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Random fuzzy bilevel linear programming through possibility-based value at risk model

  • Hideki KatagiriEmail author
  • Takeshi Uno
  • Kosuke Kato
  • Hiroshi Tsuda
  • Hiroe Tsubaki
Original Article

Abstract

This article considers bilevel linear programming problems where random fuzzy variables are contained in objective functions and constraints. In order to construct a new optimization criterion under fuzziness and randomness, the concept of value at risk and possibility theory are incorporated. The purpose of the proposed decision making model is to optimize possibility-based values at risk. It is shown that the original bilevel programming problems involving random fuzzy variables are transformed into deterministic problems. The characteristic of the proposed model is that the corresponding Stackelberg problem is exactly solved by using nonlinear bilevel programming techniques under some convexity properties. A simple numerical example is provided to show the applicability of the proposed methodology to real-world hierarchical problems.

Keywords

Bilevel linear programming Random fuzzy variable Stackelberg solutions Possibility Value at risk 

References

  1. 1.
    Stackelberg H (1952) The theory of market economy. Oxford University Press, OxfordGoogle Scholar
  2. 2.
    Bracken J, McGill J (1973) Mathematical programs with optimization problems in the constraints. Oper Res 21:37–44CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bracken J, McGill J (1974) Defense applications of mathematical programs with optimization problems in the constraints. Oper Res 22:1086–1096CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bracken J, McGill J (1978) Production and marketing decisions with multiple objectives in a competitive environment. J Optim Theory Appl 24:449–458CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Candler W, Norton R (1977) Multilevel programming. Technical Report, vol 20. World Bank Development Research Center, Washington, DCGoogle Scholar
  6. 6.
    Sherali HD, Soyster AL, Murphy FH (1983) Stackelberg–Nash–Cournot equilibria: characterizations and computations. Oper Res 31:253–276CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Ackere AV (1993) The principal/agent paradigm: characterizations and computations. Eur J Oper Res 70:83–103CrossRefzbMATHGoogle Scholar
  8. 8.
    Migdalas A (1995) Bilevel programming in traffic planning: models, methods and challenge. J Global Optim 7:381–405CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cote J-P, Marcotte P, Savard G (2003) A bilevel modeling approach to pricing and fare optimization in the airline industry. J Revenue Pricing Manag 2:23–36CrossRefGoogle Scholar
  10. 10.
    Kara BY, Verter V (2004) Designing a road network for hazardous materials transportation. Transp Sci 38:188–196CrossRefGoogle Scholar
  11. 11.
    Nicholls MG (1996) The applications of non-linear bi-level programming to the aluminum industry. J Global Optim 8:245–261CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Amouzegar MA, Moshirvaziri K (1999) Determining optimal pollution control policies: an application of bilevel programming. Eur J Oper Res 119:100–120CrossRefzbMATHGoogle Scholar
  13. 13.
    Dempe S, Bard JF (2001) Bundle trust-region algorithm for bilinear bilevel programming. J Optim Theory Appl 110:265–288CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fampa M, Barroso LA, Candal D, Simonetti L (2008) Bilevel optimization applied to strategic pricing in competitive electricity markets. Comput Optim Appl 39:121–142CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Karlof JK, Wang W (1996) Bilevel programming applied to the flow shop scheduling problem. Comput Oper Res 23:443–451CrossRefzbMATHGoogle Scholar
  16. 16.
    Roghanian E, Sadjadi SJ, Aryanezhad MB (2007) A probabilistic bi-level linear multi-objective programming problem to supply chain planning. Appl Math Comput 188:786–800CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Miller T, Friesz T, Tobin R (1992) Heuristic algorithms for delivered price spatially competitive network facility location problems. Ann Oper Res 34:177–202CrossRefzbMATHGoogle Scholar
  18. 18.
    Uno T, Katagiri H (2008) Single- and multi-objective defensive location problems on a network. Eur J Oper Res 188:76–84CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Uno T, Katagiri H, Kato K (2008) An evolutionary multi-agent based search method for Stackelberg solutions of bilevel facility location problems. Int J Innov Comput Inf Control 4:1033–1042Google Scholar
  20. 20.
    Uno T, Katagiri K, Kato K (2011) A multi-dimensionalization of competitive facility location problems. Int J Innov Comput Inf Control 7:2593–2601Google Scholar
  21. 21.
    Birge JR, Louveaux F (2011) Introduction to stochastic programming. Springer, New YorkCrossRefzbMATHGoogle Scholar
  22. 22.
    Dantzig GB (1955) Linear programming under uncertainty. Manag Sci 1:197–206CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Infanger G (eds) (2011) Stochastic programming. Springer, New YorkGoogle Scholar
  24. 24.
    Patriksson M, Wynter L (1997) Stochastic nonlinear bilevel programming. Technical report. PRISM, Universite de Versailles-SaintQuentin en Yvelines, Versailles, FranceGoogle Scholar
  25. 25.
    Ozaltin OY, Prokopyev OA, Schaefer AJ (2010) The bilevel knapsack problem with stochastic right-hand sides. Oper Res Lett 38:328–333CrossRefMathSciNetGoogle Scholar
  26. 26.
    Kalashnikov VV, Perez-Valdes GA, Tomasgard A, Kalashnykova NI (2010) Natural gas cash-out problem: bilevel stochastic optimization approach. Eur J Oper Res 206:18–33CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kosuch S, Bodic PL (2011) On a stochastic bilevel programming problem. Networks 59:107–116CrossRefGoogle Scholar
  28. 28.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    C. Kahraman (Eds.), (2008) Fuzzy multi-criteria decision making. Springer, New YorkGoogle Scholar
  30. 30.
    Lodwick WA, Kacprizyk J (eds) (2010) Fuzzy optimization. Springer, BerlinGoogle Scholar
  31. 31.
    Tsuda H, Saito S (2010) Application of fuzzy theory to the investment decision process. In: Lodwick WA, Kacprzyk J (eds) Fuzzy optimization. Springer, Berlin, pp 365–387CrossRefGoogle Scholar
  32. 32.
    Verdegay J-L (2003) Fuzzy sets based heuristics for optimization. Springer, BerlinCrossRefzbMATHGoogle Scholar
  33. 33.
    Yano H (2009) Interactive decision making for multiobjective programming problems with fuzzy domination structures. Int J Innov Comput Inf Control 5:4867–4875Google Scholar
  34. 34.
    Zimmermann H-J (1985) Applications of fuzzy sets theory to mathematical programming. Inf Sci 36:29–58CrossRefzbMATHGoogle Scholar
  35. 35.
    Dempe S, Starostina T (2007) On the solution of fuzzy bilevel programming. Working Paper. Department of Mathematics and Computer Science, TU Bergakademie FreibergGoogle Scholar
  36. 36.
    Liu B (2004) Uncertainty theory. Springer, BerlinCrossRefzbMATHGoogle Scholar
  37. 37.
    Luhandjula MK (1996) Fuzziness and randomness in an optimization framework. Fuzzy Sets Syst 77:291–297CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Luhandjula MK (2006) Fuzzy stochastic linear programming: survey and future research directions. Eur J Oper Res 174:1353–1367CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Luhandjula MK, Joubert JW (2010) On some optimisation models in a fuzzy-stochastic environment. Eur J Oper Res 207:1433–1441CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Luhandjula MK, Gupta MM (1996) On fuzzy stochastic optimization. Fuzzy Sets Syst 81:47–55CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Hirota K (1981) Concepts of probabilistic sets. Fuzzy Sets Syst 5:31–46CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Buckley JJ (2006) Fuzzy probability and statistics. Springer, BerlinzbMATHGoogle Scholar
  44. 44.
    Kato K, Sakawa M, Katagiri H, Wasada K (2004) An interactive fuzzy satisficing method based on a probability maximization model for multiobjective linear programming problems involving random variable coefficients. Electron Commun Jpn Part 3 87:67–76CrossRefGoogle Scholar
  45. 45.
    Kato K, Katagiri H, Sakawa M, Wang J (2006) Interactive fuzzy programming based on a probability maximization model for two-level stochastic linear programming problems. Electron Commun Jpn Part 3 89:33–42CrossRefGoogle Scholar
  46. 46.
    Katagiri H, Sakawa M (2011) Interactive multiobjective fuzzy random programming through the level set-based probability model. Inf Sci 181:1641–1650CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Katagiri H, Sakawa M, Ishii H (2005) Studies of stochastic programming models using possibility and necessity measures for linear programming problems with fuzzy random variable coefficients. Electron Commun Jpn Part 3 88:68–75CrossRefGoogle Scholar
  48. 48.
    Katagiri H, Sakawa M, Kato K, Ohsaki S (2005) An interactive fuzzy satisficing method based on the fractile optimization model using possibility and necessity measures for a fuzzy random multiobjective linear programming problem. Electron Commun Jpn Part 3 88:20–28CrossRefGoogle Scholar
  49. 49.
    Katagiri H, Sakawa M, Kato K, Nishizaki I (2004) A fuzzy random multiobjective 0–1 programming based on the expectation optimization model using possibility and necessity measures. Math Comput Model 40:411–421CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    Katagiri H, Sakawa M, Kato K, Nishizaki I (2008) Interactive multiobjective fuzzy random linear programming: maximization of possibility and probability. Eur J Oper Res 188:530–539CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Kwakernaak H (1978) Fuzzy random variables—I. Definitions and theorems. Inf Sci 15:1–29CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114:409–422CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Wang GY, Qiao Z (1993) Linear programming with fuzzy random variable coefficients. Fuzzy Sets Syst 57:295–311CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Liu B (2002) Random fuzzy dependent-chance programming and its hybrid intelligent algorithm. Inf Sci 141:259–271CrossRefzbMATHGoogle Scholar
  55. 55.
    Katagiri H, Ishii H, Sakawa M (2002) Linear programming problems with random fuzzy variable coefficients. In: Proceedings of 5th Czech–Japan seminar on data analysis and decision making under uncertainty, vol 1, pp 55–58Google Scholar
  56. 56.
    Hasuike T, Katagiri K, Ishii H (2009) Portfolio selection problems with random fuzzy variable returns. Fuzzy Sets Syst 160:2579–2596CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    Huang X (2007) Optimal project selection with random fuzzy parameters. Int J Prod Econ 106:513–522CrossRefGoogle Scholar
  58. 58.
    Wen M, Iwamura K (2008) Facility location–allocation problem in random fuzzy environment: Using \((\alpha, \beta )\)-cost minimization model under the Hurewicz criterion. Comput Math Appl 55:704–713CrossRefzbMATHMathSciNetGoogle Scholar
  59. 59.
    Katagiri H, Niwa K, Kubo D, Hasuike T (2010) Interactive random fuzzy two-level programming through possibility-based fractile criterion optimality. In: Proceedings of the international multiConference of engineers and computer scientists 2010, vol 3, pp 2113–2118Google Scholar
  60. 60.
    Dubois D, Prade H (2001) Possibility theory, probability theory and multiple-valued logics: a clarification. Ann Math Artif Intell 32:35–66CrossRefMathSciNetGoogle Scholar
  61. 61.
    Zadeh LA (1978) Fuzzy sets as the basis for a theory of possibility. Fuzzy Sets Syst 1:3–28CrossRefzbMATHMathSciNetGoogle Scholar
  62. 62.
    Jorion PH (1996) Value at risk: a new benchmark for measuring derivatives risk. Irwin Professional Publishers, New YorkGoogle Scholar
  63. 63.
    Pritsker M (1997) Evaluating value at risk methodologies. J Financial Serv Res 12:201–242CrossRefGoogle Scholar
  64. 64.
    Kataoka S (1963) A stochastic programming model. Econometrica 31:181–196CrossRefzbMATHMathSciNetGoogle Scholar
  65. 65.
    Geoffrion AM (1967) Stochastic programming with aspiration or fractile criteria. Manag Sci 13:672–679CrossRefzbMATHMathSciNetGoogle Scholar
  66. 66.
    Nahmias S (1978) Fuzzy variables. Fuzzy Sets Syst 1:97–110CrossRefzbMATHMathSciNetGoogle Scholar
  67. 67.
    Loridan P, Morgan J (1996) Weak via strong Stackelberg problem: new results. J Global Optim 8:263–287CrossRefzbMATHMathSciNetGoogle Scholar
  68. 68.
    Gümüs ZH, Floudas CA (2001) Global optimization of nonlinear bilevel programming problems. J Global Optim 20:1–31CrossRefzbMATHMathSciNetGoogle Scholar
  69. 69.
    Savard G, Gauvin J (1994) The steepest descent direction for the nonlinear bilevel programming problem. Oper Res Lett 15:265–272CrossRefzbMATHMathSciNetGoogle Scholar
  70. 70.
    Edmunds TA, Bard JF (1991) Algorithms for nonlinear bilevel mathematical programs. IEEE Trans Syst Man Cybern 21:83–89CrossRefMathSciNetGoogle Scholar
  71. 71.
    Falk JE, Liu J (1995) On bilevel programming, Part I : general nonlinear cases. Math Program 70:47–72zbMATHMathSciNetGoogle Scholar
  72. 72.
    Colson B, Marcotte P, Savard G (2005) A trust-region method for nonlinear bilevel programming: algorithm and computational experience. Comput Optim Appl 30:211–227CrossRefzbMATHMathSciNetGoogle Scholar
  73. 73.
    Colson BP, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153:235–256CrossRefzbMATHMathSciNetGoogle Scholar
  74. 74.
    Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52:333–359CrossRefzbMATHMathSciNetGoogle Scholar
  75. 75.
    Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: a bibliography review. J Global Optim 5:291–306CrossRefzbMATHMathSciNetGoogle Scholar
  76. 76.
    Kupiec PH (1998) Stress testing in a value at risk framework. J Deriv 6:7–24CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Hideki Katagiri
    • 1
    Email author
  • Takeshi Uno
    • 2
  • Kosuke Kato
    • 3
  • Hiroshi Tsuda
    • 4
  • Hiroe Tsubaki
    • 5
  1. 1.Faculty of EngineeringHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Institute of Socio-Arts and SciencesThe University of TokushimaTokushimaJapan
  3. 3.Faculty of Applied Information ScienceHiroshima Institute of TechnologyHiroshimaJapan
  4. 4.Faculty of Science and EngineeringDoshisha UniversityKyotanabeJapan
  5. 5.The Institute of Statistical MathematicsTachikawaJapan

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