Kybernetika 62 no. 4, 584-617, 2026

Solving neutrosophic bi-objective transportation problem using neutrosophic TOPSIS approach

S. Sandhiya and Anuradha DhanapalDOI: 10.14736/kyb-2026-4-0584

Abstract:

The Transportation Problem is an optimization problem which aims to optimize the objective function of the products from various sources to destinations. In real-world situations, decision makers face a variety of conflicting objectives such as transportation costs, time, deterioration rates and so on which yields them with comprehensive profits. In these situations, the transportation problem becomes Multi-objective Transportation Problem. In real life, the parameters are unpredictable due to factors such as incomplete input data, market fluctuations, environmental influences and so on. To address these uncertainties, the parameters are expressed as single-valued trapezoidal neutrosophic number. This article explores a bi-objective transportation problem in a neutrosophic environment, aiming to determine the Pareto optimal solution. First, the neutrosophic problem is transformed into its deterministic problem using the Weighted Possibility Mean. In road safety, delineator design and placement often involve conflicting objectives like visibility, cost, installation time and safety. Balancing these is challenging due to complex road conditions, traffic, geometry and weather. In this point of view, we have proposed the approach namely, Neutrosophic TOPSIS approach for single valued trapezoidal neutrosophic numbers to transform the bi-objective problem into its single objective problem. Then the problem is solved by LINGO software to determine the Pareto optimal solution. Our approach provides a systematized method to assess alternatives and identify the optimal delineator design by balancing conflicting objectives, aiding decision makers in selecting the most suitable road configuration. A Sensitivity Analysis is performed to determine the sensitivity ranges for all the parameters of the problem. To illustrate the applicability, effectiveness and usefulness of the approach, a numerical example is provided and the comparison was made with the existing approaches. It is observed that the obtained Pareto optimal solution extracted from the proposed approach provides better result than the existing approaches. Finally, the conclusions are drawn along with the future work.

Keywords:

bi-objective transportation problem, single-valued trapezoidal neutrosophic numbers, weighted possibility mean, neutrosophic TOPSIS approach, sensitivity analysis

Classification:

90B06, 90C29, 90C70

paper.pdf

References:

  1. M. A. Abo-Sinna, A. H. Amer and A. S. Ibrahim: Extensions of TOPSIS for large scale multi-objective non-linear programming problems with block angular structure. Appl. Math. Model.32 (2008), 3, 292-302.   DOI:10.1111/j.1467-7717.2008.01040.x
  2. F. Ahmad and A. Y. Adhami: Neutrosophic programming approach to multi-objective nonlinear transportation problem with fuzzy parameters. Int. J. Manag. Sci. Eng. Manag. 14 (2019), 3, 218-229.   CrossRef
  3. Y. P. Aneja and K. P. Nair: Bicriteria transportation problem. Manag. Sci. 25 (1979), 1, 73-78.   DOI:10.1287/mnsc.25.1.73
  4. K. T. Atanassov: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20 (1986), 1, 87-96.   DOI:10.1016/S0165-0114(86)80034-3
  5. M. Bagheri, A. Ebrahimnejad, S. Razavyan, F. Hosseinzadeh Lotfi and N. Malekmohammadi: Solving the fully fuzzy multi-objective transportation problem based on the common set of weights in DEA. J. Intell. Fuzzy Syst. 39 (2020), 3, 3099-3124.   DOI:10.3233/JIFS-191560
  6. G. Bai and L. Yao: A simple algorithm for the multi-objective transportation model. In: 2011 International Conference on Business Management and Electronic Information, IEEE 2011, pp. 479-482.   CrossRef
  7. R. E. Bellman and L. A. Zadeh: Decision-making in a fuzzy environment. Management science. Manag. Sci. 17 (1970), 4, 8141-8164.   CrossRef
  8. R. K. Bera and S. K. Mondal: A multi-objective transportation problem under quantity dependent credit period and cost structure policies in triangular intuitionistic fuzzy environment. Eng. Appl. Artif. Intell. 123 (2023), 106396.   DOI:10.1016/j.engappai.2023.106396
  9. A. K. Bind, D. Rani, A. Ebrahimnejad and J. L. Verdegay: New strategy for solving multi-objective green four dimensional transportation problems under normal type-2 uncertain environment. Eng. Appl. Artif. Intell. 137 (2024), 109084.   CrossRef
  10. S. G. Bodkhe: Multi-objective transportation problem using fuzzy programming techniques based on exponential membership functions. Int. J. Stat. Appl. Math. 8 (2024), 5, 20-24.   CrossRef
  11. B. C. Cuong and V. Kreinovich: Picture fuzzy sets-a new concept for computational intelligence problems. In: 2013 third world congress on information and communication technologies (WICT 2013), IEEE 2013, pp. 1-6.   CrossRef
  12. I. Deli and Y. Subas: Single valued neutrosophic numbers and their applications multicriteria decision making problem. Neutrosophic Sets Syst. 2 (2014), 1, 1-13.   CrossRef
  13. S. Dhanasekar, S. Hariharan and P. Sekar: Fuzzy Hungarian MODI algorithm to solve fully fuzzy transportation problems. Int. J. Fuzzy Syst. 19 (2017), 1479-1491.   DOI:10.1007/s40815-016-0251-4
  14. S. Ghosh, K. H. Küfer, S. K. Roy and G. W. Weber: Type-2 zigzag uncertain multi-objective fixed-charge solid transportation problem: time window vs. preservation technology. Cent. Eur. J. Oper. Res.31 (2023), 1, 337-362.   DOI:10.1007/s10100-022-00811-7
  15. S. Ghosh, S. K. Roy, A. Ebrahimnejad and J. L. Verdegay: Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem. Complex Intell. Syst. 7 (2021), 1009-1023.   DOI:10.1007/s40747-020-00251-3
  16. A. Gupta, A. Kumar and A. Kaur: Mehar’s method to find exact fuzzy optimal solution of unbalanced fully fuzzy multi-objective transportation problems. Optim. Lett. 6 (2012), 1737-1751.   DOI:10.1007/s11590-011-0367-2
  17. B. I. N. A. Gupta and R. E. E. T. A. Gupta: Multi-criteria simplex method for a linear multiple objective transportation problem. Indian J. Pure Appl. Math. 14 (1983), 2, 222-232.   CrossRef
  18. P. Hansen, M. Labbé and R. E. Wendell: Sensitivity analysis in multiple objective linear programming: The tolerance approach. Eur. J. Oper. Res. 38 (1989), 1, 63-69.   DOI:10.1080/09578818908426501
  19. F. L. Hitchcock: The distribution of a product from several sources to numerous localities. J. Math. Phys. 20 (1941), 1-4, 224-230.   DOI:10.1002/sapm1941201224
  20. E. Hosseinzadeh and J. Tayyebi: A compromise solution for the neutrosophic multi-objective linear programming problem and its application in transportation problem. J. Appl. Res. Ind. Eng. 10 (2023), 1, 1-10.   CrossRef
  21. C. L. Hwang and K. Yoon: Multiple Attribute Decision Making: Methods and Applications. Springer, New York 1981.   CrossRef
  22. H. Isermann: The enumeration of all efficient solutions for a linear multiple‐objective transportation problem. Nav. Res. Logist. Q. 26 (1979), 1, 123-139.   DOI:10.1002/nav.3800260112
  23. Y. Kacher and P. Singh: Fuzzy harmonic mean technique for solving fully fuzzy multi-objective transportation problem. J. Comput. Sci.63 (2022), 101782.   DOI:10.1016/j.jocs.2022.101782
  24. I. Kandasamy: Double-valued neutrosophic sets, their minimum spanning trees, and clustering algorithm. J. Intell. Syst. 27 (2018), 2, 163-182.   DOI:10.1515/para-2018-0012
  25. N. Kartli, E. Bostanci and M. S. Guzel: Heuristic algorithm for an optimal solution of fully fuzzy transportation problem. Computing 106 (2024), 10, 3195-3227.   DOI:10.1007/s00607-024-01319-5
  26. H. S. Kasana and K. D. Kumar: An efficient algorithm for multi-objective transportation problems. Asia Pac. J. Oper. Res. 17 (2000), 1, 27.   CrossRef
  27. L. Kaur, M. Rakshit and S. Singh: A new approach to solve multi-objective transportation problem. Appl. Appl. Math. 13 (2018), 1, 10.   CrossRef
  28. K. Kavitha and D. Anuradha: Heuristic algorithm for finding sensitivity analysis of a more for less solution to transportation problems. Global J. Pure Appl. Math. 11 (2015), 1, 479-485.   CrossRef
  29. K. Kavitha and D. Anuradha: Sensitivity analysis for spanning tree. In: Advances in Algebra and Analysis: International Conference on Advances in Mathematical Sciences, Vellore, Volume I, Springer International Publishing 2018, pp. 227-232.   CrossRef
  30. T. C. Koopmans: Optimum utilization of the transportation system. Econometrica (1949), 136-146.   CrossRef
  31. A. Kuiri and B. Das: An application of FISM and TOPSIS to a multi-objective multi-item solid transportation problem. Opsearch. 57 (2020), 4, 1299-1318.   DOI:10.1007/s12597-020-00456-7
  32. S. M. Lee and L. J. Moore: Optimizing transportation problems with multiple objectives. AIIE Trans. 5 (1973), 4, 333-338.   DOI:10.1016/0001-8716(73)80005-7
  33. K. M. Lee: Bipolar-valued fuzzy sets and their operations. In: Proc. Int. Conf. on Intelligent Technologies, Bangkok 2000, pp. 307-312.   CrossRef
  34. S. Mahajan and S. K. Gupta: On fully intuitionistic fuzzy multi-objective transportation problems using different membership functions. Ann. Oper. Res. 296 (2021), 211-241.   DOI:10.1007/s10479-019-03318-8
  35. M. M. Miah, A. AlArjani, A. Rashid, A. R. Khan, M. S. Uddin and E. A. Attia: Multi-objective optimization to the transportation problem considering non-linear fuzzy membership functions. AIMS Math. 8 (2023), 5, 10397-10419.   DOI:10.3934/math.2023527
  36. S. Midya and S. K. Roy: Multi-objective fixed-charge transportation problem using rough programming. Int. J. Oper. Res.37 (2020), 3, 377-395.   DOI:10.1504/ijor.2020.105444
  37. S. Midya, S. K. Roy and V. F. Yu: Intuitionistic fuzzy multi-stage multi-objective fixed-charge solid transportation problem in a green supply chain. Int. J. Mach. Learn. Cybern. 12 (2021), 699-717.   DOI:10.1007/s13042-020-01197-1
  38. M. Mondal, G. Mandal, A. A. Shaikh and A. K. Bhunia: Investigation of a multi-objective fixed charge transportation problem with quantity dependent transportation cost and discount policy via metaheuristics. Inf. Syst. Oper. Res. (2025), 1-30.   DOI:10.1080/03155986.2025.2492739
  39. M. Niksirat: A new approach to solve fully fuzzy multi-objective transportation problem. Fuzzy Inf. Eng. (2022), 1-12.   DOI:10.1080/16168658.2022.2152836
  40. K. P. O. Niluminda and E. M. U. S. B. Ekanayake: The multi-objective transportation problem solve with geometric mean and penalty methods. Indones. J. Innov. Appl. Sci. 3 (2023), 1, 74-85.   CrossRef
  41. M. A. Nomani, I. Ali and A. Ahmed: A new approach for solving multi-objective transportation problems. Int. J. Manag. Sci. Eng. Manag. 12 (2017), 3, 165-173.   DOI:10.1080/17509653.2016.1172994
  42. P. Pandian and G. Natarajan: A new method for finding an optimal solution for transportation problems. Int. J. Math. Sci. Eng. Appl. 4 (2010), 2, 59-65.   CrossRef
  43. P. Pandian and D. Anuradha: A new method for solving bi-objective transportation problems. Aust. J. Basic Appl. Sci. 5 (2011), 10, 67-74.   CrossRef
  44. R. Rajbhar and L. Das: Advancements in and multi-objective transportation problems: techniques and comparative analysis. Microsyst. Technol. (2025), 1-14.   CrossRef
  45. A. N. Revathi, S. Mohanaselvi and B. Said: An efficient neutrosophic technique for uncertain multi objective transportation problem. Neutrosophic Sets Syst. 53 (2023), 1, 27.   CrossRef
  46. J. L. Ringuest and D. B. Rinks: Interactive solutions for the linear multi-objective transportation problem. Eur. J. Oper. Res. 32 (1987), 1, 96-106.   DOI:10.1016/0377-2217(87)90274-8
  47. R. M. Rizk-Allah, A. E. Hassanien and M. Elhoseny: A multi-objective transportation model under neutrosophic environment. Comput. Electr. Eng. 69 (2018), 705-719.   CrossRef
  48. S. K. Roy, S. Midya and G. W. Weber: Multi-objective multi-item fixed-charge solid transportation problem under twofold uncertainty. Neural Comput. Appl. 31 (2019), 8593-8613.   DOI:10.1007/s00521-019-04431-2
  49. S. K. Roy and S. Midya: Multi-objective fixed-charge solid transportation problem with product blending under intuitionistic fuzzy environment. Appl. Intell. 49 (2019), 3524-3538.   DOI:10.1007/s10489-019-01466-9
  50. S. K. Roy, A. Ebrahimnejad, J. L. Verdegay and S. Das: New approach for solving intuitionistic fuzzy multi-objective transportation problem. Sadhana 43 (2019), 1, 1-12.   CrossRef
  51. S. Sandhiya and A. Dhanapal: Solving bi-objective transportation problem under neutrosophic environment. J. Appl. Math. Informatics 42 (2024), 4, 831-854.   CrossRef
  52. S. Sandhiya and A. Dhanapal: Solving neutrosophic multi-dimensional fixed charge transportation problem. Contemp. Math.2024 (2024), 3601-3624.   DOI:10.1093/imrn/rnad027
  53. A. A. Salama, H. E. Khalid, A. K. Essa, I. Yasser and A. A. Abd El-Khalek: Integrating neutrosophic logic with fuzzy inference systems: A comprehensive survey on enhancing decision-making under uncertainty. Neutrosophic Sets Syst. 91 (2025), 851-868.   CrossRef
  54. S. Samanta, D. Chakraborty and D. K. Jana: Neutrosophic multi-period two stage four-dimensional transportation problem for breakable items. Expert Syst. Appl. 246 (2024), 123266.   DOI:10.1016/j.eswa.2024.123266
  55. Shivani and D. Rani: Multi-objective multi-item four dimensional green transportation problem in interval-valued intuitionistic fuzzy environment. Int. J. Syst. Assur. Eng. Manag. 15 (2024), 2, 727-744.   CrossRef
  56. K. P. Sikkannan and V. Shanmugavel: Unraveling neutrosophic transportation problem using costs mean and complete contingency cost table. Neutrosophic Sets Syst. 29 (2019), 165-173.   CrossRef
  57. P. Singh, S. Kumari and P. Singh: Fuzzy efficient interactive goal programming approach for multi-objective transportation problems. Int. J. Appl. Comput. Math. 3 (2017), 505-525.   DOI:10.1007/s40819-016-0155-x
  58. V.E. Sobana and D. Anuradha: On solving bi-objective fuzzy transportation problem. In: Advances in Algebra and Analysis: International Conference on Advances in Mathematical Sciences, Vellore, India 2017, Volume I, Springer International Publishing 2018, pp. 233-242.   CrossRef
  59. F. Smarandache: A unifying field in logics. Neutrosophy: Neutrosophic Probability, Set and Logic. American Research Press, Rehoboth 105, 1999.   CrossRef
  60. F. Smarandache: Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset. Similarly for Neutrosophic Over-/Under-/Off-Logic, Probability, and Statistics. Infinite Study, 2016.   CrossRef
  61. F. Smarandache: Hyperuncertain, Superuncertain, and Superhyperuncertain Sets/Logics/Probabilities/Statistics. Infinite Study, 2017.   CrossRef
  62. F. Smarandache: Plithogeny, Plithogenic Set, Logic, Probability, and Statistics. Infinite Study. Pons Editions, Brussels 2017.   CrossRef
  63. F. Smarandache: Neutrosophic set is a generalization of intuitionistic fuzzy set, inconsistent intuitionistic fuzzy set (picture fuzzy set, ternary fuzzy set), pythagorean fuzzy set, spherical fuzzy set, and q-rung orthopair fuzzy set, while neutrosophication is a generalization of regret theory, grey system theory, and three-ways decision. J. New Theory 29 (2019), 1-13.   DOI:10.7748/phc.29.2.13.s11
  64. F. Smarandache and K. Mohana: Fuzzy, Intuitionistic and Neutrosophic Set Theories and their Applications in Decision Analysis. Infinite Study, 2025.   CrossRef
  65. V. Torra: Hesitant fuzzy sets. Int. J. Intell. Syst. 25 (2010), 6, 529-539.   CrossRef
  66. R. M. Umamageswari and G. Uthra: Generalized single valued neutrosophic trapezoidal numbers and their application to solve transportation problem. Studia Rosenthaliana 11 (2020), 164-170.   CrossRef
  67. H. Wang, F. Smarandache, Y. Zhang and R. Sunderraman: Single valued neutrosophic sets. Tech. Sci. Appl. Math. 16 (2010), 1, 10-14.   DOI:10.1525/gfc.2010.10.2.14
  68. L. A. Zadeh: Fuzzy sets. Inf. Control 8 (1965), 3, 338-353.   DOI:10.1016/S0019-9958(65)90241-X
  69. P. Zhu, G. Chen, Y. Chen and A. Zhang: On the pure fixed charge transportation problem. Discret. Optim. 55 (2025), 100875.   DOI:10.1016/S0255-0857(25)00088-X
  70. H. J. Zimmermann: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1 (1978), 1, 45-55.   DOI:10.1016/0165-0114(78)90031-3