Solving linear fractional programming problem in uncertainty environment with grey parameters

Author
farid pourofoghi
Department of Mathematics, Payame Noor University, Tehran, Iran
Abstract
The fractional programming problem is an important nonlinear programming tool used in various fields such as resource allocation, transportation, production planning, etc. Due to the uncertainty in real-world problems, it is very difficult to determine the definite coefficients for the mathematical model of the problems. Therefore, the coefficients of the mathematical model of the problems are considered as uncertain. One of the approaches to dealing with uncertainty, in addition to probability, stochastic, fuzzy theory, is the theory of grey systems. In this paper, a linear fractional programming problem with grey coefficients in the objective function is considered. To solve these problems, the grey parameter whitening method is often used, which causes the solution obtained in this method not to reflect the uncertainty of the grey parameters in the optimal solution. To solve this problem, in this paper, an algorithm is presented, based on which, according to the type of objective function of the grey linear fractional programming problem, it is converted into two grey linear programming subproblems, and then the solution of the grey linear fractional programming problem is determined by solving those two subproblems. By implementing the proposed method, the solution of the grey linear fractional programming problem is determined as interval grey numbers, and as a result, the uncertainty in the objective function is reflected in the final result. Finally, to demonstrate the effectiveness of the proposed method, an example has been solved with the proposed method. To demonstrate the effectiveness of the proposed method, the solution obtained from the proposed method was evaluated with the solutions obtained from other methods, for example, the two grey number ranking methods of Hu and Wang and the center and grey degree. It was shown that the solution obtained from the proposed method is better than other solution methods.
Keywords
Uncertainty
Fractional Programming
Grey System
Grey Interval Numbers
1. Charnes A., Cooper W.W., Rhodes E., "Measuring the efficiency of decision making units", European journal of operational research, 2 (1978), 429‌-444.



2. Craven B., "Fractional Programming". Heldermann Verlag, (1988).



3. Schaible S., "Fractional programming, in Handbook of global optimization", Springer. (1995).



4. Das S. K., Mandal, T., Edalatpanah S. A., "Optimal solution of neutrosophic linear fractional programming problems with mixed constraints", Research Square, 26, 17 (2022), 8699-8707.



5. Borza M., Sham Rambely A., Edalatpanah S. A., "A Linearization to the Multi-objective Linear Plus Linear Fractional Program", SN Operations Research Forum, Springer, 4 (2023), 1-22.



6. Bajalinov E.B., "Linear‌ Fractional Programming Theory, Methods, Applications and Software", Springer Science and Business Media, (2013).



7. Bitran G., Novaes A., "Linear programming with a fractional objective function", Operations Research, 21 (1973), 22–29.



8. Charnes A., Cooper W.W., "Programming with linear fractional functionals", Naval Research logistics quarterly, 9 (1962), 181–186.



9. Chinnadurai V., Muthukumar S., "Solving the linear fractional programming problem in a fuzzy environment: Numerical approach", Applied Mathematical Modelling, 40 (2016), 6148-‌6164.



10. Stancu‌-Minasian I. M., "Fractional programming: Theory, methods and applications", Dordrecht: Kluwer Academic Publishers, (1997).



11. Omer Osman O., Sulaiman N., "Solving Linear Fractional Programming Problems Via Revised Simplex Method", EURASIAN JOURNAL OF SCIENCE AND ENGINEERING, 10 (2025), 54-62.

12. Khastan A., Jimenez B.H. Moreno A.B., "On the new solution to interval linear fractional programming problems", Evolutionary Intelligence, 17 (2024), 4001-4005.

13.Das S.K., Mandal T., "A single stage single constraints linear fractional programming problem: An approach". Operation Researh and Appllication: An International Journal, 1 (2015), 1‌-5.



14. Zadeh L.A., "Fuzzy sets as a basis for a theory of possibility", Fuzzy sets and systems, 1 (1978), 3-‌28.



15. Das S. K., Edalatpanah S. A. Mandal, T., "A proposed model for solving fuzzy linear fractional programming problem: Numerical Point of View", Journal of Computational Science, 25 (2018), 367‌-375.



16. Das S. K., Mandal T., Behera D., "A new approach for solving fully fuzzy linear programming problem", International Journal of Mathematics in Operational Research, 15 (2019), 296-‌309.



17. Das S. K., Edalatpanah S. A., Mandal, T., "Development of Unrestricted Fuzzy Linear Fractional Programming Problems Applied in Real Case", Fuzzy Information and Engineering, 13 (2021), 184-195.



18. Maiti I., Pramanik S., Mandal T., Das S. K., "Solving Multi‌objective linear fractional programming problem based on Stanojevic’s normalization technique under fuzzy environment". International Journal of Operation Research, 42 (2021), 543-564.



19. Das S. K., Mandal T. Edalatpanah S. A., "A new approach for solving fully fuzzy linear fractional programming problems using the Multi objective linear programming problem", RAIRO Operation Research, 51 (2017), 285-‌297.



20. Srinivasan R., "On solving fuzzy linear fractional programming in material aspects", Materials Today: Proceedings, 21 (2019), 155‌-157.



21. Deng J.L., "The control problems of grey systems", Systems and Control Letters, 1 (1982), 288‌-294.



22. Liu S., Yang Y., Xie N., Forrest J., "New progress of grey system theory in the new millennium", Grey Systems: Theory and Application, 6 (2016), 2-‌31.



23. Pourofoghi F., Darvishi D., "Solving Linear Fractional Programming Problems in Uncertain Environments: A Novel Approach with Grey Parameters", Control and Optimization in Applied Mathematics – COAM, 9 (2024), 169-183.

24. Nasseri S.H., Yazdani A.B., Darvishi D., "A primal simplex algorithm for solving linear programming problem with grey cost coefficients", Journal of New Researches in Mathematics, 1 (2016), 121-‌142.



25. Saffar Ardabili J., Darvishi D., Pourofoghi F., "Application of center and width concepts to solving grey linear programming", International Journal of Applied and Computational Mathematics, 6 (2020), 1‌-12.



26. Darvishi D., Nasseri S.H., "A dual simplex method for grey linear programming problems based on duality results", Grey Systems: Theory and Application, 30 (2018), 127-142.



27. Liu S., Yang Y., Forrest J., "Grey Data Analysis, Springer". Singapore. (2017).



28. Nasseri S.H., Darvishi D., Yazdani A., "A new approach for solving grey assignment problems", Control and Optimization in Applied Mathematics, 2 (2017), 15‌-28.



29. Nasseri S.H., Darvishi D., "Duality results on grey linear programming problems", The Journal of Grey System, 30 (2018), 127-‌142.



30. Shafiei Nikabadi M., Helalian L., "Rating Supply Chain Risks Using the Combined Approach of FMEA Optimization and Gray Theory (Case Study: Mashhad Food Industry Units)", Journal of Decisions and operation research, 8 (2023), 872-885.



31. Pourofoghi F., Darvishi D., Saffar Ardabili J., " A New Approach to Finding the Answer of Transportation Problems with Grey Parameters", Operational Research and Its Applications, 18 (2021), 59‌-73.



32. Pourofoghi F., Saffar Ardabili J. Darvishi D., "A New Approach for Finding an Optimal Solution for Grey Transportation Problem", Nonlinear Analysis in Engineering and Sciences, 10 (2019), 83-‌95.



33. Pourofoghi F., Darvishi D., "Applying Duality Results to Solve the Linear Programming Problems with Grey Parameters", Control and Optimization in Applied Mathematics (COAM), 5 (2021), 15‌-28.



34. Pourofoghi F., "Solving the Problem of Linear Programming with Cost Coefficients and Grey Resources Using the Primal Simplex Algorithm", Big Data and Computing Visions, 2 (2022), 80-89.



35. Mahmoudi A., Liu S., Javed SA., Abbasi M., "A novel method for solving linear programming with grey parameters", Journal of Intelligent & Fuzzy Systems, 36 (2019), 161-172.



36. Baidya A., Bera U.K., Maiti M., "The grey linear programming approach and its application to multi‌-objective multi‌-stage solid transportation problem", Operation research, 53 (2016), 500–522.



37. Moore R.E., Kearfott R.B., Cloud M.J., "Introduction to Interval Analysis, SIAM Press", Philadelphia. (2009).



38. Darvishi D., Forrest J., Liu S., "A comparative analysis of grey ranking approaches", Grey Systems: Theory and Application, 9 (2019), 472‌-487.



39. Hu B.Q., Wang S., "A novel approach in uncertain programming part I: new arithmetic and order relation for interval numbers", Journal of Industrial and Management Optimization, 2 (2006), 351-‌371.