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Interval transportation problem: feasibility, optimality and the worst optimal value

Author

Listed:
  • Elif Garajová

    (Charles University
    Prague University of Economics and Business)

  • Miroslav Rada

    (Prague University of Economics and Business
    Prague University of Economics and Business)

Abstract

We consider the model of a transportation problem with the objective of finding a minimum-cost transportation plan for shipping a given commodity from a set of supply centers to the customers. Since the exact values of supply and demand and the exact transportation costs are not always available for real-world problems, we adopt the approach of interval programming to represent such uncertainty, resulting in the model of an interval transportation problem. The interval model assumes that lower and upper bounds on the data are given and the values can be independently perturbed within these bounds. In this paper, we provide an overview of conditions for checking basic properties of the interval transportation problems commonly studied in interval programming, such as weak and strong feasibility or optimality. We derive a condition for testing weak optimality of a solution in polynomial time by finding a suitable scenario of the problem. Further, we formulate a similar condition for testing strong optimality of a solution for transportation problems with interval supply and demand (and exact costs). Moreover, we also survey the results on computing the best and the worst optimal value. We build on an exact method for solving the NP-hard problem of computing the worst (finite) optimal value of the interval transportation problem based on a decomposition of the optimal solution set by complementary slackness. Finally, we conduct computational experiments to show that the method can be competitive with the state-of-the-art heuristic algorithms.

Suggested Citation

  • Elif Garajová & Miroslav Rada, 2023. "Interval transportation problem: feasibility, optimality and the worst optimal value," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 31(3), pages 769-790, September.
    • Handle: RePEc:spr:cejnor:v:31:y:2023:i:3:d:10.1007_s10100-023-00841-9
    DOI: 10.1007/s10100-023-00841-9
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    References listed on IDEAS

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    1. Liu, Shiang-Tai, 2003. "The total cost bounds of the transportation problem with varying demand and supply," Omega, Elsevier, vol. 31(4), pages 247-251, August.
    2. Juman, Z.A.M.S. & Hoque, M.A., 2014. "A heuristic solution technique to attain the minimal total cost bounds of transporting a homogeneous product with varying demands and supplies," European Journal of Operational Research, Elsevier, vol. 239(1), pages 146-156.
    3. D’Ambrosio, C. & Gentili, M. & Cerulli, R., 2020. "The optimal value range problem for the Interval (immune) Transportation Problem," Omega, Elsevier, vol. 95(C).
    4. Wlodzimierz Szwarc, 1971. "The transportation paradox," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 18(2), pages 185-202, June.
    5. Carrabs, Francesco & Cerulli, Raffaele & D’Ambrosio, Ciriaco & Della Croce, Federico & Gentili, Monica, 2021. "An improved heuristic approach for the interval immune transportation problem," Omega, Elsevier, vol. 104(C).
    6. Xie, Fanrong & Butt, Muhammad Munir & Li, Zuoan & Zhu, Linzhi, 2017. "An upper bound on the minimal total cost of the transportation problem with varying demands and supplies," Omega, Elsevier, vol. 68(C), pages 105-118.
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