Heterogeneous Penalties in Unbalanced Transportation: Exact Linear Programming, Dummy-Node Equivalence, and a Penalty-Aware Ant Colony Heuristic

Authors

  • Ali Mahmoud Assabri Department of Engineering Management, College of Technical Sciences, Bani Walid, Libya Author
  • Mohamed Al-Issawi Kajman Department of Computing and Information Technology, College of Technical Sciences, Bani Walid, Libya Author
  • Abdulbasat Abdulghafar Alfeed Department of Engineering Management, College of Technical Sciences, Bani Walid, Libya Author
  • Naji Abdalaziz Ali Department of Mechanical Engineering, College of Technical Sciences, Bani Walid, Libya Author

Keywords:

unbalanced transportation, heterogeneous penalties, linear programming, dummy-node equivalence, Ant Colony Optimization, minimum-cost flow, reproducible computational research

Abstract

Unbalanced transportation models are often balanced with a dummy source or destination. That transformation is exact only when dummy-arc costs preserve the location-specific consequences of surplus or shortage; replacing a penalty vector with its arithmetic mean defines a different optimization problem. This paper formulates both imbalance directions as continuous linear programs, establishes heterogeneous dummy-node equivalence, and evaluates a penalty-aware Ant Colony Optimization (ACO) construction against exact linear programming and five alternative methods. Fully specified examples are complemented by a controlled synthetic benchmark spanning several network dimensions. The results show that uniform penalties can materially increase the true heterogeneous objective, while penalty-aware ACO improves on a cost-only construction. Variable Neighborhood Search achieves the strongest overall rank in the expanded comparison, whereas penalty-aware ACO remains competitive with other stochastic methods under a common evaluation budget. The conclusions are restricted to the continuous model, the stated synthetic generator, the supplied cases, and the declared computational protocols. Synthetic instances are used as algorithmic stress tests rather than as observations of industrial transportation systems.

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Published

2026-08-10

Issue

Volume 2, No. 2, 2026

Section

Articles

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This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Ali Mahmoud Assabri, Mohamed Al-Issawi Kajman, Abdulbasat Abdulghafar Alfeed, & Naji Abdalaziz Ali. (2026). Heterogeneous Penalties in Unbalanced Transportation: Exact Linear Programming, Dummy-Node Equivalence, and a Penalty-Aware Ant Colony Heuristic. The Open European Journal of Applied Sciences (OEJAS), 2(2), 34-49. https://easdjournals.com/index.php/oejas/article/view/90