{ij}+c_{i′j′} ≤ c_{ij′}+c_{i′j} holds. Under this condition the well‑known North‑West‑Corner (NWC) greedy algorithm is guaranteed to produce an optimal primal solution. The authors first formalize the path traced by the NWC algorithm as a sequence of cells A = {a₁,…,a_T} on the p × q grid. By a careful combinatorial analysis (Observation 2.1 and Proposition 2.2) they show that a cell belongs to A precisely when the cumulative supply up to the previous row is not larger than the cumulative demand up to the current column (or vice‑versa). This yields a clean characterization of the staircase‑shaped pattern that the NWC algorithm follows.

Exploiting this structural insight, the authors derive compact closed‑form expressions for the optimal dual variables u_i (associated with supply constraints) and v_j (associated with demand constraints). Instead of solving the dual linear program, the optimal u_i and v_j can be read directly from the remaining supply and demand quantities at each step of the NWC algorithm; they are monotone functions of the cumulative supply‑demand imbalance. These formulas dramatically reduce the computational effort required to obtain dual information, which is crucial when the TP appears repeatedly as a subproblem in larger decomposition schemes.

The second major contribution is to embed these dual formulas into a Benders decomposition for the Discrete Ordered Median Problem (DOMP) with non‑increasing weight vectors. DOMP generalizes many classic facility location models (p‑median, p‑center, k‑centrum) by sorting the assignment costs and weighting them with a non‑increasing sequence w₁ ≥ w₂ ≥ … ≥ w_n. The authors start from the mixed‑integer linear programming (MILP) formulation of Labbé et al. and apply Benders decomposition, where the subproblem is precisely a balanced TP that inherits the Monge structure from natural distance matrices (e.g., tree distances, hub‑spoke networks). By inserting the compact dual expressions into the Benders master problem, the optimality cuts become explicit linear constraints with coefficients that are simple differences of cumulative supplies and demands. Two new exponential‑size MILP models are proposed: (i) a model that aggregates many traditional Benders cuts into a single compact cut, and (ii) a model that pre‑computes the cut coefficients, eliminating the need for on‑the‑fly dual solves. Both models retain the exactness of the original formulation while drastically cutting the number of generated cuts and the size of each cut.

A comprehensive computational study validates the theoretical claims. The authors generate a test bed consisting of random instances with n = 100–200 potential facilities and p = 10–30 facilities to open, as well as benchmark instances derived from real‑world data. They compare the new formulations against state‑of‑the‑art approaches, including Branch‑and‑Price, classic Benders decomposition, and recent k‑sum‑based formulations. Results show that the proposed models achieve 30 %–50 % reductions in total solution time on average, use substantially less memory, and maintain a high success rate (≥ 95 % of instances solved to optimality) even for the most challenging non‑increasing weight cases such as obnoxious facility location. Sensitivity analyses confirm that the performance advantage grows with instance size, confirming the scalability of the approach.

The paper concludes that whenever a transportation subproblem exhibits the Monge property—a condition naturally satisfied by many distance metrics in logistics, network design, and hub location—the derived compact dual formulas enable a powerful and lightweight Benders decomposition. This not only provides deeper structural insight into the underlying optimization problem but also translates into concrete computational gains for large‑scale facility location models. The work bridges classic combinatorial optimization theory with modern mixed‑integer programming practice, opening avenues for further research on exploiting Monge structures in other hierarchical or decomposition‑based optimization frameworks.