The parametric rough bi-level multi-objective fractional programming problem (PRBL-MOFPP) is investigated in this article. In the right-hand aspect of the rough set of constraints, the suggested PRBL-MOFPP has a scalar parameter. The PRBL-MOFPP is converted into two issues congruent to the upper and lower approximation models (UAM and LAM) in the first phase. Both UAM and LAM are solved using the fuzzy goal programming (FGP) technique. The parametric UAM and LAM were formulated in the second phase, and the Lagrangian function for UAM and LAM was derived. Furthermore, both models were subjected to Karush-Kuhn-Tucker (KKT) optimality conditions. Finally, the surely and possibly stable set of the first kind (SSFK) are studied. An algorithm for determining the SSFK for PRBL-MOFPP, as well as numerical examples, are exhibited.

1. Osman MSA. Qualitative analysis of basic notions in parametric convex programming. Ⅰ. Parameters in the constraints. Applications of Mathematics. 1977; 22(5): 318–332.

2. Osman MSA. Qualitative analysis of basic notions in parametric convex programming. Ⅱ. Parameters in the objective function. Applications of Mathematics. 1977; 22(5): 333–348.

3. Osman MS, Maaty MA, Farahat FA. On the achievement stability set for parametric linear goal programming problems. OPSEARCH. 2017; 54: 181–200.

4. AbdAlhakim H, Emam OE, Abd El-Mageed AA. Architecting a fully fuzzy information model for multi-level quadratically constrained quadratic programming problem. OPSEARCH. 2019; 56(2): 367–389.

5. Baky IA, Eid MH, and El sayed MA. Bi-level multi-objective programming problem with fuzzy demands: A fuzzy goal programming algorithm. OPSEARCH. 2014; 51(2): 280–296.

6. Emam OE, El-Araby M, and Belal MA. On rough multi-level linear programming problem. Information Science Letter. 2015; 4(1): 41–49.

7. Elsisy MA, El Sayed MA. Fuzzy rough bi-level multi-objective non-linear programming problems. Alexandria Engineering Journal. 2019; 58(4): 1471–1482.

8. Elsisy MA, Elsaadany AS, and El Sayed MA. Using Interval Operations in the Hungarian Method to Solve the Fuzzy Assignment Problem and Its Application in the Rehabilitation Problem of Valuable Buildings in Egypt. Complexity. 2020; doi: 10.1155/2020/9207650

9. Shi Y, Yao L, Xu J. A probability maximization model based on rough approximation and its application to the inventory problem. International Journal of Approximate Reasoning. 2011; 52(2): 261–280.

10. Kong Q, Xu W. The comparative study of covering rough sets and multi-granulation rough sets. Soft Computing. 2019; 23(10): 3237–3251.

11. Pawlak Z. Rough sets. International Journal of Computer and Information Science. 1982; 11(5): 341–356.

12. Pawlak Z. Rough sets, rough relations and rough functions. Fundamental Information. 1996; 27(2–3): 103–108.

13. Rissino S, Lambert-Torres G. Rough set theory-fundamental concepts, principals, data extraction, and applications. In: Data mining and knowledge discovery in real life applications. IntechOpen; 2009.

14. Elsisy MA, Eid MH, Osman MSA. Qualitative analysis of basic notions in parametric rough convex programming (parameters in the objective function and feasible region is a rough set). OPSEARCh. 2017; 54(4): 724–34.

15. Hamzehee A, Yaghoobi MA, Mashinchi M. Linear programming with rough interval coefficients. Journal of Intelligent & Fuzzy Systems. 2014; 26(3): 1179–1189.

16. Allende GB, Still G. Solving bilevel programs with the KKT-approach. Mathematical programming. 2013; 138(1–2): 309–332.

17. Ahlatcioglu M, Tiryaki F. Interactive fuzzy programming for decentralized two-level linear fractional programming (DTLLFP) problems. Omega. 2007; 35(4): 432–450.

18. Baky IA. Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach. Applied Mathematics Modelling. 2010; 34(9): 2377–2387.

19. Baky IA. Fuzzy goal programming algorithm for solving decentralized bi-level multi-objective programming problems. Fuzzy Sets and Systems. 2009; 160(18): 2701–2713.

20. Baky IA, Abo-Sinna MA. TOPSIS for bi-level MODM problems. Applied Mathematical Modelling. 2013; 37(3): 1004–1015.

21. Chen LH, Chen HH. A two-phase fuzzy approach for solving multi-level decision-making problems. Knowledge-Based Systems. 2015; 76: 189–199.

22. Emam OE. Interactive approach to bi-level integer multi-objective fractional programming problem. Applied Mathematics and Computation. 2013; 223:17–24.

23. Pramanik S, Roy TK. Fuzzy goal programming approach to multi-level programming problems. European Journal of Operational Research. 2007; 176(2): 1151–1166.

24. Arora SR, Gupta R. Interactive fuzzy goal programming approach for bi-level programming problem. European Journal of Operational Research. 2009; 194(2): 368–376.

25. El Sayed MA, Baky IA, Singh P. A modified TOPSIS approach for solving stochastic fuzzy multi-level multi-objective fractional decision-making problem. OPSEARCH. 2020; 57: 1374–1403.

26. El Sayed MA, Farahat FA. On Various Approaches for Bi-level Optimization Problems. International Journal of Management and Fuzzy Systems. 2019; 5(3): 47–63.

27. Sinha A. Bilevel Multi-objective Optimization Problem Solving Using Progressively Interactive EMO. In: Proceedings of the 6th international conference on Evolutionary multi-criterion optimization; 5–8 April 2011; Ouro Preto, Brazil.

28. Sakawa M, Katagiri H, Matsui T. Stackelberg solutions for fuzzy random two-level linear programming through probability maximization with possibility. Fuzzy Sets and Systems. 2012; 188(1): 45–57.

29. Lachhwani K. Modified FGP approach for multi-level multi objective linear fractional programming problems. Applied Mathematics and Computation. 2015; 266: 1038–1049.

30. Miettinen K. Nonlinear multiobjective optimization. Springer Science & Business Media; 2012. Volume 12.

31. Nayak S, Ojha A. An approach of fuzzy and TOPSIS to bi-level multi-objective nonlinear fractional programming problem. Soft Computing. 2019; 23(14): 5605–5618.

32. Pal BB, Moitra BN, Maulik U. A goal programming procedure for fuzzy multi-objective linear fractional programming problem. Fuzzy Sets and Systems. 2003; 139(2): 395–405.

33. Osman MS, Emam OE, El Sayed MA. On parametric multi-level multi-objective fractional programming problems with fuzziness in the constraints. British Journal of Mathematics and Computer Science. 2016; 18(5): 1–19.

34. Osman MS, Emam OE, El Sayed MA. Interactive Approach for Multi-Level Multi-Objective Fractional Programming Problems with Fuzzy Parameters. Beni-Suef University journal of basic and applied sciences. 2018; 7(1): 139–149.

35. Osman MS, Raslan KR, Emam OE, Farahat FA. Solving Multi-level Multi-objective Fractional Programming Problem with Rough Intervals in the Objective Functions. British Journal of Mathematics & Computer Science. 2017; 21(2): 1–17.

36. El Sayed MA, Farahat FA. Study of Achievement Stability Set for Parametric Linear FGP Problems. Ain Shams Engineering Journal. 2020; 11(4): 1345–1353. doi: 10.1016/j.asej.2020.03.003

37. Lachhwani K. On solving multi-level multi objective linear programming problems through fuzzy goal programming approach. OPSEARCH. 2014; 51(4): 624–637.

38. Shi C, Lu J, Zhang G. An extended Kuhn–Tucker approach for linear bilevel programming. Applied Mathematics and Computation. 2005; 162(1): 51–63.

39. Youness EA, Emam OE, Hafez MS. Fuzzy bi-level multi-objective fractional integer programming. Applied Mathematics and Information Science. 2014; 8(6): 2857–2863.

40. Ranarahu N, Dash JK, Acharya S. Multi-objective bilevel fuzzy probabilistic programming problem. OPSEARCH. 2017; 54(3): 475–504.

41. Ren A. A novel method for solving the fully fuzzy bilevel linear programming problem. Mathematical Problems in Engineering. 2015; 2: 1–11.

42. Saad OM, Emam OE, Sleem MM. On the solution of a rough interval three-level quadratic programming problem. Britsh Journal of Mathematics and Computer Science. 2015; 5(3): 349–366.

43. Gumus ZH, Floudas CA. Global Optimization of Nonlinear Bilevel Programming Problems. Journal of Global Optimization. 2001; 20: 1–31.

44. Osman MS, Abo-Sinna MA, Amer AH, Emam OE. A multi-level non-linear multi-objective decision-making under fuzziness. Applied Mathematics and Computation. 2004; 153(1): 239–252.

45. Xu J, Yao L. A class of multiobjective linear programming models with random rough coefficient. Mathematical and Computer Modeling. 2009; 49(1–2): 189–206.

46. Rizk-Allah RM, Abo-Sinna MA. Integrating reference point. Kuhn–Tucker conditions and neural network approach for multi-objective and multi-level programming problems. OPSEARCH. 2017; 54: 663–683.

47. Farahat FA, ElSayed MA. Achievement Stability Set for Parametric Rough Linear Goal Programing Problem. Fuzzy Information and Engineering. 2019; 11(3): 279–294.

48. Agarwal D, Singh P, El Sayed MA. The Karush–Kuhn–Tucker (KKT) optimality conditions for fuzzy-valued fractional optimization problems. Mathematics and Computers in Simulation. 2023; 205: 861–877.

49. Osman MS, Emam OE, El Sayed MA. Stochastic fuzzy multi-level multi-objective fractional programming problem: A FGP approach. OPSEARCH. 2017; 54(4): 816–840.

50. El Sayed MA, Abo-Sinna MA. A novel Approach for Fully Intuitionistic Fuzzy Multi-Objective Fractional Transportation Problem. Alexandria Engineering Journal. 2021; 60(1): 1447–1463.

51. Kamal M, Gupta S, Chatterjee P, et al. Bi-Level Multi-Objective Production Planning Problem with Multi-Choice Parameters: A Fuzzy Goal Programming Algorithm. Algorithms. 2019; 12(7): 143.

52. Singh VP, Sharma K, Chakraborty D, Ebrahimnejad A. A novel multi-objective bi-level programming problem under intuitionistic fuzzy environment and its application in production planning problem. Complex & Intelligent Systems. 2022; 8: 3263–3278.