Advertisement

A robust optimization model for p-median problem with uncertain edge lengths

  • Mohammad Ebrahim Nikoofal
  • Seyed Jafar Sadjadi
ORIGINAL ARTICLE

Abstract

In this paper, we propose a p-median problem with uncertain edge lengths where uncertainty is characterized by given intervals. The uncertainty in edge lengths may appear in transportation costs or travel times along the edges in any network location problem. Minimax regret approach is a promising tool to cope with uncertainty in network location problems. However, minimax regret algorithms normally suffer from complexity, and they are time consuming. We propose a robust optimization approach to obtain the robust linear counterpart for the same class of the nominal p-median problem. The performance of the proposed model is compared with minimax regret approach through a simple but illustrative example, and results are discussed in more details.

Keywords

Location Networks Robust optimization Linearization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hakimi SL (1964) Optimum locations of switching centers and the absolute centers and medians of a graph. Oper Res 12:450–459zbMATHCrossRefGoogle Scholar
  2. 2.
    Hakimi SL (1965) Optimum distributions of switching centers in a communications network and some related graph theoretic problems. Oper Res 13:462–475zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Hakimi SL, Maheshwari SN (1972) Optimum locations of centers in networks. Oper Res 20:967–973zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Wendell RE, Hurter AP (1973) Location theory, dominance and convexity. Oper Res 21:314–320zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Handler GY, Mirchandani PB (1979) Location on networks, theory and applications. MIT Press, Cambridge, MAGoogle Scholar
  6. 6.
    Berman O, Krass D (2002) Facility location problems with stochastic demands and congestion. In: Drezner Z, Hamacher HW (eds) Location analysis: applications and theory, 329–371 (Chapter 11)Google Scholar
  7. 7.
    Berman O, Wang J (2004) Probabilistic location problems with discrete demand weights. Networks 57:47–57CrossRefMathSciNetGoogle Scholar
  8. 8.
    Berman O, Wang J (2006) The 1-median and 1-antimedian with uniform distributed demand. INFORGoogle Scholar
  9. 9.
    Drezner T, Drezner Z, Shiode S (2002) A threshold satisfying competitive location model. J Reg Sci 42:287–299CrossRefGoogle Scholar
  10. 10.
    Frank H (1966) Optimum locations on graphs with correlated normal demands. Oper Res 15:552–557CrossRefGoogle Scholar
  11. 11.
    Drezner Z (1985) Sensitivity analysis of the optimal location of a facility. Nav Res Logist Q 32:209–224zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Current JR, Ratick S, ReVelle CS (1998) Dynamic facility location when the total number of facilities is uncertain: a decision analysis approach. Eur J Oper Res 110:597–609zbMATHCrossRefGoogle Scholar
  13. 13.
    Berman O, Drezner Z (2008) The p-median problem under uncertainty. Eur J Oper Res 189:19–30zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kouvelis P, Vairaktarakis G, Yu G (1994) Robust 1-median location on a tree in the presence of demand and transportation cost uncertainty. Working Paper 93/94-3-4, Department of Management Science and Information Systems, Graduate School of Business, The University of Texas at Austin, Austin, TXGoogle Scholar
  15. 15.
    Chen B, Lin C (1997) Minmax-regret robust one-median location on a tree. Networks 31:93–103CrossRefGoogle Scholar
  16. 16.
    Averbakh I, Berman O (2000) Minmax regret median location on a network under uncertainty. INFORMS J Comput 12:104–110zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Averbakh I, Berman O (2000) Algorithms for the robust 1-center problem. Eur J Oper Res 123:292–302zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Averbakh I, Berman O (2003) An Improved algorithm for the minmax regret median problem on a tree. Networks 41(2):97–103zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Burkard R, Dollani H (2001) Robust location problems with pos/neg weights on a tree. Networks 38:102–113zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Serra D, Marianov V (1998) The p-median problem in a changing network: the case of Barcelona. Location Sci 6:383–394CrossRefGoogle Scholar
  21. 21.
    Yaman H, Karaşan OE, Pinar MÇ (2007) Restricted robust uniform matroid maximization under interval uncertainty. Math Program 110:430–441CrossRefGoogle Scholar
  22. 22.
    Beltran C, Tadonki C, Vial JPH (2006) Solving the p-median problem with a semi-lagrangian relaxation. Comput Optim Appl 35:239–260zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Jabalameli MS, Ghaderi A (2008) Hybrid algorithms for the uncapacitated continuous location-allocation problem. The International Journal of Advanced Manufacturing 37:202–209CrossRefGoogle Scholar
  24. 24.
    Wu TH, Low C, Wu WT (2004) A tabu search approach to the cell formation problem. The International Journal of Advanced Manufacturing 23:916–924Google Scholar
  25. 25.
    Kim CO, Baek J-G, Jun J (2005) A machine cell formation algorithm for simultaneously minimising machine workload imbalances and inter-cell part movements. The International Journal of Advanced Manufacturing 3:268–275CrossRefGoogle Scholar
  26. 26.
    Avella P, Sassano A, Vasil’ev I (2007) Computational study of large-scale p-median problems. Math Program 109:89–114zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Szwarc D, Rajamani D, Bector CR (1997) Cell formation considering fuzzy demand and machine capacity. The International Journal of Advanced Manufacturing 13:134–147CrossRefGoogle Scholar
  28. 28.
    Silva EF, Wood RK (2006) Solving a class of stochastic mixed-integer programs with branch and price. Math Program 108:395–418zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Soyster AL (1973) Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper Res 21:1154–1157zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Ben-Tal A, Nemirovski A (1998) Robust convex optimization. Math Oper Res 23:769–805zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Ben-Tal A, Nemirovski A (1999) Robust solutions to uncertain programs. Oper Res Lett 25:1–13zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Snyder LV (2006) Facility location under uncertainty: a review. IIE Trans 38(7):537–554CrossRefMathSciNetGoogle Scholar
  33. 33.
    ReVelle C, Swain RW (1970) Central facilities location. Geogr Anal 2:30–42Google Scholar
  34. 34.
    Bertsimas D, Thiele A (2006) A robust optimization approach to inventory theory. Operations Research, Informs 54(1):150–168zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Averbakh I (2003) Complexity of robust single facility location problems on networks with uncertain edge lengths. Discrete Appl Math 127:505–522zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  • Mohammad Ebrahim Nikoofal
    • 1
  • Seyed Jafar Sadjadi
    • 1
  1. 1.Department of Industrial EngineeringIran University of Science and TechnologyNarmak, TehranIran

Personalised recommendations