Abstract
The main objective of this paper is to present an improved solution approach for a fully neutrosophic generalized multi-level linear programming (MLP) problem in which de-neutrosophication of coefficients and parameters are carried out with the proportional probability density functions of each N Number and the use of Laplace transform. The present paper describes a unique solution methodology for generalized multi-level linear programming involving coefficient parameters in objectives at each level as well as constraints as interval-valued trapezoidal neutrosophic numbers (IVTrpN numbers), based on Laplace Transform. In this approach, we first propose to associate the probability density function to each membership function of each IVTrpN number and obtain an equivalent output value of each N Number using Laplace transform. The proposed algorithm is novel and unique for solving the generalized MLLP problem under N Numbers environment, which converts the neutrosophic problem into an equivalent crisp problem. After that, the multi-level structure of the crisp problem is tackled by formulating separate membership functions for each objective function at each level and decision variables up to the (T-1) level with their best values. A simple solution model is formulated to obtain a satisfactory solution to MLLP problem under the neutrosophic environment with the help of usual goal programming. Further, a comparative study is also carried out between the use of Laplace transform and Melin transform (as suggested by Tamilarasi and Paulraj (SC 26:8497–8507, 2022)) for de-neutrosophication of N numbers in the context of the present problem. Numerical example and complex real problem are illustrated to show the functionality and applicability of the proposed improved approach.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Lachhwani, K., Dwivedi, A.: Bi-level and multi-level programming problems: taxonomy of literature review and research issues. Archiv. Comput. Methods Eng. 25, 847–877 (2017)
Pramanik, S., Banerjee, D., Giri, B.: Multi-level multi-objective linear plus linear fractional programming problem based on FGP approach. Int. J. Innov. Sci. Eng. Technol. 2, 171–177 (2015)
Lachhwani, K.: Modified FGP approach for multi-level multi objective linear fractional programming problems. Appl. Math. Comput. 266, 1038–1049 (2015)
Osman, M.S., Emam, O.E., El Sayed, M.A.: Solving multi-level multi-objective fractional programming problems with fuzzy demands via FGP approach. Int. J. Appl. Comput. Math. 4, 41 (2017). https://doi.org/10.1007/s40819-017-0467-5
Liu, Q., Yang, Y.: Interactive programming approach for solving multi-level multi-objective linear programming problem. J. Intell. Fuzzy Syst. 35, 55–61 (2018)
Peraei, E.Y., Maleki, H.R., Mashinchi, M.: A method for solving a fuzzy linear programming. Korean J. Comput. Appl. Math. 8, 347–356 (2001)
Alaulden, N.A., Sanar, M.Y.: Solving fuzzy network problems by defuzzification techniques. Int. J. Innov. Res. Sci. Eng. Technol. 03, 17114–17123 (2014)
Zadeh, L.A.: Fuzzy sets. Inf. Control. 8, 338–353 (1965)
Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)
Smarandache, F.: A unifying field in logics: neutrosophic logic. Neutrosophic set, neutrosophic probability and statistics. American Research Press, Rehoboth (1998)
Ye, J.: Neutrosophic number linear programming method and its application under neutrosophic number environments. Soft. Comput. 22, 4639–4646 (2017)
Mohamed, M., Abdel-Basset, M., Zaied, A., Smarandache, F.: Neutrosophic integer programming problem. Neutrosophic Sets Syst. 15, 3–7 (2017)
Abdel-Basset, M., Gunasekaran, M., Mohamed, M., Smarandache, F.: A novel method for solving the fully neutrosophic linear programming problems. Neural Comput. Appl. 31, 1595–1605 (2019)
Pramanik, S., Banerjee, D.: Neutrosophic number goal programming for multi-objective linear programming problem in neutrosophic number environment. MOJ Curr. Res. Rev. 1, 135–141 (2018). https://doi.org/10.15406/mojcrr.2018.01.00021
Pramanik, S., Dey, P.: Bi-level linear programming problem with neutrosophic numbers. Neutrosophic Sets Syst. 21, 110–121 (2018)
Bera, T., Mahapatra, N.K.: An approach to solve the linear programming problem using single valued trapezoidal neutrosophic number. Int. J. Neutrosophic Sci. 2, 54–66 (2020)
Darehmiraki, M.: A solution for the neutrosophic linear programming problem with a new ranking function. Optim. Theory Based Neutrosophic Plithogenic Sets (2020). https://doi.org/10.1016/B978-0-12-819670-0.00011-1
Basumatary, B., Broumi, S.: Interval-valued triangular neutrosophic linear programming problem. Int. J. Neutrosophic Sci. 10, 105–115 (2020)
Khatter, K.: Interval valued trapezoidal neutrosophic set: multi-attribute decision making for prioritization of non-functional requirements. J. Ambient. Intell. Humaniz. Comput. 12, 1039–1055 (2020)
Maiti, I., Mandal, T., Pramanik, S.: Neutrosophic goal programming strategy for multi-level multi-objective linear programming problem. J. Ambient. Intell. Humaniz. Comput. 11, 3175–3186 (2020)
Lachhwani, K.: Solving the general fully neutrosophic multi-level multiobjective linear programming problems. Opsearch (2021). https://doi.org/10.1007/s12597-021-00522-8
Ahmad, F.: Interactive neutrosophic optimization technique for multiobjective programming problems: an application to pharmaceutical supply chain management. AN Numberals Oper. Res. (2021). https://doi.org/10.1007/s10479-021-03997-2
Tamilarasi, G., Paulraj, S.: An improved solution for the neutrosophic linear programming problems based on Mellin’s transform. Soft. Comput. 26, 8497–8507 (2022). https://doi.org/10.1007/s00500-022-07252-z
Smarandache, F.: Introduction of Neutrosophic Statistics. Sitech and Education Publisher, Craiova (2013)
Smarandache, F.: (t, i, f)-Neutrosophic structures & I-neutrosophic structures (revisited). Neutrosophic Sets Syst. 8, 3–9 (2015)
Luo, S., Pedrycz, W., Xing, L.: Interactive multilevel programming approaches in neutrosophic environments. J. Ambient. Intell. Humaniz. Comput. 13, 2143–2159 (2021). https://doi.org/10.1007/s12652-021-02975-7
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
KL: Contributed as being the sole author of this research work.
Corresponding author
Ethics declarations
Conflict of interest
The author declares that there is no competing interest related to publication of this manuscript.
Ethics approval and consent to participate
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lachhwani, K. An improved solution for fully neutrosophic multi-level linear programming problem based on Laplace transform. OPSEARCH (2023). https://doi.org/10.1007/s12597-023-00715-3
Accepted:
Published:
DOI: https://doi.org/10.1007/s12597-023-00715-3