An improved solution for fully neutrosophic multi-level linear programming problem based on Laplace transform

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.