A fuzzy based mathematical model on Lagrangian multiplier conditions has been proposed to address the Non-linear Programming (NLP) with equality constraints. Furthermore, the model demonstrates how multivariable optimization issues can be solved using membership functions. This model is excellent for problem-solving because there is no need to explicitly solve the conditions and utilize them to eliminate additional variables. Then the sufficient conditions for a constrained local maximum or minimum can be stated in terms of a sequence of principal minors of the bordered Hessian matrix of second derivatives of the Lagrangian expression. It also demonstrates how the Lagrange multiplier method can be used for proving the Jacobian matrix. Additionally, the model can be considered in three stages: that is, mathematical formulation, computational procedures, and numerical illustration with comparative analysis. Likewise, the model illustrates the considered problem using two distinct approaches, namely membership functions (MF) and robust ranking index. Finally, the comparison analysis provides detailed results and discussion that justify the optimal outcome in order to address the vagueness of certain NLPPs.