Advertisement

Soft Computing

, Volume 23, Issue 1, pp 197–210 | Cite as

Duality-based branch–bound computational algorithm for sum-of-linear-fractional multi-objective optimization problem

  • Deepika Agarwal
  • Pitam SinghEmail author
  • Deepak Bhati
  • Saru Kumari
  • Mohammad S. Obaidat
  • Fellow of IEEE and Fellow of SCS
Methodologies and Application
  • 22 Downloads

Abstract

Optimizing the sum-of-fractional functions under the bounded feasible space is a very difficult optimization problem in the research area of nonlinear optimization. All the existing solution methods in the literature are developed to find the solution of single-objective sum-of-fractional optimization problems only. Sum-of-fractional multi-objective optimization problem is not attempted to solve much by the researchers even when the fractional functions are linear. In the present article, a duality-based branch and bound computational algorithm is proposed to find a global efficient (non-dominated) solution for the sum-of-linear-fractional multi-objective optimization (SOLF-MOP) problem. Charnes–Cooper transformation technique is applied to convert the original problem into non-fractional optimization problem, and equivalence is shown between the original SOLF-MOP and non-fractional MOP. After that, weighted sum method is applied to transform MOP into a single-objective problem. The Lagrange weak duality theorem is used to develop the proposed algorithm. This algorithm is programmed in MATLAB (2016b), and three numerical illustrations are done for the systematic implementation. The non-dominance of obtained solutions is shown by comparison with the existing algorithm and by taking some feasible solution points from the feasible space in the neighborhood of obtained global efficient solution. This shows the superiority of the developed method.

Keywords

Multi-objective programming Sum-of-ratio Multi-objective linear fractional programming Duality Branch and bound 

Notes

Acknowledgements

This work is financially supported by DST-SERB, Government of India, Vide Sanction No. SB/EMEQ - 049/2014. Also, Authors would like to acknowledge the help of Mr. Abhishek Chaurasiya, B.Tech. final year student of MNNIT Allahabad for his help in developing the MATLAB code of the method.

Compliance with ethical standards

Conflict of interest

All the authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Benson HP (2001) Global optimization algorithm for the non-linear sum of ratios problem. J Math Anal Appl 263:301–315MathSciNetCrossRefGoogle Scholar
  2. Benson HP (2002) Global optimization algorithm for the non-linear sum of ratios problem. J Optim Theory Appl 112(1):1–29MathSciNetCrossRefzbMATHGoogle Scholar
  3. Benson HP (2007a) A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratio problem. Eur J Oper Res 182:597–611MathSciNetCrossRefzbMATHGoogle Scholar
  4. Benson HP (2007b) Solving sum of ratios fractional programs via concave minimization. J Optim Theory Appl 135:1–17MathSciNetCrossRefzbMATHGoogle Scholar
  5. Benson HP (2010) Branch-and-bound outer approximation algorithms for sum-of-ratios fractional programs. J Optim Theory Appl 146:1–18MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bhati D, Singh P (2016) Branch and bound computational method for multi-objective linear fractional optimization problem. Neural Comput Appl.  https://doi.org/10.1007/s00521-016-2243-6 Google Scholar
  7. Chen HJ (2009) Generic algorithm for generalized fractional programming. J Optim Theory Appl 141:93–105MathSciNetCrossRefzbMATHGoogle Scholar
  8. Freund RW, Jarre F (2001) Solving the sum-of-ratios problem by an interior-point method. J Glob Optim 19:83–102MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gao Y, Jin S (2013) A global optimization algorithm for sum of linear ratios problem. J Appl Math.  https://doi.org/10.1155/2013/276245 MathSciNetzbMATHGoogle Scholar
  10. Horst R, Tuy H (1996) Global optimization: deterministic approaches. Springer, BerlinCrossRefzbMATHGoogle Scholar
  11. Jaberipour M, Khorram E (2010) Solving the sum-of-ratios problem by a harmony search algorithm. J Comput Appl Math 234:733–742MathSciNetCrossRefzbMATHGoogle Scholar
  12. Jiao HW, Liu SY (2015) A practicable branch and bound algorithm for sum of linear ratios problem. Eur J Oper Res 243:723–730MathSciNetCrossRefzbMATHGoogle Scholar
  13. Jiao H, Shen P (2007) A note on the paper global optimization of non-linear sum of ratios. Appl Math Comput 188:1812–1815MathSciNetzbMATHGoogle Scholar
  14. Jin L, Hou XP (2014) Global optimization for a class non-linear sum of ratios problems. Probl Eng.  https://doi.org/10.1155/2014/103569 Google Scholar
  15. Kanno H, Tsuchiya K, Yamamoto R (2007) Minimization of ratio of function defined as sum of the absolute values. J Optim Theory Appl 135:399–410MathSciNetCrossRefzbMATHGoogle Scholar
  16. Qu SJ, Zhang KC, Zhao JK (2007) An efficient algorithm for globally minimizing sum of quadratics ratios problem with non-convex quadratics constraints. Appl Math Comput 189:1624–1636MathSciNetzbMATHGoogle Scholar
  17. Schaible S (1977) A note on the sum of a linear and linear fractional functions. Naval Res Logist Q 24:61–963CrossRefGoogle Scholar
  18. Schaible S, Shi J (2003) Fractional programming: the sum-of-ratio case. Optim Method Softw 18(2):219–229MathSciNetCrossRefzbMATHGoogle Scholar
  19. Scott CH, Jefferson TR (1998) Duality of non-convex sum of ratios. J Optim Theory Appl 98(1):151–159MathSciNetCrossRefzbMATHGoogle Scholar
  20. Shen PP, Jin L (2010) Using canonical partition to globally maximizing the non-linear sum of ratios. Appl Math Model 34:2396–2413MathSciNetCrossRefzbMATHGoogle Scholar
  21. Shen PP, Wang CF (2006) Global optimization for sum of ratios problem with coefficient. Appl Math Comput 176:219–229MathSciNetzbMATHGoogle Scholar
  22. Shen PP, Wang CF (2008) Global optimization for sum of generalization fractional functions. J Comput Appl Math 214:1–12MathSciNetCrossRefzbMATHGoogle Scholar
  23. Shen PP, Chen Y, Yuan M (2009a) Solving sum of quadratic ratios fractional programs via monotonic function. Appl Math Comput 212:234–244MathSciNetzbMATHGoogle Scholar
  24. Shen PP, Duan YP, Pei YG (2009b) A simplicial branch and duality bound algorithm for the sum of convex–convex ratios problem. J Comput Appl Math 223:145–158MathSciNetCrossRefzbMATHGoogle Scholar
  25. Shen PP, Li W, Bai X (2013) Maximizing for the sum of ratios of two convex functions over a convex set. Comput Oper Res 40:2301–2307MathSciNetCrossRefzbMATHGoogle Scholar
  26. Singh P, Dutta D (2012) Sum of ratios multi-objective programming problem: a fuzzy goal programming approach. Nonlinear Dyn Syst Theory 12(3):289–302MathSciNetzbMATHGoogle Scholar
  27. Singh S, Gupta P (2010) On multiparametric analysis in sum-of-ratios programming. In: Proceeding of the international multiconference of engineers and computer scientist, IMECS-2010, Hong Kong, 17–19 Mar 2010Google Scholar
  28. Wang YJ, Zhang KC (2004) Global optimization of non-linear sum of ratios problem. Appl Math Appl 158:319–330MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Deepika Agarwal
    • 1
  • Pitam Singh
    • 1
    Email author
  • Deepak Bhati
    • 2
  • Saru Kumari
    • 3
  • Mohammad S. Obaidat
    • 4
    • 5
  • Fellow of IEEE and Fellow of SCS
  1. 1.Department of MathematicsMotilal Nehru National Institute of Technology AllahabadAllahabadIndia
  2. 2.Department of Mathematics, Shyama Prasad Mukharji CollegeUniversity of DelhiNew DelhiIndia
  3. 3.Department of MathematicsChaudhary Charan Singh UniversityMeerutIndia
  4. 4.ECE DepartmentNazarbayev UniversityAstanaKazakhstan
  5. 5.King Abdullah II School of Information Technology The University of JordanAmmanJordan

Personalised recommendations