Iranian Journal of Numerical Analysis and Optimization

Iranian Journal of Numerical Analysis and Optimization

A solution approach to the multi-level linear fractional programming problems

Document Type : Research Article

Authors
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
Abstract
In this paper, we consider multi-level linear fractional programming problems over a bounded polytope set. We present a characterization of the optimum solution to the $n$-level linear fractional programming problem for case $n > 2$. Then, we propose an extension of the $K$th-best algorithm, for solving the $n$-level linear fractional programming problem with $n > 2$, and prove its convergence. Furthermore, we consider a previously published paper on such problems. It is shown that some results and proofs presented in that paper are incorrect by providing a counterexample. Finally, some numerical examples are presented, and the results are compared to those obtained from existing methods to show the accuracy and efficiency of the proposed algorithm.
Keywords
Subjects

[1] Alguacil, N., Delgadillo, A. and Arroyo, J.M. A trilevel programming approach for electric
grid defense planning, Comput. Oper. Res., 41 (2014), 282–290.
[2] Bhargava, Sh. Solving linear fractional multi-level programs, Oper. Res. Decis., 1 (2014),
5–21.
[3] Calvete, H.I. and Galé, C. Solving linear fractional bilevel programs, Oper. Res. Lett., 32
(2004), 143–151.
[4] Cormen, T., Leiserson, C., Rivest, R. and Stein, C. Introduction to algorithms, 3rd ed., MIT
Press, Cambridge, MA, 2009.
[5] Dempe, S. and Zemkoho, A. Bilevel Optimization: Advances and challenges, Springer,
Cham, Switzerland, 2020.
[6] Fakhry, R., Hassini, E., Ezzeldin, M. and El-Dakhakhni, W. Trilevel mixed-binary linear
programming solution approaches and application in defending critical infrastructure, Eur.
J. Oper. Res., 298(3) (2022), 1114–1131.
[7] Fathy, E., Ammar, E. and Helmy, M.A. Fully intuitionistic fuzzy multilevel linear fractional
programming problem, Alexandria Eng. J., 77 (2023), 684–694.
[8] Florensa, G., Garcia-Herreros, P., Mirsa, P. et al. Capacity planning with competitive
decision-makers: Trilevel MILP formulation, degeneracy, and solution approaches, Eur.
J. Oper. Res., 262 (2017), 449–463.
[9] Han, J., Lu, J. and Zhang, G. Tri-level decision-making for decentralized vendor-managed
inventory, Inf. Sci., 421 (2017), 85–103.
[10] Henke, D., Lefebvre, H. and Schmidt, M. On coupling constraints in linear bilevel optimiza-
tion, Optim. Lett., 19(3) (2025), 689–697.
[11] Ke, G.Y. and Bookbinder, J.H. Coordinating the discount policies for retailer, wholesaler,
and less-than-truckload carrier under price-sensitive demand: A trilevel optimization ap-
proach, Int. J. Prod. Econ., 196 (2018), 82–100.
[12] Lachwani, K. and Nehra, S. Modified FGP approach and MATLAB program for solving
multi-level linear fractional programming problems, J. Ind. Eng. Int., 11 (2015), 15–36.
[13] Liu, Y.H. and Hart, S.M. Characterizing an optimal solution to the linear bilevel program-
ming problem, Eur. J. Oper. Res., 73(1) (1994), 164–166.
[14] Martos, B. Nonlinear programming: Theory and methods, North-Holland, Amsterdam, 1975.
[15] Murty, K.G. Linear programming, John Wiley & Sons, New York, 1983.
[16] Ozkok, B.A. An iterative algorithm to solve a linear fractional programming problem, Com-
put. Ind. Eng., 140 (2020), 1–15.
[17] Pal, B., Kumar, M. and Sen, S. Priority based fuzzy goal programming approach for fractional
multilevel programming problems, Int. Rev. Fuzzy Math., 6 (2011), 1–14.
[18] Sadeghi, H. and Esmaeili, M. A modification of the trilevel Kth-best algorithm, AIMS Math.,
3(4) (2018), 524–538.
[19] Sadeghi, H. and Esmaeili, M. On the quasi-concave multi-level programming problem, Asia-
Pac. J. Oper. Res., 39(3) (2022), 2150026.
[20] Sana, S.S. A production-inventory model of imperfect quality products in a three-layer supply
chain, Decis. Support Syst., 50 (2011), 539–547.
[21] Stancu-Minasian, I.M. A ninth bibliography of fractional programming, Optim., 68 (2019),
2125–2169.
[22] Wang, G., Jiang, B., Zhu, K. and Wan, Z. Global convergent algorithm for the bilevel linear
fractional programming based on modified convex simplex method, J. Syst. Eng. Electron.,
21(2) (2010), 239–243.
[23] Xu, X., Meng, Z. and Shen, R. A tri-level programming model based on conditional value-
at-risk for three-stage supply chain management, Comput. Ind. Eng., 66 (2013), 470–475.
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