Books in English

  1. Liu B, Uncertainty Theory: An Introduction to its Axiomatic Foundations, Springer-Verlag, Berlin, 2004.

  2. Liu B, Theory and Practice of Uncertain Programming, Physica-Verlag, Heidelberg, 2002.

  3. Liu B and Esogbue A.O, Decision Criteria and Optimal Inventory Processes, Kluwer, Boston, 1999.

  4. Liu B, Uncertain Programming, Wiley, New York, 1999.

  Books in Chinese

  1. Liu B, Peng J, A Course in Uncertainty Theory, Tsinghua University Press, Beijing, 2005.
  2. Liu B, Zhao R, and Wang G, Uncertain Programming with Applications, Tsinghua University Press, Beijing, 2003.

  3. Liu B and Zhao R, Stochastic Programming and Fuzzy Programming, Tsinghua University Press, Beijing, 1998.

  Selected Papers

     2006

  1. Liu B., A survey of credibility theory, Fuzzy Optimization and Decision Making, Vol.5, No.4, 387-408, 2006.

  2. Zheng Y. and Liu B., Fuzzy vehicle routing model with credibility measure and its hybrid intelligent algorithm, Applied Mathematics and Computation, Vol.176, No.2, 673-683, 2006.

  3. Yang L. and Liu B., On continuity theorem for characteristic function of fuzzy variable, Journal of Intelligent & Fuzzy Systems, Vol.17, No.3, 325-332, 2006.

  4. Li X. and Liu B., A sufficient and necessary condition for credibility measures, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Vol.14, No.5, 2006.

  5. Esogbue A.O. and Liu B., Reservoir operations optimization via fuzzy criterion decision processes, Fuzzy Optimization and Decision Making, Vol.5, No.3, 289-305, 2006.

  6. Feng X, Liu Y, Measurability criteria for fuzzy random vectors, Fuzzy Optimization and Decision Making, Vol.5, No.3, 245-253, 2006.

  7. Liu Y, Convergent results about the use of fuzzy simulation in fuzzy optimization problems, IEEE Transactions on Fuzzy Systems, Vol.14, No.2, 295-304, 2006.

  8. Yang L, and Liu B, A sufficient and necessary condition for chance distribution of birandom variable, Information, Vol.9, No.1, 33-36, 2006.

  9. Zhao R, Tang W, and Yun H, Random Fuzzy Renewal Process, European Journal of Operational Research, Vol. 169,  189-201, 2006.

  10. Liu L, The fuzzy quadratic assignment problem with penalty: New models and genetic algorithm, Applied Mathematics and Computation, Vol.174, 1229-1244, 2006.

  11. Ji X and Shao Z, Model and algorithm for bilevel Newsboy problem with fuzzy demands and discounts, Applied Mathematics and Computation, Vol. 172, 163-174, 2006.

  12. Zhu Y, Solutions to variational inequalities of parabolic type, Journal of Mathematical Analysis and Applications, Vol. 321, 24-31, 2006.

  13. Zhu Y, Convergence of Random Fuzzy Sequences in Distribution, WSEAS Transactions on Information Science & Applications, Vol. 3, No. 7, 1179-1186, 2006.

  14. Zhao R and Tang W, Some Properties of Fuzzy Random Processes, IEEE Transactions on Fuzzy Systems, Vol.14, No.2, 173-179, 2006.

    2005

  15. Liu Y, and Liu B, On minimum-risk problems in fuzzy random decision systems, Computers & Operations Research, Vol.32, No.2, 257-283, 2005.

  16. Liu Y, Fuzzy programming with recourse, nternational Journal of Uncertainty Fuzziness Knowledge-Based Systems, Vol.13, No.4, 381-413, 2005.

  17. Liu Y., and Liu B., Fuzzy random programming with equilibrium chance constraints, Information Sciences, Vol.170, Nos.2-4, 363-395, 2005.

  18. Zhu Y, Convex hull and closure of a fuzzy set, Journal of Intelligent and Fuzzy Systems, Vol. 16, No. 1, 67-73, 2005.

  19. Lai Y, Zhu Y and Deng Y, Fixed point index approach for solutions of variational inequalities, International Journal of Mathematics and Mathematical Sciences, Vol. 12, 1879-1887, 2005.

  20. Zhu Y and Liu B, Some inequalities of random fuzzy variables with application to moment convergence, Computers and Mathematics with Applications, Vol. 50, Nos. 5-6, 719-727, 2005.

  21. Zhao R, Liu B, Standby Redundancy Optimization Problems with Fuzzy Lifetimes, Computers & Industrial Engineering, Vol.49, 318-338, 2005.

  22. Gao J and Liu B, Fuzzy multilevel programming with a hybrid intelligent algorithm, Computer & Mathematics with Applications, Vol.49, Nos.9-10, 1539-1548, 2005.

  23. Gao J and Lu M, Fuzzy quadratic minimum spanning tree problem, Applied Mathematics and Computation, Vol. 164, No. 3, 773-788, 2005.

  24. Gao J and Lu M, On the Randic Index of Unicyclic Graphs, MATCH Communications in Mathematical and in Computer Chemistry, Vol.53, No.2, 377-384, 2005.

  25. Ji X, Models and Algorithm for Stochastic Shortest Path Problem, Applied Mathematics and Computation, Vol. 170, No. 1, 503-514, 2005.

  26. Lai Y and Zhu Y, Existence theorems for solutions of variational inequalities, Acta Mathematica Hungarica, Vol.108(1-2), 95-103, 2005.

    200
    4
     

  27. Gao J, Liu B, and Gen M, A hybrid intelligent algorithm for stochastic multilevel programming, IEEJ Transactions on Electronics, Information and Systems, Vol.124-C, No.10, 1991-1998, 2004.

  28. Peng J, and Liu B, Parallel machine scheduling models with fuzzy processing times, Information Sciences, Vol.166, Nos.1-4, 49-66, 2004.

  29. Zhou J, and Liu B, Analysis and algorithms of bifuzzy systems, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Vol.12, No.3, 357-376, 2004.

  30. Zhao R, and Liu B, Redundancy optimization problems with uncertainty of combining randomness and fuzziness, European Journal of Operational Research, Vol.157, No.3, 716-735, 2004.

  31. Liu H, Lu M and Tian F, On the Laplacian Spectral Radius of a Graph, Linear Algebra and its Applications, Vol. 376, 135-141, 2004.

  32. Lu M, Liu H and Tian F, The Connectivity, MATCH Communications in Mathematical and in Computer Chemistry, Vol. 51, 149-154, 2004.

  33. Liu H, Lu M and Tian F, Neighborhood Unions and Cyclability of Graphs, Discrete Applied Mathematics, Vol. 140, 91-101, 2004.

  34. Yu A, Lu M and Tian F, On the Spectral Radius of Graphs, Linear Algebra and its Applications, Vol. 387, 41-49, 2004.

  35. Liu H, Lu M and Tian F, On the Spectral Radius of Graphs with Cut Edges, Linear Algebra and its Applications, Vol. 389, 139-145, 2004.

  36. Zhao R, W. Li, and Tang W, Fuzzy Random Block Replacement Policies, Asian Information-Science-Life, Vol. 2, 1-5, 2004.

  37. Zhu Y and Liu B, Continuity theorems and chance distribution of random fuzzy variables, Proceedings of the Royal Society London Series A, Vol. 460, 2505-2519, 2004.

    2003
     

  38. Zhao R, and Liu B, Renewal process with fuzzy interarrival times and rewards, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Vol.11, No.5, 573-586, 2003.

  39. Liu Y. and Liu B, A class of fuzzy random optimization: Expected value models, Information Sciences, Vol.155, No.1-2, 89-102, 2003.

  40. Zhao R. and Liu B, Stochastic programming models for general redundancy optimization problems, IEEE Transactions on Reliability, Vol.52, No.2, 181-191, 2003.

  41. Zhou J. and Liu B, New stochastic models for capacitated location-allocation problem, Computers & Industrial Engineering, Vol.45, No.1, 111-125, 2003.

  42. Liu Y. and Liu B, Expected value operator of random fuzzy variable and random fuzzy expected value models, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Vol.11, No.2, 195-215, 2003.

  43. Liu Y. and Liu B, Fuzzy random variables: A scalar expected value operator, Fuzzy Optimization and Decision Making, Vol.2, No.2, 143-160, 2003.

  44. Liu B, Inequalities and convergence concepts of fuzzy and rough variables, Fuzzy Optimization and Decision Making, Vol.2, No.2, 87-100, 2003.

  45. Lu M, On Crisp Equivalents and Solutions of Fuzzy Programming with Different Chance Measures, Information: An International Journal, Vol. 6, No. 2, 125-134, 2003.

  46. Zhao R, Liu B, Stochastic Programming Models for General Redundancy Optimization Problems, IEEE Transactions on Reliability, Vol. 52, No. 2, 181-191, 2003.

  47. Zhao R, and Liu B, Renewal Process with Fuzzy Interarrival Times and Rewards, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Vol.11, 573-586, 2003.

  48. Zhu Y., Fuzzy differential inclusions with fuzzy constraints, Information: An International Journal, Vol. 6, No. 4, 373-378, 2003.

  49.  Lai Y., Zhu Y., and Deng Y., The existence of nonzero solutions for a class of variational inequalities by index, Applied Mathematics Letters, Vol. 16, 839-845, 2003.

    2002
     

  50. Liu B, Toward fuzzy optimization without mathematical ambiguity, Fuzzy Optimization and Decision Making, Vol.1, No.1, 43-63, 2002.

  51. Liu B, Random fuzzy dependent-chance programming and its hybrid intelligent algorithm, Information Sciences, Vol.141, No.3-4, 259-271, 2002.

  52. Liu B and Lai K.K, Stochastic programming models for vehicle routing problems, Asian Information-Science-Life, Vol.1, No.1, 13-28, 2002.

  53. Liu B and Liu Y, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, Vol.10, No.4, 445-450, 2002.

  54. Liu Y. and Liu B, Random fuzzy programming with chance measures defined by fuzzy integrals, Mathematical and Computer Modelling, Vol.36, No.4-5, 509-524, 2002.

  55. Liu Y. and Liu B, Fuzzy random programming problems with multiple criteria, Asian Information-Science-Life, Vol.1, No.3, 249-256, 2002.

  56. Peng J, and Liu B, Stochastic goal programming models for parallel machine scheduling problems, Asian Information-Science-Life, Vol.1, No.3, 257-266, 2002.

  57. Li G, Lu M, Liu Z, Hamiltonian cycles in 3-connected claw-free graphs, Discrete Mathematics, Vol. 250, 137-151. 2002.

  58.  Lu M and Gao J, Stochastic Quadratic Minimum Spanning Tree Problem, Asian Information-Science-Life, Vol. 1, No. 3, 1218-1220, 2002.

  59. Zhou J, Fuzzy programming models for minimax location problem, Asian Journal of Information-Science-Life, Vol. 1, No. 3, 231-240, 2002.

  60. Wang G. and Iwamura K, Yield management with random fuzzy demand, Asian Information-Science-Life, Vol.1, No.3, 27-31, 2002.



  61. 2001
     
  62. Liu B, Uncertain programming: A unifying optimization theory in various uncertain environments, Applied Mathematics and Computation, Vol.120, Nos.1-3, 227-234, 2001.

  63. Liu Y. and Liu B, The relationship between structural characteristics of fuzzy measure and convergences of sequences of measurable functions, Fuzzy Sets and Systems, Vol.120, No.3, 511-516, 2001.

  64. Liu B and Iwamura K, Fuzzy programming with fuzzy decisions and fuzzy simulation-based genetic algorithm, Fuzzy Sets and Systems, Vol.122, No.2, 253-262, 2001.

  65. Liu B, Fuzzy random chance-constrained programming, IEEE Transactions on Fuzzy Systems, Vol.9, No.5, 713-720, 2001.

  66. Liu B, Fuzzy random dependent-chance programming, IEEE Transactions on Fuzzy Systems, Vol.9, No.5, 721-726, 2001.

  67. Gao J. and Liu B, New primitive chance measures of fuzzy random event, International Journal of Fuzzy Systems, Vol.3, No.4, 527-531, 2001.

  68. Liu Y, The completion of a fuzzy measure and its applications, Fuzzy Sets and Systems, Vol. 123, No. 2, 137-145, 2001.

  69. Lu M and Yu Z, Cycles of length 1 modulo 3 in graph, Discrete Applied Mathematics, Vol. 113, 329-336, 2001.



  70. 2000
     
  71. Xia Y, Liu B, Wang S, and Lai K.K, A model for portfolio selection with order of expected returns, Computers & Operations Research, Vol.27, 409-422, 2000.

  72. Liu B, Dependent-chance programming in fuzzy environments, Fuzzy Sets and Systems, Vol.109, No.1, 97-106, 2000.

  73. Liu B and Iwamura K, Topological optimization models for communication network with multiple reliability goals, Computers & Mathematics with Applications, Vol.39, 59-69, 2000.

  74. Liu Y, On the convergence of measurable set-valued function sequence on fuzzy measure space, Fuzzy Sets and Systems, Vol. 112, No. 2, 241-249, 2000.

  75. Zhu Y and Rao L, Differential inclusions for fuzzy maps, Fuzzy Sets and Systems, Vol. 112, 257-261, 2000.



  76. 1999
     
  77. Gen M. and Liu B, A genetic algorithm for nonlinear goal programming, Evolutionary Optimization, Vol.1, No.1, 65-76, 1999.

  78. Liu B, Fuzzy criterion models for inventory systems with partial backorders, Annals of Operations Research, Vol.87, 117-126, 1999.

  79. Liu B and Odanaka T, Dynamic fuzzy criterion model for reservoir operations and a case study, Computers & Mathematics with Applications, Vol.37, Nos.11-12, 65-75, 1999.

  80. Esogbue A.O. and Liu B, Cluster validity for fuzzy criterion clustering, Computers & Mathematics with Applications, Vol.37, Nos.11-12, 95-100, 1999.

  81. Liu B, Dependent-chance programming with fuzzy decisions, IEEE Transactions on Fuzzy Systems, Vol.7, No.3, 354-360, 1999.

  82. Iwamura K. and Liu B, Dependent-chance integer programming applied to capital budgeting, Journal of the Operations Research Society of Japan, Vol.42, No.2, 117-127, 1999.

  83. Iwamura K. and Liu B, Stochastic operation models for open inventory networks, Journal of Information & Optimization Sciences, Vol.20, No.3, 347-363, 1999.

  84. Liu Y, and Zhang G, On the completeness of fuzzy measure-space, Fuzzy Sets and Systems, Vol. 102, No. 2, 345-351, 1999.

  85. Liu L, A note on the lower bounds of signed domination number of a graph, Discrete Mathematics, Vol. 195, 295-298, 1999.

    1998
     

  86. Liu B and Iwamura K, Chance constrained programming with fuzzy parameters, Fuzzy Sets and Systems, Vol.94, No.2, 227-237, 1998.

  87. Iwamura K. and Liu B, Chance constrained integer programming models for capital budgeting in fuzzy environments, Journal of the Operational Research Society, Vol.49, No.8, 854-860, 1998.

  88. Liu B, Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms, Computers & Mathematics with Applications, Vol.36, No.7, 79-89, 1998.

  89. Liu B and Iwamura K, A note on chance constrained programming with fuzzy coefficients, Fuzzy Sets and Systems, Vol.100, Nos.1-3, 229-233, 1998.

  90. Liu B, Minimax chance constrained programming models for fuzzy decision systems, Information Sciences, Vol.112, Nos.1-4, 25-38, 1998.

  91. Zhu Y, Positive solutions to a system of variational inequalities, Applied Mathematics Letters, Vol. 11, No. 4, 63-70, 1998.

  92. Li H, Lu M, Sun Z, Hamiltonicity in 2-connected graphs with claw, Discrete Mathematics, Vol. 183, 223-236, 1998.

  93. Liu L, A note on the total chromatics number of Halin-graphs, Applied Mathematics Letters, Vol. 11, No. 5, 23-27, 1998.

  94. Zhao R, K.Iwamura and Liu B, A Genetic Algorithm for Multivariate Isotonic Regression, Journal of Information & Optimization Sciences, Vol. 19, No. 2, 273-284, 1998.



  95. 1997
     
  96. Gen M and Liu B, Evolution algorithm for optimal capacity expansion, Journal of the Operations Research Society of Japan, Vol.40, No.1, 1-9, 1997.

  97. Liu B and Iwamura K, Modelling stochastic decision systems using dependent-chance programming, European Journal of Operational Research, Vol.101, No.1, 193-203, 1997.

  98. Liu B, Dependent-chance programming: A class of stochastic optimization, Computers & Mathematics with Applications, Vol.34, No.12, 89-104, 1997.

  99. Li H, Lu M, Tian F, Wei B, Hamiltonicity in 2-connected claw-center independent grephs, Discrete Mathematics, Vol. 165/166, 469-479, 1997.



  100. 1996
     
  101. Liu B and Esogbue A.O, Fuzzy criterion set and fuzzy criterion dynamic programming, Journal of Mathematical Analysis and Applications, Vol.199, No.1, 293-311, 1996.

  102. Gen M, Liu B and Ida K, Evolution program for deterministic and stochastic optimizations, European Journal of Operational Research, Vol.94, No.3, 618-625, 1996.

  103. Liu B, Dependent-chance goal programming and its genetic algorithm based approach, Mathematical and Computer Modelling, Vol.24, No.7, 43-52, 1996.

  104. Iwamura K. and Liu B, A genetic algorithm for chance constrained programming, Journal of Information & Optimization Sciences, Vol.17, No.2, 409-422, 1996.

  105.