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

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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.

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References

  1. Stackelberg H (1952) The theory of market economy. Oxford University Press, Oxford

    Google Scholar 

  2. Bracken J, McGill J (1973) Mathematical programs with optimization problems in the constraints. Oper Res 21:37–44

    Article  MATH  MathSciNet  Google Scholar 

  3. Bracken J, McGill J (1974) Defense applications of mathematical programs with optimization problems in the constraints. Oper Res 22:1086–1096

    Article  MATH  MathSciNet  Google Scholar 

  4. Bracken J, McGill J (1978) Production and marketing decisions with multiple objectives in a competitive environment. J Optim Theory Appl 24:449–458

    Article  MATH  MathSciNet  Google Scholar 

  5. Candler W, Norton R (1977) Multilevel programming. Technical Report, vol 20. World Bank Development Research Center, Washington, DC

    Google Scholar 

  6. Sherali HD, Soyster AL, Murphy FH (1983) Stackelberg–Nash–Cournot equilibria: characterizations and computations. Oper Res 31:253–276

    Article  MATH  MathSciNet  Google Scholar 

  7. Ackere AV (1993) The principal/agent paradigm: characterizations and computations. Eur J Oper Res 70:83–103

    Article  MATH  Google Scholar 

  8. Migdalas A (1995) Bilevel programming in traffic planning: models, methods and challenge. J Global Optim 7:381–405

    Article  MATH  MathSciNet  Google Scholar 

  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 2:23–36

    Article  Google Scholar 

  10. Kara BY, Verter V (2004) Designing a road network for hazardous materials transportation. Transp Sci 38:188–196

    Article  Google Scholar 

  11. Nicholls MG (1996) The applications of non-linear bi-level programming to the aluminum industry. J Global Optim 8:245–261

    Article  MATH  MathSciNet  Google Scholar 

  12. Amouzegar MA, Moshirvaziri K (1999) Determining optimal pollution control policies: an application of bilevel programming. Eur J Oper Res 119:100–120

    Article  MATH  Google Scholar 

  13. Dempe S, Bard JF (2001) Bundle trust-region algorithm for bilinear bilevel programming. J Optim Theory Appl 110:265–288

    Article  MATH  MathSciNet  Google Scholar 

  14. Fampa M, Barroso LA, Candal D, Simonetti L (2008) Bilevel optimization applied to strategic pricing in competitive electricity markets. Comput Optim Appl 39:121–142

    Article  MATH  MathSciNet  Google Scholar 

  15. Karlof JK, Wang W (1996) Bilevel programming applied to the flow shop scheduling problem. Comput Oper Res 23:443–451

    Article  MATH  Google Scholar 

  16. Roghanian E, Sadjadi SJ, Aryanezhad MB (2007) A probabilistic bi-level linear multi-objective programming problem to supply chain planning. Appl Math Comput 188:786–800

    Article  MATH  MathSciNet  Google Scholar 

  17. Miller T, Friesz T, Tobin R (1992) Heuristic algorithms for delivered price spatially competitive network facility location problems. Ann Oper Res 34:177–202

    Article  MATH  Google Scholar 

  18. Uno T, Katagiri H (2008) Single- and multi-objective defensive location problems on a network. Eur J Oper Res 188:76–84

    Article  MATH  MathSciNet  Google Scholar 

  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 4:1033–1042

    Google Scholar 

  20. Uno T, Katagiri K, Kato K (2011) A multi-dimensionalization of competitive facility location problems. Int J Innov Comput Inf Control 7:2593–2601

    Google Scholar 

  21. Birge JR, Louveaux F (2011) Introduction to stochastic programming. Springer, New York

    Book  MATH  Google Scholar 

  22. Dantzig GB (1955) Linear programming under uncertainty. Manag Sci 1:197–206

    Article  MATH  MathSciNet  Google Scholar 

  23. Infanger G (eds) (2011) Stochastic programming. Springer, New York

  24. Patriksson M, Wynter L (1997) Stochastic nonlinear bilevel programming. Technical report. PRISM, Universite de Versailles-SaintQuentin en Yvelines, Versailles, France

    Google Scholar 

  25. Ozaltin OY, Prokopyev OA, Schaefer AJ (2010) The bilevel knapsack problem with stochastic right-hand sides. Oper Res Lett 38:328–333

    Article  MathSciNet  Google Scholar 

  26. Kalashnikov VV, Perez-Valdes GA, Tomasgard A, Kalashnykova NI (2010) Natural gas cash-out problem: bilevel stochastic optimization approach. Eur J Oper Res 206:18–33

    Article  MATH  MathSciNet  Google Scholar 

  27. Kosuch S, Bodic PL (2011) On a stochastic bilevel programming problem. Networks 59:107–116

    Article  Google Scholar 

  28. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MATH  MathSciNet  Google Scholar 

  29. C. Kahraman (Eds.), (2008) Fuzzy multi-criteria decision making. Springer, New York

  30. Lodwick WA, Kacprizyk J (eds) (2010) Fuzzy optimization. Springer, Berlin

  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 365–387

    Chapter  Google Scholar 

  32. Verdegay J-L (2003) Fuzzy sets based heuristics for optimization. Springer, Berlin

    Book  MATH  Google Scholar 

  33. Yano H (2009) Interactive decision making for multiobjective programming problems with fuzzy domination structures. Int J Innov Comput Inf Control 5:4867–4875

    Google Scholar 

  34. Zimmermann H-J (1985) Applications of fuzzy sets theory to mathematical programming. Inf Sci 36:29–58

    Article  MATH  Google Scholar 

  35. Dempe S, Starostina T (2007) On the solution of fuzzy bilevel programming. Working Paper. Department of Mathematics and Computer Science, TU Bergakademie Freiberg

    Google Scholar 

  36. Liu B (2004) Uncertainty theory. Springer, Berlin

    Book  MATH  Google Scholar 

  37. Luhandjula MK (1996) Fuzziness and randomness in an optimization framework. Fuzzy Sets Syst 77:291–297

    Article  MATH  MathSciNet  Google Scholar 

  38. Luhandjula MK (2006) Fuzzy stochastic linear programming: survey and future research directions. Eur J Oper Res 174:1353–1367

    Article  MATH  MathSciNet  Google Scholar 

  39. Luhandjula MK, Joubert JW (2010) On some optimisation models in a fuzzy-stochastic environment. Eur J Oper Res 207:1433–1441

    Article  MATH  MathSciNet  Google Scholar 

  40. Luhandjula MK, Gupta MM (1996) On fuzzy stochastic optimization. Fuzzy Sets Syst 81:47–55

    Article  MATH  MathSciNet  Google Scholar 

  41. Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427

    Article  MATH  MathSciNet  Google Scholar 

  42. Hirota K (1981) Concepts of probabilistic sets. Fuzzy Sets Syst 5:31–46

    Article  MATH  MathSciNet  Google Scholar 

  43. Buckley JJ (2006) Fuzzy probability and statistics. Springer, Berlin

    MATH  Google Scholar 

  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 87:67–76

    Article  Google Scholar 

  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 89:33–42

    Article  Google Scholar 

  46. Katagiri H, Sakawa M (2011) Interactive multiobjective fuzzy random programming through the level set-based probability model. Inf Sci 181:1641–1650

    Article  MATH  MathSciNet  Google Scholar 

  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 88:68–75

    Article  Google Scholar 

  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 88:20–28

    Article  Google Scholar 

  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 40:411–421

    Article  MATH  MathSciNet  Google Scholar 

  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 188:530–539

    Article  MATH  MathSciNet  Google Scholar 

  51. Kwakernaak H (1978) Fuzzy random variables—I. Definitions and theorems. Inf Sci 15:1–29

    Article  MATH  MathSciNet  Google Scholar 

  52. Puri ML, Ralescu DA (1986) Fuzzy random variables. J Math Anal Appl 114:409–422

    Article  MATH  MathSciNet  Google Scholar 

  53. Wang GY, Qiao Z (1993) Linear programming with fuzzy random variable coefficients. Fuzzy Sets Syst 57:295–311

    Article  MATH  MathSciNet  Google Scholar 

  54. Liu B (2002) Random fuzzy dependent-chance programming and its hybrid intelligent algorithm. Inf Sci 141:259–271

    Article  MATH  Google Scholar 

  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 55–58

  56. Hasuike T, Katagiri K, Ishii H (2009) Portfolio selection problems with random fuzzy variable returns. Fuzzy Sets Syst 160:2579–2596

    Article  MATH  MathSciNet  Google Scholar 

  57. Huang X (2007) Optimal project selection with random fuzzy parameters. Int J Prod Econ 106:513–522

    Article  Google Scholar 

  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 55:704–713

    Article  MATH  MathSciNet  Google Scholar 

  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 2113–2118

  60. Dubois D, Prade H (2001) Possibility theory, probability theory and multiple-valued logics: a clarification. Ann Math Artif Intell 32:35–66

    Article  MathSciNet  Google Scholar 

  61. Zadeh LA (1978) Fuzzy sets as the basis for a theory of possibility. Fuzzy Sets Syst 1:3–28

    Article  MATH  MathSciNet  Google Scholar 

  62. Jorion PH (1996) Value at risk: a new benchmark for measuring derivatives risk. Irwin Professional Publishers, New York

    Google Scholar 

  63. Pritsker M (1997) Evaluating value at risk methodologies. J Financial Serv Res 12:201–242

    Article  Google Scholar 

  64. Kataoka S (1963) A stochastic programming model. Econometrica 31:181–196

    Article  MATH  MathSciNet  Google Scholar 

  65. Geoffrion AM (1967) Stochastic programming with aspiration or fractile criteria. Manag Sci 13:672–679

    Article  MATH  MathSciNet  Google Scholar 

  66. Nahmias S (1978) Fuzzy variables. Fuzzy Sets Syst 1:97–110

    Article  MATH  MathSciNet  Google Scholar 

  67. Loridan P, Morgan J (1996) Weak via strong Stackelberg problem: new results. J Global Optim 8:263–287

    Article  MATH  MathSciNet  Google Scholar 

  68. Gümüs ZH, Floudas CA (2001) Global optimization of nonlinear bilevel programming problems. J Global Optim 20:1–31

    Article  MATH  MathSciNet  Google Scholar 

  69. Savard G, Gauvin J (1994) The steepest descent direction for the nonlinear bilevel programming problem. Oper Res Lett 15:265–272

    Article  MATH  MathSciNet  Google Scholar 

  70. Edmunds TA, Bard JF (1991) Algorithms for nonlinear bilevel mathematical programs. IEEE Trans Syst Man Cybern 21:83–89

    Article  MathSciNet  Google Scholar 

  71. Falk JE, Liu J (1995) On bilevel programming, Part I : general nonlinear cases. Math Program 70:47–72

    MATH  MathSciNet  Google Scholar 

  72. Colson B, Marcotte P, Savard G (2005) A trust-region method for nonlinear bilevel programming: algorithm and computational experience. Comput Optim Appl 30:211–227

    Article  MATH  MathSciNet  Google Scholar 

  73. Colson BP, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153:235–256

    Article  MATH  MathSciNet  Google Scholar 

  74. Dempe S (2003) Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52:333–359

    Article  MATH  MathSciNet  Google Scholar 

  75. Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: a bibliography review. J Global Optim 5:291–306

    Article  MATH  MathSciNet  Google Scholar 

  76. Kupiec PH (1998) Stress testing in a value at risk framework. J Deriv 6:7–24

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima, 739-8527, Japan

    Hideki Katagiri

  2. Institute of Socio-Arts and Sciences, The University of Tokushima, 1-1, Minamijosanjima-cho, Tokushima, Tokushima, 770-8502, Japan

    Takeshi Uno

  3. Faculty of Applied Information Science, Hiroshima Institute of Technology, 2-1-1 Miyake, Saeki-ku, Hiroshima, 731-5193, Japan

    Kosuke Kato

  4. Faculty of Science and Engineering, Doshisha University, Tatara Miyakodani 1-3, Kyotanabe, 610-0394, Japan

    Hiroshi Tsuda

  5. The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo, 190-8562, Japan

    Hiroe Tsubaki

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  1. Hideki Katagiri
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  2. Takeshi Uno
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  3. Kosuke Kato
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  4. Hiroshi Tsuda
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Correspondence to Hideki Katagiri.

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Katagiri, H., Uno, T., Kato, K. et al. Random fuzzy bilevel linear programming through possibility-based value at risk model. Int. J. Mach. Learn. & Cyber. 5, 211–224 (2014). https://doi.org/10.1007/s13042-012-0126-4

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  • DOI: https://doi.org/10.1007/s13042-012-0126-4

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