Authors: Nguyen Dinh Phuª, Nguyen Nhut Hung^b, Ali Ahmadian^c,*, Soheil Salahshour^d and Norazak Senu^e

Affiliations: [a] Faculty of Engineering Technology, Quang Trung University, Quy Nhon City, Vietnam | [b] Faculty of Engineering Technology, Quang Trung University, Vietnam | [c] Institute of IR 4.0, The National University of Malaysia, UKM, Bangi, Selangor, Malaysia | [d] Bahcesehir University, Faculty of Engineering and Natural Sciences, Istanbul, Turkey | [e] Institute for Mathematical Research (INSPEM), University Putra Malaysia, UPM, Selangor, Malaysia

Journal: , vol. 40, no. 1, pp. 1-12, 2021

Article type: Research Article

Abstract: This study presents a possible relationship between two main objects, which are three-dimensional copulas (3D-Cs) and geometric picture fuzzy numbers (GPFNs). This opens up a potential field for future studies for these two objects that three-dimensional copulas can become useful tools for handling uncertainty information in the form of a picture fuzzy set (PFS). Specifically, we define a GPFN as a base element of the PFS and a defined domain of three-dimensional copulas that contains a set of GPFNs, then we show some examples of three-dimensional copulas identified on this domain. In thi framework, we present the theorems related to these two objects. At the same time, we provide some examples for three-dimensional semi-copulas, three-dimensional quasi-copulas, and three-dimensional empirical copulas defined on D, which is a defined domain of a three-dimensional copula and contains a set of GPFNs D∗g. In addition, we also introduce a new approach to non-linear programming problems.

Keywords: Three-dimensional distribution functions, three-dimensional copulas (3D-Cs), geometric picture fuzzy numbers (GPFNs), additional set of geometric picture fuzzy numbers (Ad-GPFNs), non-linear programming approach