Non-linear programming method for multi-criteria decision making problems under interval neutrosophic set environment

Interval neutrosophic sets ( INSs) capturing the uncertainties by characterizing into the intervals of the truth, the indeterminacy, and the falsity membership degrees, is a more flexible way to explore the decision- making applications. In this paper, we develop a nonlinear programming ( NP) model based on the technique for order preference by similarity to ideal solution ( TOPSIS), to solve decision- making problems in which criterion values and their importance are given in the form of interval neutrosophic numbers ( INNs). Based on the concept of closeness coefficient, we firstly construct a pair of the nonlinear fractional programming model and then transform it into the linear programming model. Furthermore, to determine the ranking of considered alternatives, likelihood- based comparison relations are constructed. Finally, an illustrative example demonstrates the applicability of the proposed method for dealing with decision making problems with incomplete knowledge.