Skip to main content
SpringerLink
Log in

Foundations of Random-Like Bi-Level Decision Making

  • Chapter
  • First Online:
Random-Like Bi-level Decision Making

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 688))

Abstract

The investigation of bi-level (decision-making) programming problems has been strongly motivated by real world applications (Dempe, Foundations of bilevel programming. Kluwer Academic, Dordrecht, 2002). The complex nature of decision-making requires practitioners to make decisions based on a wide variety of cost considerations, benefit analyses and purely technical considerations. As we have entered the era of Big Data, which is the next frontier for innovation, competition and productivity, a new scientific revolution is about to begin. Fortunately, we will be in a position to witness the coming technological leapfrog. Many fields and sectors, ranging from economic and business activities to public administration, from national security to scientific research, are involved in Big Data problems. Because of the excessive data, inherent laws cannot be identified or the inherent data laws are changing so dynamically with “uncertainty” an outward manifestation. Therefore, practical situations are usually too complicated to be described using determinate variables and different random phenomena need to be considered for different practical problems. The development of analytical approaches, such as mathematical models with random-like uncertainty and algorithms able to solve the bi-level decision making problems are still largely unexplored. This book is an attempt to elucidate random-like bi-level decision making to all these levels. In this chapter, we present some elements of random sets theory, which includes some fundamental concepts, definitions of random variables, fuzzy variables, random-overlapped random (Ra-Ra) variables and fuzzy-overlapped random (Ra-Fu) variables. Subsequently, the elements of bi-level programming are introduced. This part is the prerequisite for reading this book. Since these results are well-known, we only introduce the results and readers can refer to correlative research such as Billingsley (Probability and measure, John Wiley & Sons, New York, 1965), Cohn (Measure theory, Birkhäuser, Boston, 1980), Halmos (Measure theory, Van Nostrad Reinhold Company, New York, 1950), and Krickeberg (Probability theory, Addison-Wesley, Reading, 1965).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abass SA (2005) Bilevel programming approach applied to the flow shop scheduling problem under fuzziness. Comput Manag Sci 4(4):279–293

    Article  MathSciNet  MATH  Google Scholar 

  2. Aiyoshi E, Shimizu K (1981) Hierarchical decentralized systems and its new solution by a barrier method. IEEE Trans Syst Man Cybern 11(6):444–449

    Article  MathSciNet  Google Scholar 

  3. Aiyoshi E, Shimizu K (1984) A solution method for the static constrained Stackelberg problem via penalty method. IEEE Trans Autom Control 29(12):1111–1114

    Article  MathSciNet  MATH  Google Scholar 

  4. Aliakbarian N, Dehghanian F, Salari M (2015) A bi-level programming model for protection of hierarchical facilities under imminent attacks. Comput Oper Res 64(3):210–224

    Article  MathSciNet  Google Scholar 

  5. Al-Khayyal FA, Horst R, Pardalos PM (1992) Global optimization of concave functions subject to quadratic constraints: an application in nonlinear bilevel programming. Ann Oper Res 34(1):125–147

    Article  MathSciNet  MATH  Google Scholar 

  6. Bard J, Falk J (1982) An explicit solution to the multi-level programming problem. Comput Oper Res 9(1):77–100

    Article  MathSciNet  Google Scholar 

  7. Bard J, Moore J (1990) A branch and bound algorithm for the bilevel programming problem. SIAM J Sci Stat Comput 11:281

    Article  MathSciNet  MATH  Google Scholar 

  8. Bard JF (1985) Geometric and algorithmic developments for a hierarchical planning problem. Eur J Oper Res 19(3):372–383

    Article  MathSciNet  MATH  Google Scholar 

  9. Bard J.F (1988) Convex two-level optimization. Math Program 40(1):15–27

    Article  MathSciNet  MATH  Google Scholar 

  10. Bard JF, Falk J (1982) An explicit solution to the multi-level programming problem. Comput Oper Res 9:77–100

    Article  MathSciNet  Google Scholar 

  11. Bard JF, Moore J (1990) A branch and bound algorithm for the bilevel programming problem. SIAM J Sci Stat Comput 11:281–292

    Article  MathSciNet  MATH  Google Scholar 

  12. Ben-Ayed O (1993) Bilevel linear programming. Comput Oper Res 20(5):485–501

    Article  MathSciNet  MATH  Google Scholar 

  13. Ben-Ayed O, Blair CE, Boyce DE, LeBlanc LJ (1992) Construction of a real-world bilevel linear programming model of the highway network design problem. Ann Oper Res 34(1): 219–254

    Article  MathSciNet  MATH  Google Scholar 

  14. Berberian S (1965) Measure and integration. The Macmillan Co., New York; Collier-Macmillan Ltd., London

    Google Scholar 

  15. Beyer I (2016) Information technology-based logistics planning: approaches to developing a coordination mechanism for decentralized planning. Commun Llma. doi:10.1016/j.neucom.2016.01.031

    Google Scholar 

  16. Bi Z, Calamai PH, Conn AR (1989) An exact penalty function approach for the linear bilevel programming problem. Tech. rep., Department of Systems Design Engineering, University of Waterloo. Technical Report #167-O-310789

    Google Scholar 

  17. Bi Z, Calamai PH, Conn AR (1989) An exact penalty function approach for the nonlinear bilevel programming problem. University of Waterloo. Technical Report No. 167-0-310789

    Google Scholar 

  18. Bialas W, Karwan M (1984) Two-level linear programming. Manag Sci 30:1004–1020

    Article  MathSciNet  MATH  Google Scholar 

  19. Bialas W, Karwan M, Shaw J (1980) A parametric complementary pivot approach for two-level linear programming. State University of New York at Buffalo

    Google Scholar 

  20. Bianco L, Caramia M, Giordani S (2009) A bilevel flow model for Hazmat transportation network design. Transp Res Part C Emerg Technol 17(2):175–196

    Article  Google Scholar 

  21. Billingsley P (1965) Probability and measure. John Wiley & Sons, New York

    MATH  Google Scholar 

  22. Blackwell D (1956) On a class of probability spaces. In: Proceedings of the third Berkeley symposium on mathematical statistics and probability: contributions to econometrics industrial research and psychometry. University of California Press, Berkeley, p 1

    Google Scholar 

  23. Bracken J, Falk JE, Miercort FA (1977) A strategic weapons exchange allocation model. Oper Res 25(6):968–976

    Article  MathSciNet  MATH  Google Scholar 

  24. Buckley J (2006) Fuzzy probability and statistics. Springer, Berlin

    MATH  Google Scholar 

  25. Calamai P, Vicente L (1993) Generating bilevel and linear-quadratic programming problems. SIAM J Sci Stat Comput 14:770–782

    Article  MathSciNet  MATH  Google Scholar 

  26. Campos L, Verdegay J (1989) Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets Syst 32(1):1–11

    Article  MathSciNet  MATH  Google Scholar 

  27. Candler W, Fortuny-Amat J, McCarl B (1981) The potential role of multilevel programming in agricultural economics. Am J Agric Econ 63(3):521–531

    Article  Google Scholar 

  28. Candler W, Townsley R (1982) Linear two-level programming problem. Comput Oper Res 9(1):59–76

    Article  MathSciNet  Google Scholar 

  29. Cao Y, Sun D (2012) A parallel computing framework for large-scale air traffic flow optimization. IEEE Trans Intell Transp Syst 13(4):1855–1864

    Article  MathSciNet  Google Scholar 

  30. Case LM (1999) An ℓ 1 penalty function approach to the nonlinear bilevel programming problem. Ph.D. thesis, University of Waterloo, Ontario

    Google Scholar 

  31. Cassidy RG, Kirby MJL, Raike WM (1971) Efficient distribution of resources through three levels of government. Manag Sci 17(8):462–473

    Article  Google Scholar 

  32. Chang TS, Luh P (1984) Derivation of necessary and sufficient conditions for single-stage stackelberg games via the inducible region concept. IEEE Trans Autom Control 29(1):63–66

    Article  MathSciNet  MATH  Google Scholar 

  33. Clauset A, Newman MEJ, Moore C (2004) Finding community structure in very large networks. Phys Rev E 70(6):066, 111

    Google Scholar 

  34. Cohn D (1980) Measure theory. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  35. Colson B, Marcotte P, Savard G (2005) Bilevel programming: a survey. 4OR: Q J Oper Res 3(2):87–107

    Article  MathSciNet  MATH  Google Scholar 

  36. Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153(1):235–256

    Article  MathSciNet  MATH  Google Scholar 

  37. Deng X (1998) Complexity issues in bilevel linear programming. In: Multilevel optimization: algorithms and applications. Kluwer Academic, Dordrecht, pp 149–164

    Chapter  Google Scholar 

  38. Dempe S (2002) Foundations of bilevel programming. Kluwer Academic, Dordrecht

    MATH  Google Scholar 

  39. Desilva AH (1978) Sensitivity formulas for nonlinear factorable programming and their application to the solution of an implicitly defined optimization model of US crude oil production. Ph.D. thesis, George Washington University, Washington, DC

    Google Scholar 

  40. Dimitriou L, Tsekeris T, Stathopoulos A (2008) Genetic computation of road network design and pricing stackelberg games with multi-class users. In: Giacobini M, Brabazon A, Cagnoni S, Di Caro GA, Drechsler R, Ekárt A, Esparcia-Alcázar AI, Farooq M, Fink A, McCormack J, O’Neill M, Romero J, Rothlauf F, Squillero G, Uyar AŞ, Yang S (eds) EvoWorkshops 2008. LNCS, vol 4974. Springer, Heidelberg, pp 669–678

    Google Scholar 

  41. Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9(6):613–626

    Article  MathSciNet  MATH  Google Scholar 

  42. DuBois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New York

    MATH  Google Scholar 

  43. Dubois D, Prade H (1987) Fuzzy numbers: an overview. Anal Fuzzy Inf Math Logics 1:3–39

    MathSciNet  MATH  Google Scholar 

  44. Dubois D, Prade H (1988) Possibility theory: an approach to computerized processing of uncertainty. Plenum Press, New York

    Book  MATH  Google Scholar 

  45. Durrett R, Durrett R (2010) Probability: theory and examples. Cambridge University Press, Cambridge/New York

    Book  MATH  Google Scholar 

  46. Edmunds TA, Bard JF (1991) Algorithms for nonlinear bilevel mathematical programs. IEEE Trans Syst Man Cybern 21(1):83–89

    Article  MathSciNet  Google Scholar 

  47. Eichfelder G (2007) Solving nonlinear multiobjective bilevel optimization problems with coupled upper level constraints. Technical Report Preprint No. 320, Preprint-Series of the Institute of Applied Mathematics, University of Erlangen-Nürnberg

    Google Scholar 

  48. Eichfelder G (2008) Multiobjective bilevel optimization. Math Program. doi:10.1007/s10107-008-0259-0

    MathSciNet  MATH  Google Scholar 

  49. Evans L, Gariepy R (1992) Measure theory and fine properties of functions. CRC, Boca Raton

    MATH  Google Scholar 

  50. Falk JE (1973) A linear max–min problem. Math Program 5(1):169–188

    Article  MathSciNet  MATH  Google Scholar 

  51. Falk JE, Liu J (1995) On bilevel programming, Part I: general nonlinear cases. Math Program 70(1):47–72

    MathSciNet  MATH  Google Scholar 

  52. 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  MathSciNet  MATH  Google Scholar 

  53. Ferrero A, Prioli M, Salicone S (2015) Conditional random-fuzzy variables representing measurement results. IEEE Trans Instrum Meas 64(5):1170–1178

    Article  Google Scholar 

  54. Feller W (1971) An introduction to probability theory and its application. Tome II, John Wiley & Sons, New York

    MATH  Google Scholar 

  55. Fernández-Blanco R, Arroyo J, Alguacil N (2012) A unified bilevel programming framework for price-based market clearing under marginal pricing. IEEE Trans Power Syst 27(11): 517–525

    Article  Google Scholar 

  56. Fliege J, Vicente LN (2006) Multicriteria approach to bilevel optimization. J Optim Theory Appl 131(2):209–225

    Article  MathSciNet  MATH  Google Scholar 

  57. Fortuny-Amat J, McCarl B (1981) A representation and economic interpretation of a two-level programming problem. J Oper Res Soc 32:783–792

    Article  MathSciNet  MATH  Google Scholar 

  58. Gaur A, Arora SR (2008) Multi-level multi-attributemulti-objective integer linear programming problem. AMO-Adv Model Optim 10(2):297–322

    MathSciNet  MATH  Google Scholar 

  59. González A (1990) A study of the ranking function approach through mean values. Fuzzy Sets Syst 35(1):29–41

    Article  MathSciNet  MATH  Google Scholar 

  60. Guo XN, Hua TS, Zhang T, Lv YB (2012) Bilevel model for multi-reservoir operating policy in inter-basin water transfer-supply project. J Hydrol 424:252–263

    Article  Google Scholar 

  61. Halmos P (1950) Measure theory. Van Nostrad Reinhold Company, New York

    Book  MATH  Google Scholar 

  62. Halmos P (1974) Naive set theory. Springer, New York

    Book  MATH  Google Scholar 

  63. Hansen P, Jaumard B, Savard G (1992) New branch-and-bound rules for linear bilevel programming. SIAM J Sci Stat Comput 13:1194

    Article  MathSciNet  MATH  Google Scholar 

  64. Halter W, Mostaghim S (2006) Bilevel optimization of multicomponent chemical systems using particle swarm optimization. In: Proceedings of world congress on computational intelligence (WCCI 2006), pp 1240–1247

    Google Scholar 

  65. Harel D, Koren Y (2001) A fast multi-scale method for drawing large graphs//Graph drawing. Springer, Berlin/Heidelberg

    MATH  Google Scholar 

  66. Herskovits J, Leontiev A, Dias G, Santos G (2000) Contact shape optimization: a bilevel programming approach. Struct Multidisc Optim 20:214–221

    Article  Google Scholar 

  67. Ho YC, Luh P, Muralidharan R (1981) Information structure, Stackelberg games, and incentive controllability. IEEE Trans Autom Control 26(2):454–460

    Article  MathSciNet  MATH  Google Scholar 

  68. Ishizuka Y, Aiyoshi E (1992) Double penalty method for bilevel optimization problems. Ann Oper Res 34(1):73–88

    Article  MathSciNet  MATH  Google Scholar 

  69. Jacobs K, Kurzweil J (1978) Measure and integral. Academic Press, New York

    Google Scholar 

  70. Judice J, Faustino A (1988) The solution of the linear bilevel programming problem by using the linear complementarity problem. Investigação Operacional 8:77–95

    MATH  Google Scholar 

  71. Júdice J, Faustino AM (1992) A sequential LCP method for bilevel linear programming. Ann Oper Res 34(1):89–106

    Article  MathSciNet  MATH  Google Scholar 

  72. Katsoulakos NM, Kaliampakos DC (2016) Mountainous areas and decentralized energy planning: insights from Greece. Energy Policy 91:174–188

    Article  Google Scholar 

  73. Kaufmann A (1975) Introduction to the theory of fuzzy subsets. Academic Press, New York

    MATH  Google Scholar 

  74. Klement E, Puri M, Ralescu D (1986) Limit theorems for fuzzy random variables. Proc R Soc Lond Ser A Math Phys Sci 407(1832):171–182

    Article  MathSciNet  Google Scholar 

  75. Koh A (2007) Solving transportation bi-level programs with differential evolution. In: IEEE congress on evolutionary computation (CEC 2007), pp 2243–2250. IEEE Press

    Google Scholar 

  76. Kolmogorov A (1950) Foundations of the theory of probability. Chelsea Publishing Company. New York

    MATH  Google Scholar 

  77. Kolstad CD, Lasdon LS (1990) Derivative estimation and computational experience with large bilevel mathematical programs. J Optim Theory Appl 65:485–499

    Article  MathSciNet  MATH  Google Scholar 

  78. Kong YN, Yang QF (2011) A political economic analysis on multilateral agricultural negotiation: based on two-level interactive evolutional game model. J Int Trade 6(3)

    Google Scholar 

  79. Krickeberg K (1965) Probability theory. Addison-Wesley, Reading

    MATH  Google Scholar 

  80. Kruse R, Meyer K (1987) Statistics with vague data. Springer, Dordrecht

    Book  MATH  Google Scholar 

  81. Kuhn H, Tucker A (1951) Non linear programming. In: Neyman J (ed) 2nd Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, pp 481–492. University of California

    Google Scholar 

  82. Kwakernaak H (1978) Fuzzy random variables-I. Definitions and theorems. Information Sciences 15(1):1–29

    Article  MathSciNet  MATH  Google Scholar 

  83. Kwakernaak H (1979) Fuzzy random variables-II. Algorithms and examples for the discrete case. Inf Sci 17(3):253–278

    Article  MathSciNet  MATH  Google Scholar 

  84. Kydland F (1975) Hierarchical decomposition in linear economic models. Manag Sci 21(9):1029–1039

    Article  MathSciNet  MATH  Google Scholar 

  85. Laha R, Rohatgi V (1985) Probability Theory. John Wiley & Sons, New York

    MATH  Google Scholar 

  86. Li X, Tian P, Min X (2006) A hierarchical particle swarm optimization for solving bilevel programming problems. In: Rutkowski L, Tadeusiewicz R, Zadeh LA, Żurada JM (eds) ICAISC 2006. LNCS (LNAI), vol 4029. Springer, Heidelberg, pp 1169–1178

    Google Scholar 

  87. Li H, Wang Y (2007) A genetic algorithm for solving a special class of nonlinear bilevel programming problems. In: Shi Y, van Albada GD, Dongarra J, Sloot PMA (eds) ICCS 2007, Part IV. LNCS, vol 4490. Springer, Heidelberg, pp 1159–1162

    Chapter  Google Scholar 

  88. Liu Y, Liu B (2003) Expected value operator of random fuzzy variable and random fuzzy expected value models. Int J Uncertain Fuzziness Knowl Based Syst 11(2):195–216

    Article  MathSciNet  MATH  Google Scholar 

  89. Lu J, Atamturktur S, Huang Y (2016) Bi-level resource allocation framework for retrofitting bridges in a transportation network. Transp Res Rec J Transp Res Board 2550:31–37

    Article  Google Scholar 

  90. Mathieu R, Pittard L, Anandalingam G (1994) Genetic algorithm based approach to bi-level linear programming. Oper Res 28(1):1–21

    MathSciNet  MATH  Google Scholar 

  91. Meijboom BR (1987) Planning in decentralized firms—a contribution to the theory on multilevel decisions. Springer, Berlin/New York

    Book  Google Scholar 

  92. Nahmias S (1978) Fuzzy variables. Fuzzy Sets Syst 1(2):97–110

    Article  MathSciNet  MATH  Google Scholar 

  93. Näther W (2000) On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data. Metrika 51(3):201–221

    Article  MathSciNet  MATH  Google Scholar 

  94. Oduguwa V, Roy R (2002) Bi-level optimisation using genetic algorithm. In: IEEE International Conference on Artificial Intelligence Systems (ICAIS 2002), pp 322–327

    Google Scholar 

  95. Peng J, Liu B (2007) Birandom variables and birandom programming. Comput Ind Eng 53(3):433–453

    Article  MathSciNet  Google Scholar 

  96. Pfeiffer P (1990) Probability for applications. Springer

    Book  MATH  Google Scholar 

  97. Puri M, Ralescu D (1986) Fuzzy random variables. J Math Anal Appl 114(2):409–422

    Article  MathSciNet  MATH  Google Scholar 

  98. Sakawa M, Nishizaki I (2002) Interactive fuzzy programming for decentralized two-level linear programming problems. Fuzzy Set Syst 125(3):301–315

    Article  MathSciNet  MATH  Google Scholar 

  99. Savard G, Gauvin J (1994) The steepest descent direction for the nonlinear bilevel programming problem. Oper Res Lett 15(5):265–272

    Article  MathSciNet  MATH  Google Scholar 

  100. Shi X, Xia H (1997) Interactive bilevel multi-objective decision making. Journal of the Operational Research Society 48(9):943–949

    Article  MATH  Google Scholar 

  101. Shimizu K, Aiyoshi E (1981) A new computational method for Stackelberg and min-max problems by use of a penalty method. IEEE Trans Autom Control 26:460–466

    Article  MathSciNet  MATH  Google Scholar 

  102. Simaan M, Cruz J.B (1973) On the Stackelberg strategy in nonzero-sum games. J Optim Theory Appl 11(5):533–555

    Article  MathSciNet  MATH  Google Scholar 

  103. Srivastava S (1998) A course on Borel sets. Springer, New York

    Book  MATH  Google Scholar 

  104. Stoilov T, Stoilova K (2012) Portfolio risk management modelling by bi-level optimization. Handbook on Decision Making, pp 91–110

    Google Scholar 

  105. Sun D, Benekohal RF, Waller ST (2006) Bi-level programming formulation and heuristic solution approach for dynamic traffic signal optimization. Comput-Aided Civil Infrastruct Eng 21(5):321–333

    Article  Google Scholar 

  106. Talbi E (2014) Metaheuristics for bilevel optimization, Springer, Berlin/New York

    Google Scholar 

  107. Talbi E (2009) Metaheuristics: from design to implementation. John Wiley & Sons, Hoboken

    Book  MATH  Google Scholar 

  108. Thoai NV, Yamamoto Y, Yoshise A (2005) Global optimization method for solving mathematical programs with linear complementarity constraints. J Optim Theory Appl 124(2):467–490

    Article  MathSciNet  MATH  Google Scholar 

  109. Tuy H, Migdalas A, Värbrand P (1993) A global optimization approach for the linear two-level program. J Global Optim 3(1):1–23

    Article  MathSciNet  MATH  Google Scholar 

  110. Vallejo JFC, Sánchez RM (2013) A path based algorithm for solve the hazardous materials transportation bilevel problem. Appl Mech Mater 253–255:1082–1088

    Google Scholar 

  111. Vicente L, Savard G, Judice J (1996) Discrete linear bilevel programming problem. J Optim Theory Appl 89(3):597–614

    Article  MathSciNet  MATH  Google Scholar 

  112. Vicente LN, Calamai PH (2004) Bilevel and multilevel programming: a bibliography review. J Global Optim 5(3):291–306

    Article  MathSciNet  MATH  Google Scholar 

  113. Wang HF (2011) Multi-level subsidy and penalty strategy for a green industry sector. In: IEEE ninth international conference on dependable, autonomic and secure computing (DASC 2011), pp 776–783

    Google Scholar 

  114. Wen UP, Hsu ST (1991) Linear bi-level programming problems—a review. J Oper Res Soc 42:125–133

    MATH  Google Scholar 

  115. White DJ, Anandalingam G (1993) A penalty function approach for solving bi-level linear programs. J Global Optim 3(4):397–419

    Article  MathSciNet  MATH  Google Scholar 

  116. Xu J, He Y, Gen M (2009) A class of random fuzzy programming and its application to supply chain design. Comput Ind Eng 56(3):937–950

    Article  Google Scholar 

  117. Xu J, Liu Q, Wang R (2008) A class of multi-objective supply chain networks optimal model under random fuzzy environment and its application to the industry of Chinese liquor. Inf Sci 178(8):2022–2043

    Article  MATH  Google Scholar 

  118. Xu JP, Tu Y, Zeng ZQ (2012) Bi-level optimization of regional water resources allocation problem under fuzzy random environment. J Water Resour Plan Manag 139(3):246–264

    Article  Google Scholar 

  119. Xu JP, Yao LM (2011) Random-like multiple objective decision making. Springer, Berlin/Heidelberg

    Google Scholar 

  120. Xu JP, Wang Z, Zhang M, et al (2016) A new model for a 72-h post-earthquake emergency logistics location-routing problem under a random fuzzy environment. Transp Lett Int J Transp Res. doi:10.1080/19427867.2015.1126064

    Google Scholar 

  121. Yager R (1981) A procedure for ordering fuzzy subsets of the unit interval. Inf Sci 24(2):143–161

    Article  MathSciNet  MATH  Google Scholar 

  122. Yager R (2002) On the evaluation of uncertain courses of action. Fuzzy Optim Decis Making 1(1):13–41

    Article  MathSciNet  MATH  Google Scholar 

  123. Yates J, Casas I (2010) Role of spatial data in the protection of critical infrastructure and homeland defense. Appl Spat Anal Policy 5:1–23

    Article  Google Scholar 

  124. Ye J (2006) Constraint qualifications and KKT conditions for bilevel programming problems. Math Opera Res 31(4):811–824

    Article  MathSciNet  MATH  Google Scholar 

  125. Yin Y (2000) Genetic algorithm based approach for bilevel programming models. J Transp Eng 126(2):115–120

    Article  Google Scholar 

  126. Zadeh L (1965) Fuzzy sets. Inf control 8(3):338–353

    Article  MathSciNet  MATH  Google Scholar 

  127. Zadeh L (1972) A fuzzy-set-theoretic interpretation of linguistic hedges. Cybern Syst 2(3): 4–34

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  129. Zadeh L, et al (1975) Calculus of fuzzy restrictions. Fuzzy sets and their applications to cognitive and decision processes. Academic Press, New York, pp 1–39

    Google Scholar 

  130. Zannetos ZS (1965) On the theory of divisional structures: aspects of centralization and decentralization of control and decision-making. Manag Sci 12(4):49–68

    Article  Google Scholar 

  131. Zhang W, Fan J (2015) Cloud architecture intrusion detection system based on KKT condition and hyper-sphere incremental SVM algorithm. J Comput Appl 35(10):2886–2890

    MathSciNet  Google Scholar 

  132. Zhou X, Xu J (2009) A class of integrated logistics network model under random fuzzy environment and its application to Chinese beer company. Int J Uncertain Fuzziness Knowl-Based Syst 17(6):807–831

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Business School, Sichuan University, Chengdu, Sichuan, China

    Jiuping Xu, Zongmin Li & Zhimiao Tao

Authors
  1. Jiuping Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zongmin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhimiao Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media Singapore

About this chapter

Cite this chapter

Xu, J., Li, Z., Tao, Z. (2016). Foundations of Random-Like Bi-Level Decision Making. In: Random-Like Bi-level Decision Making. Lecture Notes in Economics and Mathematical Systems, vol 688. Springer, Singapore. https://doi.org/10.1007/978-981-10-1768-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-1768-1_1

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-1767-4

  • Online ISBN: 978-981-10-1768-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation