Public Transport

, Volume 11, Issue 2, pp 321–340 | Cite as

Locating key stations of a metro network using bi-objective programming: discrete and continuous demand mode

  • Seyed Sina MohriEmail author
  • Meisam Akbarzadeh
Original Paper


This study proposes two bi-objective optimization problems for locating key stations of a metro network in both discrete and continuous demand modes. Traditionally, designing a metro network based on optimization techniques consists of two approaches. The first approach locates a number of alignments and their stations simultaneously, while the second approach involves locating key stations, designing a core network, and locating secondary stations. In locating key stations processed by a single objective model, the number of produced and attracted trips to the key stations is maximized. This paper considers a second objective for this stage to maximize the coverage of key stations on origin/destination (OD) trips. A fuzzy goal programming model is established to solve the bi-objective model and provide some Pareto-optimal solutions. The previous single objective model and the proposed model with continuous demand mode are applied to a real network. Results show that the proposed model significantly increases the coverage of key stations on OD trips with only a slight reduction in the number of produced and attracted trips.


Rail rapid transit Metro network design Fuzzy goal programming 

List of symbols



The set of proposed stations


The set of demand points


The set of demand areas


The set of catchment levels


The index for proposed stations


The index for demand points/areas


The index for catchment levels

Input parameters


The coverage of station s of network demand points


The coverage of stations s and \(s^{\prime}\) of demand flow i to j where s and \(s^{\prime}\) are situated around the demand points/areas i and j, respectively


The coverage of stations s and \(s^{\prime}\) of total demand flows


The number of key stations in the network


The cover intensity at the catchment level l


The total number of trip production and attraction in the demand point i


The demand flow from point/area i to point/area j


A binary input parameter, which is equal to one, if station i belongs to catchment level l of station s, and zero, otherwise


The Euclidian distance between stations s and \(s^{\prime}\)


The minimum permitted Euclidian distance between two key stations


A constant value based on the gravity model


The total area of demand area i


The common area of demand area i and catchment area l of station s

Decision variables


A binary decision variable; it is equal to one if station s is selected as key station; otherwise it is equal to zero


A binary decision variable; it is equal to one if both stations s and \(s^{\prime}\) are selected as key stations; otherwise it is equal to zero



The authors are grateful for the helpful comments of two anonymous referees that have allowed us to improve upon the original version. The authors also acknowledge the help of Dr. Hossein Haghshenas from the Isfahan University of Technology for making the data set available for this study.


  1. Alexandris G, Giannikos I (2010) A new model for maximal coverage exploiting GIS capabilities. Eur J Oper Res 202(2):328–338CrossRefGoogle Scholar
  2. Amid A, Ghodsypour S, O’Brien C (2011) A weighted max–min model for fuzzy multi-objective supplier selection in a supply chain. Int J Prod Econ 131(1):139–145CrossRefGoogle Scholar
  3. Bay P (1985) Determining cost-effectiveness of transit systems. Transportation Research Board state-of-the-art report, pp 9–12Google Scholar
  4. Berman O, Krass D (2002) The generalized maximal covering location problem. Comput Oper Res 29(6):563–581CrossRefGoogle Scholar
  5. Berman O, Krass D, Drezner Z (2003) The gradual covering decay location problem on a network. Eur J Oper Res 151(3):474–480CrossRefGoogle Scholar
  6. Blackledge D, Humphreys E (1984) The West Midland rapid transit study. In: Proceedings of the Planning and Transport Research and Computation Ltd., Sussex, pp 71–84Google Scholar
  7. Bruno G, Ghiani G, Improta G (1998) A multi-modal approach to the location of a rapid transit line. Eur J Oper Res 104(2):321–332CrossRefGoogle Scholar
  8. Bruno G, Gendreau M, Laporte G (2002) A heuristic for the location of a rapid transit line. Comput Oper Res 29(1):1–12CrossRefGoogle Scholar
  9. Chen L-H, Tsai F-C (2001) Fuzzy goal programming with different importance and priorities. Eur J Oper Res 133(3):548–556CrossRefGoogle Scholar
  10. Curtin KM, Biba S (2011) The transit route arc-node service maximization problem. Eur J Oper Res 208(1):46–56CrossRefGoogle Scholar
  11. Dufourd H, Gendreau M, Laporte G (1996) Locating a transit line using tabu search. Location Sci 4(1):1–19CrossRefGoogle Scholar
  12. Escudero L, Muñoz S (2009) An approach for solving a modification of the extended rapid transit network design problem. Top 17(2):320–334CrossRefGoogle Scholar
  13. Gutiérrez-Jarpa G, Obreque C, Laporte G, Marianov V (2013) Rapid transit network design for optimal cost and origin–destination demand capture. Comput Oper Res 40(12):3000–3009CrossRefGoogle Scholar
  14. Gutiérrez-Jarpa G, Laporte G, Marianov V (2018) Corridor-based metro network design with travel flow capture. Comput Oper Res 89:58–67CrossRefGoogle Scholar
  15. Hannan EL (1981a) Linear programming with multiple fuzzy goals. Fuzzy Sets Syst 6(3):235–248CrossRefGoogle Scholar
  16. Hannan EL (1981b) On fuzzy goal programming. Decision Sci 12(3):522–531CrossRefGoogle Scholar
  17. Hwang CL, Masud ASM (1979) Multiple objective decision making—methods and applications: a state-of-the-art survey, Lecture notes in economics and mathematical systems, vol 164. Springer, BerlinCrossRefGoogle Scholar
  18. Isfahan University of Technology (2014) Report of comprehensive transportation study in Isfahan city. Transportation Deputy of Isfahan Municipality, IsfahanGoogle Scholar
  19. Jones K, Simmons J (1993) Location, location, location: analyzing the retail environment, 2nd edn. Routledge, LondonGoogle Scholar
  20. Karlaftis MG (2004) A DEA approach for evaluating the efficiency and effectiveness of urban transit systems. Eur J Oper Res 152(2):354–364CrossRefGoogle Scholar
  21. Laporte G, Mesa JA (2015) The design of rapid transit networks. In: Laporte G, Nickel S, Saldanha da Gama F (eds) Location science. Springer, Cham, pp 581–594Google Scholar
  22. Laporte G, Pascoal MM (2015) Path based algorithms for metro network design. Comput Oper Res 62:78–94CrossRefGoogle Scholar
  23. Laporte G, Mesa JA, Ortega FA (2002) Locating stations on rapid transit lines. Comput Oper Res 29(6):741–759CrossRefGoogle Scholar
  24. Laporte G, Mesa JA, Ortega FA, Sevillano I (2005) Maximizing trip coverage in the location of a single rapid transit alignment. Ann Oper Res 136(1):49–63CrossRefGoogle Scholar
  25. Laporte G, Marín Á, Mesa JA, Ortega FA (2007) An integrated methodology for the rapid transit network design problem. In Algorithmic methods for railway optimization. Springer, Berlin, pp 187–199CrossRefGoogle Scholar
  26. Lee JM, Lee YH (2010) Tabu based heuristics for the generalized hierarchical covering location problem. Comput Ind Eng 58(4):638–645CrossRefGoogle Scholar
  27. Mesa JA, Ortega FA (2001) Park-and-ride station catchment areas in metropolitan rapid transit systems. In: Pursula M, Niittymäki J (eds) Mathematical methods on optimization in transportation systems. Applied optimization, vol 48. Springer, Boston, pp 81–93CrossRefGoogle Scholar
  28. Miettinen K (2012) Nonlinear multiobjective optimization. Springer, BerlinGoogle Scholar
  29. Murray AT, Tong D, Kim K (2010) Enhancing classic coverage location models. Int Regional Sci Rev 33(2):115–133CrossRefGoogle Scholar
  30. Narasimhan R (1980) Goal programming in a fuzzy environment. Decision Sci 11(2):325–336CrossRefGoogle Scholar
  31. Rhode M (2014) World Metro Database. Accessed 30 Jul 2014
  32. Sadigh AN, Mozafari M, Kashan AH (2010) A mixed integer linear program and tabu search approach for the complementary edge covering problem. Adv Eng Softw 41(5):762–768CrossRefGoogle Scholar
  33. Schabas M (1988) Quantitative analysis of rapid transit alignment alternatives. Transp Q 42(3):403–416Google Scholar
  34. Siegel S (1980) Major obstacles to effective LRT surface operations. A report on light rail transit: surface operations. Transportation Research Board, pp 20–25Google Scholar
  35. Straus P (1980) Issues relating to effective LRT surface operations. A report on light rail transit: surface operations. Transportation Research Board, pp 7–15Google Scholar
  36. Tiwari R, Dharmar S, Rao J (1987) Fuzzy goal programming—an additive model. Fuzzy Sets Syst 24(1):27–34CrossRefGoogle Scholar
  37. Wulkan A, Henry L (1985) Evaluation of light rail transit for Austin, Transportation Research Board state-of-the-art report, vol 2, Texas, pp 82–90Google Scholar
  38. Zimmermann H-J (1978) Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst 1(1):45–55CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Transportation EngineeringIsfahan University of TechnologyIsfahanIran

Personalised recommendations