Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems

Hamiden Abd El- Wahed Khalifa^{1, 2, *,} Ashraf Al-Quran³, Faisal Al-Sharqi ^4,8, Binyamin Yusoff ⁵, Khadiga Wadi Nahar Tajer⁶, Abeer T. Faisal⁴, Ali M. Alorsan Bany Awad⁷

¹Department of Mathematics, College of Science and Arts, Al- Badaya 51951, Qassim University, Saudi Arabia

²Department of Operations Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt

³Basic Sciences Department, Preparatory Year Deanship, King Faisal University, Al-Ahsa, Saudi Arabia

⁴University Headquarter, Department of Scholarships and Cultural Relations, University of Anbar, Ramadi

⁵Special Interest Group on Modelling and Data Analytics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Malaysia

⁶Department of Mathematics, College of Science and Arts, Qassim University, Ar Rass 51452, Saudi Arabia

⁷Deanship of Development and Quality Assurance, King Faisal University, Al-Ahsa 31982, Saudi Arabia

⁸College of Pharmacy, National University of Science and Technology, Dhi Qar, Iraq

Emails: Ha.Ahmed@qu.edu.sa; hamiden@cu.edu.eg; aalquran@kfu.edu.sa; faisal.ghazi@uoanbar.edu.iq; binyamin@umt.edu.my; khadiganahar@gmail.com; abeert2017@uoanbar.edu.iq; abanyawad@kfu.edu.sa

Abstract

This article considers a bi-level linear programming with single valued trapezoidal fuzzy neutrosophic cost coefficient matrix and Pythagorean fuzzy parameters in the set of constraints both in the right and left sides. Based on the score functions of the neutrosophic numbers and Pythagorean fuzzy numbers, the model is changed to the corresponding crisp bi-level linear programming (BLP) problem. This problem is designated as a Pythagorean fuzzy bi-level linear programming (PFBLP) problem under neutrosophic environment. Kuhn-Tucker's conditions for optimality are necessary and sufficient for the existence of the optimal solution to a BLP problem. Using the suggested methodology, the problem is formulated as a single-objective non-linear programming problem with several variables and constraints. Two typical numerical examples are examined to illustrate the proposed approach.

Keywords: Optimization; Optimization problems; Bi-level programming; Pythagorean fuzzy number;  Neutrosophic set; Single valued neutrosophic numbers; Treapezoidal neutrosophic numbers; Kuhn-Tucker's  optimality conditions; Decision Making; GAMS computer package.