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

## 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.

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