A Mathematical Model for Nonlinear Optimization Which Attempts Membership Functions to Address the Uncertainties

Abstract

The problem of optimizing an objective function that exists within the constraints of equality and inequality is addressed by nonlinear programming (NLP). A linear program exists if all of the functions are linear; otherwise, the problem is referred to as a nonlinear program. The development of highly efficient and robust linear programming (LP) algorithms and software, the advent of highspeed computers, and practitioners’ wider understanding and portability of mathematical modeling and analysis have all contributed to LP’s importance in solving problems in a variety of fields. However, due to the nature of the nonlinearity of the objective functions and any of the constraints, several practical situations cannot be completely explained or predicted as a linear program. Efforts to overcome such nonlinear problems quickly and efficiently have made rapid progress in recent decades. The past century has seen rapid progress in the field of nonlinear modeling of real-world problems. Because of the uncertainty that exists in all aspects of nature and human life, these models must be viewed through a system known as a fuzzy system. In this article, a new fuzzy model is proposed to address the vagueness presented in the nonlinear programming problems (NLPPs). The proposed model is described; its mathematical formulation and detailed computational procedure are shown with numerical illustrations by employing trapezoidal fuzzy membership functions (TFMFs). Here, the computational procedure has an important role in acquiring the optimum result by utilizing the necessary and sufficient conditions of the Lagrangian multipliers method in terms of fuzziness. Additionally, the proposed model is based on the previous research in the literature, and the obtained optimal result is justified with TFMFs. A model performance evaluation was completed with different set of inputs, followed by a comparison analysis, results and discussion. Lastly, the performance evaluation states that the efficiency level of the proposed model is of high impact. The code to solve the model is implemented in LINGO, and it comes with a collection of built-in solvers for various problems.