QUBO formulation of a two-echelon vehicle routing problem with optional activation of intermediate nodes: penalty landscape diagnostics and a hybrid algorithm

Rogulin R.S.
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The haulage of timber from remote logging areas of Siberia is characterized by high transportation costs. Long stretches of unpaved logging roads, seasonal constraints (spring thaw and winter roads), and considerable distances from cutting areas to main transport hubs (50–200 km) make systematic route planning essential.

This paper proposes a mathematical model of the two-echelon vehicle routing problem (2E-VRP) adapted to timber haulage in the macro-region of the Urals and Siberia. The model incorporates variable transportation costs (truck mileage), fixed costs of using each vehicle ($c_{fix}$ = 30 000 RUB per trip), and vehicle-capacity constraints. The problem is transformed into a QUBO (quadratic unconstrained binary optimization) formulation; capacity constraints are integrated through the penalty coefficient $\lambda_{cap}$ = 500. This representation makes the model compatible with modern specialized solvers, including quantum annealers and digital annealers.

The model is tested on the real geography of the Ural and Siberian Federal Districts: 207 nodes in the logistics network, 65 customer enterprises, and 8 main transport hubs. The proposed three-stage “Feasible-First” algorithm constructs a feasible routing plan in 1.7 s and reduces total transportation costs by 69% relative to a baseline genetic algorithm. Subsequent simulated annealing further improves the QUBO objective value by 55%.

Keywords: timber haulage, forest logistics, logging roads, two-echelon vehicle routing, optimization of timber truck routes, QUBO, Ural Federal District, Siberian Federal District, mathematical modeling of timber harvesting
Citation in English: Rogulin R.S. QUBO formulation of a two-echelon vehicle routing problem with optional activation of intermediate nodes: penalty landscape diagnostics and a hybrid algorithm // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 765-791
Citation in English: Rogulin R.S. QUBO formulation of a two-echelon vehicle routing problem with optional activation of intermediate nodes: penalty landscape diagnostics and a hybrid algorithm // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 765-791
DOI: 10.20537/2076-7633-2026-18-4-765-791
URL: http://crm-en.ics.org.ru/journal/article/3744/
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Copyright © 2026 Rogulin R.S.

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