A Solution Approach to Multi-level Nonlinear Fractional Programming Problem

  • Suvasis Nayak
  • A. K. Ojha
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 225)


This paper studies multi-level nonlinear fractional programming problem (ML-NLFPP) of maximization type and proposes a solution approach which is based on the concept of fuzzy and simultaneous minimization, maximization of the objectives from their ideal, anti-ideal values, respectively. Nonlinear polynomial functions are considered as the numerators and denominators of the fractional objectives at each level. In the objective space, distance function or Euclidean metric is implemented to measure the distances between numerators, denominators and their ideal, anti-ideal values which need to be minimized and maximized. Goals for the controlled decision variables of upper levels are ascertained from the individual best optimal solutions of the corresponding levels, and tolerances are defined by decision makers to avoid the situation of decision deadlock. Fuzzy goal programming with reduction of only under-deviation from the highest membership value derives the best compromise solution of the concerned multi-level problem. An illustrative numerical example is discussed to demonstrate the solution approach and its effectiveness.


Multi-level programming Fractional programming Distance function Fuzzy goal programming Best compromise solution 



Authors are grateful to the editor and anonymous referees for their valuable comments and suggestions to improve the quality of the paper.


  1. 1.
    Shih, H.-S., Lai, Y.-J., Lee, E.S.: Fuzzy approach for multi-level programming problems. Comput. Oper. Res. 23, 73–91 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Stancu-Minasian, I.M.: Fractional programming: Theory, Methods and Applications. Kluwer Academic Publishers (1997)Google Scholar
  3. 3.
    Bellman, R.: Dynamic Programming. Princeton University Press (1957)Google Scholar
  4. 4.
    Lachhwani, K.: Modified FGP approach for multi-level multi objective linear fractional programming problems. Appl. Math. Comput. 266, 1038–1049 (2015)MathSciNetGoogle Scholar
  5. 5.
    Lai, Y.-J.: Hierarchical optimization: a satisfactory solution. Fuzzy Sets Syst. 77, 321–335 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Zimmermann, H.-J.: Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst. 1, 45–55 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    White, D.J.: Penalty function approach to linear trilevel programming. J. Optim. Theory Appl. 93, 183–197 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Osman, M.S., Abo-Sinna, M.A., Amer, A.H., Emam, O.E.: A multi-level non-linear multi-objective decision-making under fuzziness. Appl. Math. Comput. 153, 239–252 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Zhang, G., Lu, J., Montero, J., Zeng, Y.: Model, solution concept, and Kth-best algorithm for linear trilevel programming. Inf. Sci. 180, 481–492 (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Pramanik, S., Roy, T.K.: Fuzzy goal programming approach to multilevel programming problems. Eur. J. Oper. Res. 176, 1151–1166 (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Mohamed, R.H.: The relationship between goal programming and fuzzy programming. Fuzzy Sets Syst. 89, 215–222 (1997)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Baky, I.A.: Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach. Appl. Math. Model. 34, 2377–2387 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Baky, I.A.: Interactive TOPSIS algorithms for solving multi-level non-linear multi-objective decision-making problems. Appl. Math. Model. 38, 1417–1433 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Abo-Sinna, M.A., Baky, I.A.: Interactive balance space approach for solving multi-level multi-objective programming problems. Inf. Sci. 177, 3397–3410 (2007)CrossRefzbMATHGoogle Scholar
  15. 15.
    Sinha, S.: Fuzzy programming approach to multi-level programming problems. Fuzzy Sets Syst. 136, 189–202 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lachhwani, K.: On solving multi-level multi objective linear programming problems through fuzzy goal programming approach. Opsearch 51, 624–637 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Baky, I.A., Abo-Sinna, M.A.: TOPSIS for bi-level MODM problems. Appl. Math. Model. 37, 1004–1015 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Miettinen, K.M.: Nonlinear Multiobjective Optimization. Springer Science & Business Media (2012)Google Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.School of Basic SciencesIndian Institute of Technology BhubaneswarBhubaneswarIndia

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