Submit Your Paper
  1. Home
  2. Journals
  3. International Journal of Neutrosophic Science

International Journal of Neutrosophic Science

Journal DOI

https://doi.org/10.54216/IJNS

Submit Your Paper

2690-6805ISSN (Online) 2692-6148ISSN (Print)
Submit Your Paper

Journal Menu

Homepage Journal Overview Editorial Board Submitting Articles Indexing & Abstracting

Journal Volumes

International Journal of Neutrosophic Science

Volume 23 , Issue 3 , PP: 87-96, 2024 | Cite this article as | XML | Html | PDF | Full Length Article

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

Hamiden Abd El- Wahed Khalifa 1 * , Ashraf Al-Quran 2 , Faisal Al-Sharqi 3 , Binyamin Yusoff 4 , Khadiga W. Nahar Tajer 5 , Abeer T. Faisal 6 , Ali M. Alorsan Bany Awad 7

  • 1 Department of Mathematics, College of Science and Arts, Al- Badaya 51951, Qassim University, Saudi Arabia; Department of Operations Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt - (hamiden@cu.edu.eg)
  • 2 Basic Sciences Department, Preparatory Year Deanship, King Faisal University, Al-Ahsa, Saudi Arabia - (aalquran@kfu.edu.sa)
  • 3 University Headquarter, Department of Scholarships and Cultural Relations, University of Anbar, Ramadi; College of Pharmacy, National University of Science and Technology, Dhi Qar, Iraq - (faisal.ghazi@uoanbar.edu.iq)
  • 4 Special Interest Group on Modelling and Data Analytics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Malaysia - (binyamin@umt.edu.my)
  • 5 Department of Mathematics, College of Science and Arts, Qassim University, Ar Rass 51452, Saudi Arabia - (khadiganahar@gmail.com)
  • 6 University Headquarter, Department of Scholarships and Cultural Relations, University of Anbar, Ramadi - (abeert2017@uoanbar.edu.iq)
  • 7 Deanship of Development and Quality Assurance, King Faisal University, Al-Ahsa 31982, Saudi Arabia - (abanyawad@kfu.edu.sa)

    • Doi: https://doi.org/10.54216/IJNS.230308

    Received: July 25, 2023 Revised: November 12, 2023 Accepted: January 25, 2024
    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.

    References

    [1]    Bard, J. F., and Falk, J. E. (1982). An explicit solution to the multi-level programming problem. Computers & Operations Research, (9): 77-100.

    [2]    Sinha, S., and Biswal, M. P. (2000). Fuzzy programming approach to bi-level linear programming problems. The Journal of Fuzzy Mathematics, 8(2): 337-347.

    [3]    Bialas, W.F., and Karwan, M. H. (1982). On two-level optimization, IEEE Transaction Automatic Control, AC-27: 211-214.

    [4]    Ye, J. J., and Zhu, D. L. (1995). Optimality conditions for bi-level linear programming problems. Optimization, 33(1): 9-27.

    [5]    Chen, Y., and Florian, M. (1995). The non-linear bi-level programming problem: Formulations, regularity and optimality conditions, Optimization, 32(3): 193-209.

    [6]    Zadeh, L. A. (1965). Fuzzy sets. Information Control, 8(3): 338-353.

    [7]    Dubois, D., and Prade, H. (1980). Fuzzy sets and systems: theory and applications, Academic Press, New York.

    [8]    Kaufmann, A., and Gupta, M. M. Fuzzy Mathematical Models in Engineering and Management Science, Elsevier Science Publishing Company INC, New York, 1988.

    [9]    Zimmermann, H. J. (1978). Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and System, 1(1): 45–55

    [10] Atanassov. K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20: 87-96.

    [11] F. Smarandache, A Unifying Field in Logics. Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, Rehoboth, NM, USA,1998.

    [12] Falk, J. E., and Liu, J. (1995). On bi-level programming I: General non-linear cases. Mathematical Programming, 70(1): Ser A, 47-72.

    [13] Gendreau, M. Marcotte, P., and Savard, G. (1996). A hybrid Tabu-Ascent algorithm for the linear bi-level programming problem. Journal of Operational Research Society, (32): 783-792.

    [14] Dempe, S.., and Schemidt, H. (1996). On an algorithm solving 2-level programming problems with non-unique lower level solutions. Computers& Optimization Applications, 6(3): 227-249.

    [15] Safaei, N., and Saraj, M. (2014). A new method for solving fully fuzzy linear bi-level programming problems. International Journal of Applied Operational Research, 4(1): 39-46.

    [16] Ren, A. (2015). A novel method for solving the fully fuzzy bi-level linear programming problem. Mathematical Problems in Engineering, vol. 2015, Article ID, 11 pages.

    [17] Hossein, Z. A., and Edalatpanah, S. A. (2016). A new approach for solving fully fuzzy linear programming by using the Lexicographic method. Advanced in Fuzzy Systems, (2016), 6 pages.

    [18] Rajarajeswari, P., and Sudha, A. S. (2014). Solving a fully fuzzy linear programming problem by ranking. International Journal of Mathematics Trends and Technology, 9(2): 159-164.

    [19] Khalifa, H. A., and Kumar, P. (2023). Multi-objective optimization for solving cooperative continuous static games using Karush-Kuhn-Tucker conditions. International Journal of Operations Research, 46(1): 133- 147.

    [20] Yager, R. R. (2014). Pythagorean membership grades in multi-criteria decision-making. IEEE Transection on Fuzzy Systems, 22(4): 958–965.

    [21] F. Al-Sharqi, Y. Al-Qudah and N. Alotaibi, Decision-making techniques based on similarity measures of possibility neutrosophic soft expert sets. Neutrosophic Sets and Systems, 55(1) (2023), 358-382.

    [22] M. U. Romdhini, F. Al-Sharqi, A. Nawawi, A. Al-Quran and H. Rashmanlou, Signless Laplacian Energy of Interval-Valued Fuzzy Graph and its Applications, Sains Malaysiana 52(7), 2127-2137, 2023.

    [23] F. Al-Sharqi, A. Al-Quran, M. U. Romdhini, Decision-making techniques based on similarity measures of possibility interval fuzzy soft environment, Iraqi Journal for Computer Science and Mathematics, vol. 4, pp.18–29, 2023.

    [24] Z. bin M. Rodzi et al. Integrated Single-Valued Neutrosophic Normalized Weighted Bonferroni Mean (SVNNWBM)-DEMATEL for Analyzing the Key Barriers to Halal Certification Adoption in Malaysia, Int. J. Neutrosophic Sci., vol. 21, no. 3, pp. 106–114, 2023.

    [25] F. Al-Sharqi, M. U. Romdhini, A. Al-Quran, Group decision-making based on aggregation operator and score function of Q-neutrosophic soft matrix, Journal of Intelligent and Fuzzy Systems, vol. 45, pp.305–321, 2023.

    [26] A. Al-Quran, F. Al-Sharqi, Z. M. Rodzi, M. Aladil, R. A. shlaka, M. U. Romdhini, M. K. Tahat, O.S. Solaiman. (2023). The Algebraic Structures of Q-Complex Neutrosophic Soft Sets Associated with Groups and Subgroups. International Journal of Neutrosophic Science, 22 (1), 60-76.

    [27] F. Al-Sharqi, A. Al-Quran and Z. M. Rodzi, Multi-Attribute Group Decision-Making Based on Aggregation Operator and Score Function of Bipolar Neutrosophic Hypersoft Environment, Neutrosophic Sets and Systems, 61(1), 465--492, 2023.‏

    [28] Jamiatun Nadwa Ismail et al. The Integrated Novel Framework: Linguistic Variables in Pythagorean Neutrosophic Set with DEMATEL for Enhanced Decision Support. Int. J. Neutrosophic Sci., vol. 21, no. 2, pp. 129-141, 2023.

    [29] H. Wang, F. Smarandache, Y.Q. Zhang, R. Sunderraman. (2010). "Single valued neutrosophic sets," Multistructure, 4(1): 410- 413.

    [30] Pathade PA, Ghadle KP (2018) Transportation problem with triangular mixed intuitionistic fuzzy numbers solved by BCM. Int J Fuzzy Math Arch 15(1):55–61.

    [31] Mokhtar, S., Bazaraa and Shetty, C.M. Non-Linear Programming: Theory and Algorithms, John Wiley & Sons, 1979.

     

    Cite This Article As :
    Abd, Hamiden. , Al-Quran, Ashraf. , Al-Sharqi, Faisal. , Yusoff, Binyamin. , W., Khadiga. , T., Abeer. , M., Ali. Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems. International Journal of Neutrosophic Science, vol. , no. , 2024, pp. 87-96. DOI: https://doi.org/10.54216/IJNS.230308
    Abd, H. Al-Quran, A. Al-Sharqi, F. Yusoff, B. W., K. T., A. M., A. (2024). Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems. International Journal of Neutrosophic Science, (), 87-96. DOI: https://doi.org/10.54216/IJNS.230308
    Abd, Hamiden. Al-Quran, Ashraf. Al-Sharqi, Faisal. Yusoff, Binyamin. W., Khadiga. T., Abeer. M., Ali. Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems. International Journal of Neutrosophic Science , no. (2024): 87-96. DOI: https://doi.org/10.54216/IJNS.230308
    Abd, H. , Al-Quran, A. , Al-Sharqi, F. , Yusoff, B. , W., K. , T., A. , M., A. (2024) . Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems. International Journal of Neutrosophic Science , () , 87-96 . DOI: https://doi.org/10.54216/IJNS.230308
    Abd H. , Al-Quran A. , Al-Sharqi F. , Yusoff B. , W. K. , T. A. , M. A. [2024]. Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems. International Journal of Neutrosophic Science. (): 87-96. DOI: https://doi.org/10.54216/IJNS.230308
    Abd, H. Al-Quran, A. Al-Sharqi, F. Yusoff, B. W., K. T., A. M., A. "Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems," International Journal of Neutrosophic Science, vol. , no. , pp. 87-96, 2024. DOI: https://doi.org/10.54216/IJNS.230308

    Article Statistics

    View
    55425
    Download
    518

    Download