AIMS Mathematics, 2018, 3(4): 524-538. doi: 10.3934/Math.2018.4.524.

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

A modification of the trilevel Kth-Best algorithm

Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran

In this paper, we present a modification of the traditional linear trilevel Kth-Best algorithm. The proposed modified Kth-Best algorithm considers the linear trilevel programming problems in which the middle level and the lower level problems are unbounded or their objective functions are inconsistant. These cases are not considered in the trilevel Kth-Best algorithm proposed by Zhang et al. Moreover, we discuss some geometric properties of a linear trilevel programming problem wherein each decision maker might have his (her) own restrictions and the upper level objective function contain lower level variables. Finally, a number of numerical examples are presented and the results are verified as well.
  Article Metrics

Keywords multilevel programming; trilevel programming; Kth-Best algorithm

Citation: Habibe Sadeghi, Maryam Esmaeili. A modification of the trilevel Kth-Best algorithm. AIMS Mathematics, 2018, 3(4): 524-538. doi: 10.3934/Math.2018.4.524


  • 1. N. Alguacil, A. Delgadillo, J. M. Arroyo, A trilevel programming approach for electric grid defense planning, Comput. Oper. Res., 41 (2014), 282–290.    
  • 2. J. F. Bard, An investigation of the linear three level programming problem, IEEE T. Syst. Man Cy., 14 (1984), 711–717.    
  • 3. J. F. Bard, Practical Bilevel Optimization, Kluwer, Dordrecht, Netherland, 1998.
  • 4. M. S. Bazaraa, H. D. Sheraly, C. M. Shetty, Nonlinear Programming, Theory and Algorithms, Wiley Interscience, New Jersy, 2006.
  • 5. N. P. Faisca, P. M. Saraiva, B. Rustem, E. N. Pistikopoulos, A multi-parametric programming approach for multilevel hierarchical and decentralised optimisation problems, Computational Management Science, 6 (2009), 377–397.    
  • 6. G. Florensa, P. Garcia-Herreros, P. Mirsa, et al. Capacity planning with competitive decisionmakers: Trilevel MILP formulation, degeneracy, and solution approaches, Eur. J. Oper. Res., 262 (2017), 449–463.    
  • 7. J. Han, J. Lu, Y. Hu, et al. Tri-level decision-making with multiple followers: Model, algorithm and case study, Inform. Sci., 311 (2015), 182–204.    
  • 8. J. Han, G. Zhang, Y. Hu, et al. A solution to bi/tri-level programming problems using particle swarm optimization, Inform. Sci., 370 (2016), 519–537.    
  • 9. J. Han, J. Lu, G. Zhang, Tri-level decision-making for decentralized vendor-managed inventory, Inform. Sci., 421 (2017), 85–103    
  • 10. G. Y. Ke, J. H. Bookbinder, Coordinating the discount policies for retailer, wholesaler, and less-than-truckload carrier under price-sensitive demand: A trilevel optimization approach, Int. J. Prod. Econ., 196 (2018), 82–100.    
  • 11. J. Lu, J. Han, Y. Hu, et al. Multilevel Decision-Making: A Survey, Inform. Sci., 346 (2016), 463–487.    
  • 12. R. T. Rockafellar, Convex Analysis, Prinston University Press, Prinston, USA, 1970.
  • 13. M. Sakawa, I. Nishizaki, Interactive fuzzy programming for multi-level programming problems: a review, International journal of multicriteria decision making, 2 (2012), 241–266.    
  • 14. S. S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decis. Support Syst., 50 (2011), 539–547.    
  • 15. U.Wen,W. Bialas, The hybrid algorithm for solving the three level programming problem, Comput. Oper. Res., 13 (1986), 367–377.    
  • 16. D. White, Penalty function approach to linear trilevel programming, J. Optimiz. Theory App., 93 (1997), 183–197.    
  • 17. X. Xu, Z. Meng, R. Shen, A tri-level programming model based on conditional value-at-risk for three-stage supply chain management, Comput. Ind. Eng., 66 (2013), 470–475.    
  • 18. Y. Yao, T. Edmunds, D. Papageorgiou, et al. Trilevel optimization in power network defense, IEEE T. Syst. Man Cy., 37 (2007), 712–718.    
  • 19. G. Zhang, J. lu, J. Montero, et al. Model, solution concept, and K-th Best algorithm for linear trilevel programming, Inf. Sci., 180 (2010), 481–492.    


Reader Comments

your name: *   your email: *  

© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved