Soft Computing

, Volume 22, Issue 17, pp 5901–5919 | Cite as

Two novel combined approaches based on TLBO and PSO for a partial interdiction/fortification problem using capacitated facilities and budget constraint

  • Raheleh Khanduzi
  • H. Reza Maleki
  • Reza Akbari
Methodologies and Application


In this paper, we introduce a novel continuous bi-level programming model of an interdiction/fortification problem, referred to as partial interdiction/fortification problem with capacitated facilities and Budget constraint (PIFCB). PIFCB is modeled as a Stackelberg game between a defender (leader) and an attacker (follower). The decisions of the defender are related to find optimal allocations of defensive resources as well as customer-facility assignments. So, the total system losses and demand-weighted distance are minimized. Following this action, the attacker seeks facilities to interdict for the capacity or service reduction and maximal losses resulting from a limited offensive budget. After modeling this problem, two combined methods, entitled particle swarm optimization with CPLEX (PSO-CPLEX) and teaching learning-based optimization with CPLEX (TLBO-CPLEX), are devised as solution procedures. For this bi-level programming problem, we executed two approaches which search the solution space of the defender’s subproblem, according to PSO and TLBO principles, and corresponding attacker’s subproblem is solved using CPLEX. To investigate the suggested methods, a comprehensive computational study is conducted and the effectiveness of the methods is compared together. The experimental results show the superiority of TLBO-CPLEX against PSO-CPLEX.


Continuous bi-level model Stackelberg game Capacitated facility Partial offense and defense Particle swarm optimization Teaching learning-based optimization 



The first author would like to thank Gonbad Kavous University for supporting this research work. The second and third authors would like to appreciate the research council of Shiraz University of Technology for supporting this research. Last but not least, the authors are very grateful to the editor and the reviewers for their valuable comments.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


  1. Aksen D, Piyade N, Aras N (2010) The budget constrained r-interdiction median problem with capacity expansion. Central Eur J Oper Res 18(3):269–291MathSciNetCrossRefzbMATHGoogle Scholar
  2. Aksen D, Aras N, Piyade N (2013) A bilevel p-median model for the planning and protection of critical facilities. J Heuristics 19:373–398CrossRefGoogle Scholar
  3. Aksen D, Akca SS, Aras N (2014) A bilevel partial interdiction problem with capacitated facilities and demand outsourcing. Comput Oper Res 41(1):346–358MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bard JF (1998) Practical bilevel optimization: algorithms and applications. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  5. Chen D, Zou F, Wang J, Yuan W (2015) A teaching–learning-based optimization algorithm with producer–scrounger model for global optimization. Soft Comput 19:745–762CrossRefGoogle Scholar
  6. Church RL, Scaparra MP, Middleton RS (2004) Identifying critical infrastructure: the median and covering facility interdiction problems. Ann Assoc Am Geogr 94(3):491–502CrossRefGoogle Scholar
  7. Church RL, Scaparra MP (2007) Protecting critical assets: the r-interdiction median problem with fortification. Geogr Anal 39(2):129–46CrossRefGoogle Scholar
  8. Colson B, Marcotte P, Savard G (2005) Bilevel programming: a survey. 4OR Q J Oper Res 3(2):87–107MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hakimi SL (1964) Optimum locations of switching centers and the absolute centers and medians of a graph. Oper Res 12(3):450–459CrossRefzbMATHGoogle Scholar
  10. Hansen P, Jaumard B, Savard G (1992) New branch-and-bound rules for linear bilevel programming. SIAM J Sci Stat Comput 13:1194–1217MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hassan MY, Suharto MN, Abdullah MP, Majid MS, Hussin F (2012) Application of particle swarm optimization for solving optimal generation plant location problem. Int J Electr Electron Syst Res 5:47–56Google Scholar
  12. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: IEEE international conference on neural networks, vol 4, pp 1942–1948Google Scholar
  13. Liberatore F, Scaparra MP, Daskin MS (2012) Hedging against disruptions with ripple effects in location analysis. Omega 40(10):21–30CrossRefGoogle Scholar
  14. Rao RV, Kalyankar VD (2013) Parameter optimization of modern machining processes using teaching–learning-based optimization algorithm. Eng Appl Artif Intell 26:524–531CrossRefGoogle Scholar
  15. Rao RV, Patel V (2012) An elitist teaching–learning-based optimization algorithm for solving complex constrained optimization problems. Int J Ind Eng Comput 3:535–560Google Scholar
  16. Rao RV, Patel V (2013) Comparative performance of an elitist teaching–learning-based optimization algorithm for solving unconstrained optimization problems. Int J Ind Eng Comput 4:29–50Google Scholar
  17. Rao RV, Savsani VJ, Vakharia DP (2011) Teaching-learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43:303–315CrossRefGoogle Scholar
  18. Rao RV, Savsani VJ, Vakharia DP (2012) Teaching–learning-based optimization: an optimization method for continuous non-linear large scale problem. Inf Sci 183:1–15MathSciNetCrossRefGoogle Scholar
  19. Ren A, Wang Y, Xue X (2014) An interval programming approach for the bilevel linear programming problem under fuzzy random environments. Soft Comput 18:995–1009CrossRefzbMATHGoogle Scholar
  20. Satapathy SC, Naik A (2012) Improved teaching learning based optimization for global function optimization. Decis Sci Lett 2:23–34CrossRefGoogle Scholar
  21. Scaparra MP, Church RL (2008) A bilevel mixed integer program for critical infrastructure protection planning. Comput Oper Res 35(6):1905–23CrossRefzbMATHGoogle Scholar
  22. Scaparra MP, Church RL (2012) Protecting supply systems to mitigate potential disaster: a model to fortify capacitated facilities. Int Reg Sci Rev 35:188–210CrossRefGoogle Scholar
  23. Stackelberg H (1952) The theory of market economy. Oxford University Press, OxfordGoogle Scholar
  24. Talbi EG (2013) Metaheuristics for bi-level optimization. Springer, Berlin HeidelbergCrossRefzbMATHGoogle Scholar
  25. Weber A (1909) Über den Standort der Industrien. Reine Theorie des Standorts, Mohr, Tübingen, TeilGoogle Scholar
  26. White JA, Case KE (1974) On covering problems and the central facilities location problems. Geogr Anal 6(3):281–294CrossRefGoogle Scholar
  27. Yang XS (2010) Nature-inspired metaheuristic algorithm. Luniver Press, FromeGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsGonbad Kavous UniversityGonbad KavousIran
  2. 2.Faculty of MathematicsShiraz University of TechnologyShirazIran
  3. 3.Department of Computer Engineering and Information TechnologyShiraz University of TechnologyShirazIran

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