A Transportation Problem Considering Fixed Charge and Fuzzy Shipping Costs

Esmaiel Keshavarz; Ali Mahmoodirad; Sadegh Niroomand

doi:10.31181/dma11202313

Authors

Esmaiel Keshavarz Department of Mathematics, Sirjan Branch, Islamic Azad University, Sirjan, Iran https://orcid.org/0000-0001-5603-3678
Ali Mahmoodirad Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran https://orcid.org/0000-0002-0018-5389
Sadegh Niroomand Department of Industrial Engineering, Firouzabad Institute of Higher Education, Firouzabad, Fars, Iran https://orcid.org/0000-0001-8196-3906

DOI:

https://doi.org/10.31181/dma11202313

Keywords:

Fixed charge transportation, Fuzzy shipping cost, Max-Min criterion, Mixed-integer programming, Fractional programming

Abstract

This paper considers a fixed-charge transportation problem with fuzzy shipping costs. Whereas the shipping costs of routes are fuzzy intervals, with increasing linear membership functions, fixed costs, supplies, and demands are deterministic numbers. By defining a membership function associated with the objective values and utilizing Bellman-Zadeh’s max-min criterion instead of the main problem, a new crisp nonlinear mixed-integer programming problem is formulated. To solve this non-linear programming problem, at first, we find optimality conditions for a feasible solution, then, using these optimality conditions, reformulate the non-linear programming problem as a linear mixed-integer fractional programming problem, and finally transform the fractional programming problem as a mixed-integer linear programming problem, which can be solved using the existence methods. An illustrative example is solved to explain the presented details.

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References

Adlakha, V., Kowalski, K., & Lev, B. (2010). A branching method for the fixed charge transportation problem. Omega, 38, 393–397. https://doi.org/10.1016/j.omega.2009.10.005

Molla-Alizadeh-Zavardehi, S., Mahmoodirad, A., Rahimian, M. (2014). Step fixed charge transportation problems via genetic algorithm. Indian Journal of Science and Technology, 7 (7), 949-954. • https://doi.org/10.17485/IJST/2014/V7I7/33656

Sanei, M., Mahmoodirad, A., Niroomand, S., Jamalian, A., & Gelareh, S. (2015). Step fixed-charge solid transportation problem: a lagrangian relaxation heuristic approach. Computational and Applied Mathematics, 36,1–21. https://doi.org/10.1007/s40314-015-0293-5

Mahmoodirad, A., Allahviranloo, T. & Niroomand, S. (2019). A new effective solution method for fully intuitionistic fuzzy transportation problem. Soft Computing, 23, 4521–4530. https://doi.org/10.1007/s00500-018-3115-z

Broomandi, P., Crape B., Jahanbakhshi, A., Janatian, N., Nikfal, A., Tamjidi, M., Kim, J.R. Kim, Middleton, N., Karaca, F. (2022). Assessment of the association between dust storms and COVID-19 infection rate in Southwest Iran. Environ. Environmental Science and Pollution Research, 29 (24), 36392-36411. https://doi.org/10.1007/s11356-021-18195-7

Amouei Torkmahalleh, M., Hopke, P.K., Broomandi, P., Naseri, M., Abdrakhmanov, T., Ishanov, A., Kim, J., Shah, D., Kumar, P. (2020). Exposure to particulate matter and gaseous pollutants during cab commuting in Nur-Sultan city of Kazakhstan. Atmospheric Pollution Research 11. https://doi.org/10.1016/j.apr.2020.01.016

Mahmoodird, A., & Niroomand, S. (2020a). A belief degree-based uncertain scheme for a bi-objective two-stage green supply chain network design problem with direct shipment. Soft Computing, 24(24), 18499–18519. https://doi.org/10.1007/s00500-020-05085-2

Mahmoodirad, A., & Niroomand, S. (2020b). Uncertain location-allocation decisions for a bi-objective two-stage supply chain network design problem with environmental impacts. Expert Systems, 37, e12558. https://doi.org/10.1111/exsy.12558

Broomandi, P.; Tleuken, A.; Zhaxylykov, S.; Nikfal, A.; Kim, J.R.; Karaca, F. (2021). Assessment of potential benefits of traffic and urban mobility reductions during COVID-19 lockdowns: Dose-response calculations for material corrosions on built cultural heritage. Environmental Science and Pollution Research, 29, 6491–6510. https://doi.org/10.1007/s11356-021-16078-5

Kennington, J., & Unger, E. (1976). A New Branch-and-Bound Algorithm for the Fixed-Charge Transportation Problem. Management Science, 22, 10, 1116-1126. https://www.jstor.org/stable/2629910

Morawska, L., Zhu, T., Liu, N., Amouei Torkmahalleh, M., de Fatima Andrade, M., Barratt, B., Broomandi, P., Buonanno, G., Carlos Belalcazar Ceron, L., Chen, J., Cheng, Y., Evans, G., Gavidia, M., Guo, H., Hanigan, I., Hu, M., Jeong, C.H., Kelly, F., Gallardo, L., Kumar, P., Lyu, X., Mullins, B.J., Nordstrøm, C., Pereira, G., Querol, X., Yezid Rojas Roa, N., Russell, A., Thompson, H., Wang, H., Wang, L., Wang, T., Wierzbicka, A., Xue, T., Broomandi, P, Amouei Torkmahalleh, M, Akturk, M, et al. (2020). A new exposure route to trace elements in indoor particulate matter. Indoor Air, 30, 492– 499. https://doi.org/10.1111/ina.12641

Gray, P. (1971). Exact Solution of the Fixed-Charge Transportation Problem. Operations Research, 19, 6, 1529-1538.

Mahmoodirad, A., & Sanei, M. (2016). Solving a multi-stage multi-product solid supply chain network design problem by meta-heuristics. Scientia Iranica, 23 (3), 1429-1440. https://doi.org/10.24200/sci.2016.3908

No, R. K. G., Niroomand, S., Didehkhani, H., & Mahmoodirad, A. (2021). Modified interval EDAS approach for the multi-criteria ranking problem in banking sector of Iran. Journal of Ambient Intelligence and Humanized Computing, 12, 8129-8148. https://doi.org/10.1007/s12652-020-02550-6

Bellman, R. E., & Zadeh, L. A. (1970). Decision-making in a fuzzy environment. Management Science, 17B, 141–164. https://doi.org/10.1287/mnsc.17.4.B141

Keshavarz, E ., Khorram, E. (2011). A fuzzy bi-criteria transportation problem. Computers & Industrial Engineering, 61 (4), 947-957. https://doi.org/10.1016/j.cie.2011.06.007

Keshavarz , E,. (2016). On “A fuzzy bi-criteria transportation problem”: A revised algorithm. Computers & Industrial Engineering, 98, 144-148. https://doi.org/10.1016/j.cie.2016.05.034

Charnes, A., Cooper, W. W. (1962). Programming with linear fractional functional. Naval Research Logistics Quarterly, 9, 181-185. https://doi.org/10.1002/nav.3800090303

Ye, C. (2021). The state of science on severe air pollution episodes: Quantitative and qualitative analysis. Environment International, 156, 106732. https://doi.org/10.1016/j.envint.2021.106732

Tripathy, S. P. (2023). Bi-objective covering salesman problem with uncertainty. Journal of Decision Analytics and Intelligent Computing, 3(1), 122–138. https://doi.org/10.31181/jdaic10015082023t