Skip to main content
SpringerLink
Log in

Fuzzy Multi-Criteria Optimization: Possibilistic and Fuzzy/Stochastic Approaches

  • Chapter
Multiple Criteria Decision Analysis

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 233))

Abstract

In this chapter, we review fuzzy multi-criteria optimization focusing on possibilistic treatments of objective functions with fuzzy coefficients and on interactive fuzzy stochastic multiple objective programming approaches. In the first part, treatments of objective functions with fuzzy coefficients dividing into single objective function case and multiple objective function case. In single objective function case, multi-criteria treatments, possibly and necessarily optimal solutions, and minimax regret solutions are described showing the relations to multi-criteria optimization. In multiple objective function case, possibly and necessarily efficient solutions are investigated. Their properties and possible and necessary efficiency tests are shown. As one of interactive fuzzy stochastic programming approaches, multiple objective programming problems with fuzzy random parameters are discussed. Possibilistic expectation and variance models are proposed through incorporation of possibilistic and stochastic programming approaches. Interactive algorithms for deriving a satisficing solution of a decision maker are shown.

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 349.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 449.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 449.99
Price excludes VAT (USA)
  • Durable hardcover 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. Averbakh, I., Lebedev, V.: On the complexity of minmax regret linear programming. Eur. J. Oper. Res. 160, 227–231 (2005)

    Article  Google Scholar 

  2. Benayoun, R., de Montgolfier, J., Tergny, J., Larichev, O.I.: Linear programming with multiple objective functions: STEP Method (STEM). Math. Program. 1, 366–375 (1971)

    Article  Google Scholar 

  3. Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming, 2nd edn. Springer, New York/Dordrecht/Heidelberg/London (2011)

    Book  Google Scholar 

  4. Bitran, G.R.: Linear multiple objective problems with interval coefficients. Manag. Sci. 26, 694–706 (1980)

    Article  Google Scholar 

  5. Borde, J., Crouzeix, J.P.: Convergence of a Dinkelbach-type algorithm in generalized fractional programming. Z. Oper. Res. 31, 31–54 (1987)

    Article  Google Scholar 

  6. Caballero, R., Cerdá, E., Muñoz, M.M., Rey, L., Stancu-Minasian, I.M.: Efficient solution concepts and their relations in stochastic multiobjective programming. J. Optim. Theory Appl. 110, 53–74 (2001)

    Article  Google Scholar 

  7. Chernikova, N.V.: Algorithm for finding a general formula for the nonnegative solutions of a system of linear inequalities. USSR Comput. Math. Math. Phys. 5, 228–233 (1965)

    Article  Google Scholar 

  8. Contini, B.: A stochastic approach to goal programming. Oper. Res. 16, 576–586 (1968)

    Article  Google Scholar 

  9. Dantzig, G.B.: Linear programming under uncertainty. Manag. Sci. 1, 197–206 (1955)

    Article  Google Scholar 

  10. Dubois, D.: Linear programming with fuzzy data. In: Bezdek, J.C. (ed.) Analysis of Fuzzy Information. Application in Engineering and Science, vol. III, pp. 241–263. CRC Press, Boca Raton (1987)

    Google Scholar 

  11. Dubois, D., Prade, H.: Fuzzy numbers: an overview. In: Bezdek, J.C. (ed.) Analysis of Fuzzy Information. Mathematics and Logic, vol. I, pp. 3–39. CRC Press, Boca Raton (1987)

    Google Scholar 

  12. Dubois, D., Prade, H.: Possibility Theory: An Approach to Computerized Processing of Uncertainty. Plenum Press, New York (1988)

    Book  Google Scholar 

  13. Ehrgott, M., Gandibleux, X. (eds.): Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys. Kluwer Academic Publishers, Boston (2002)

    Google Scholar 

  14. Evans, J.P., Steuer, R.E.: A revised simplex method for linear multiple objective programs. Math. Program. 5, 54–72 (1973)

    Article  Google Scholar 

  15. Furukawa, N.: Parametric orders on fuzzy numbers and their roles in fuzzy optimization problems. Optimization 40, 171–192 (1997)

    Article  Google Scholar 

  16. Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)

    Article  Google Scholar 

  17. Gil, M.A., Lopez-Diaz, M., Ralescu, D.A.: Overview on the development of fuzzy random variables. Fuzzy Sets Syst. 157, 2546–2557 (2006)

    Article  Google Scholar 

  18. Goicoecha, A., Hansen, D.R., Duckstein, L.: Multiobjective Decision Analysis with Engineering and Business Applications. Wiley, New York (1982)

    Google Scholar 

  19. Hladík, M.: On necessary efficient solutions in interval multiobjective linear programming. In: CD-ROM Proceedings of the 25th Mini-EURO Conference Uncertainty and Robustness in Planning and Decision Making URPDM 2010, pp. 1–10 (2010)

    Google Scholar 

  20. Hladík, M.: Complexity of necessary efficiency in interval linear programming and multiobjective linear programming. Optim. Lett. 6, 893–899 (2012)

    Article  Google Scholar 

  21. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd Revised and Enlarged Edition. Springer, Berlin (1995)

    Google Scholar 

  22. Ida, M.: Necessary efficient test in interval multiobjective linear programming. In: Proceedings of the Eighth International Fuzzy Systems Association World Congress, pp. 500–504 (1999)

    Google Scholar 

  23. Ida, M.: Efficient solution generation for multiple objective linear programming and uncertain coefficients. In: Proceedings of the 8th Bellman Continuum, pp. 132–136 (2000)

    Google Scholar 

  24. Infanger, G. (eds.): Stochastic Programming. Springer, New York/Dorderecht/Heidelberg/ London (2011)

    Google Scholar 

  25. Inuiguchi, M.: Multiple objective linear programming with fuzzy coefficients. In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multiple Criteria Decision Analysis: State of the Art Surveys, pp.723–760. Kluwer Academic Publishers, Boston (2002)

    Google Scholar 

  26. Inuiguchi, M.: On possibilistic/fuzzy optimization. In: Melin, P., Castillo, O., Aguilar, L.T., Kacprzyk, J., Pedrycz, W. (eds.) Foundations of Fuzzy Logic and Soft Computing. Proceedings of 12th International Fuzzy Systems Association World Congress, IFSA 2007, Cancun, June 2007. Lecture Notes in Artificial Intelligence, vol. 4529, pp. 351–360. Springer, Berlin (2007)

    Google Scholar 

  27. Inuiguchi, M.: Necessary efficiency is partitioned into possible and necessary optimalities. In: Proceedings of 15th International Fuzzy Systems Association World Congress, IFSA-NAFIPS 2013 (2013)

    Google Scholar 

  28. Inuiguchi, M., Kume, Y.: A discrimination method of possibly efficient extreme points for interval multiobjective linear programming problems (in Japanese). Trans. SICE 25, 824–826 (1989)

    Article  Google Scholar 

  29. Inuiguchi, M., Kume, Y.: Solution concepts for fuzzy multiple objective programming problems. Jpn. J. Fuzzy Theory Syst. 2, 1–22 (1990)

    Google Scholar 

  30. Inuiguchi, M., Kume, Y.: Minimax regret in linear programming problems with an interval objective function. In: Tzeng, G.H., Wang, H.F., Wen, U.P., Yu, P.L. (eds.) Multiple Criteria Decision Making, pp. 65–74. Springer, New York (1994)

    Chapter  Google Scholar 

  31. Inuiguchi, M., Ramík, J.: Possibilistic linear programming: a brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets Syst. 111, 3–28 (2000)

    Article  Google Scholar 

  32. Inuiguchi, M., Sakawa, M.: Possible and necessary optimality tests in possibilistic linear programming problems. Fuzzy Sets Syst. 67, 29–46 (1994)

    Article  Google Scholar 

  33. Inuiguchi, M., Sakawa, M.: Minimax regret solutions to linear programming problems with an interval objective function. Eur. J. Oper. Res. 86, 526–536 (1995)

    Article  Google Scholar 

  34. Inuiguchi, M., Sakawa, M.: Possible and necessary efficiency in possibilistic multiobjective linear programming problems and possible efficiency test. Fuzzy Sets Syst. 78, 231–241 (1996)

    Article  Google Scholar 

  35. Inuiguchi, M., Sakawa, M.: An achievement rate approach to linear programming problems with an interval objective function. J. Oper. Res. Soc. 48, 25–33 (1997)

    Article  Google Scholar 

  36. Inuiguchi, M., Sakawa, M.: Robust optimization under softness in a fuzzy linear programming problem. Int. J. Approx. Reason. 18, 21–34 (1998)

    Article  Google Scholar 

  37. Inuiguchi, M., Tanino, T.: On computation methods for a minimax regret solution based on outer approximation and cutting hyperplanes. Int. J. Fuzzy Syst. 3, 548–557 (2001)

    Google Scholar 

  38. Inuiguchi, M., Tanino, T.: Enumeration of all possibly optimal vertices with possible optimality degrees in linear programming problems with a possibilistic objective function. Fuzzy Optim. Decis. Making 3, 311–326 (2004)

    Article  Google Scholar 

  39. Inuiguchi, M., Tanino, T., Tanaka, H.: Optimization approaches to possibilistic linear programming problems. In: Proceedings of Joint 9th IFSA World Congress and 20th NAFIPS International Conference, pp. 2724–2729 (2001)

    Google Scholar 

  40. Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48, 219–225 (1990)

    Article  Google Scholar 

  41. Kasperski, A.: Discrete Optimization with Interval Data: Minimax Regret and Fuzzy Approach. Springer, Berlin (2008)

    Google Scholar 

  42. Katagiri, H., Ishii, H.: Linear programming problem with fuzzy random constraint. Math. Jpn. 52, 123–129 (2000)

    Google Scholar 

  43. Katagiri, H., Sakawa, M.: A study on fuzzy random linear programming problems based on possibility and necessity. In: Bilgic, T., DeBaets, B., Kaynak, O. (eds.) Fuzzy Sets and Systems, IFSA 2003: 10th International Fuzzy Systems Association World Congress, Istanbul. Lecture Notes in Computer Science, pp. 725–732. Springer, Berlin (2003)

    Chapter  Google Scholar 

  44. Katagiri, H., Ishii, H., Itoh, T.: Fuzzy random linear programming problem. In: Proceedings of Second European Workshop on Fuzzy Decision Analysis and Neural Networks for Management, Planning and Optimization, Dortmund, pp. 107–115 (1997)

    Google Scholar 

  45. Katagiri, H., Ishii, H., Sakawa, M.: Linear programming problem under fuzziness and randomness. In: Proceedings of International Conference on Applied Stochastic System Modeling, Kyoto, pp. 97–106 (2000)

    Google Scholar 

  46. Katagiri, H., Sakawa, M., Ishii, H.: Multiobjective fuzzy random linear programming using E-model and possibility measure. In: Joint 9th IFSA World Congress and 20th NAFIPS International Conference, Vancouver, pp. 2295–2300 (2001)

    Google Scholar 

  47. Katagiri, H., Ishii, H., Sakawa, M.: Linear programming problems with random fuzzy variable coefficients. In: Proceedings of 5th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty, Koyasan, vol. 1, pp. 55–58 (2002)

    Google Scholar 

  48. Katagiri, H., Sakawa, M., Ohsaki, S.: An interactive satisficing method through the variance minimization model for fuzzy random linear programming problems. In: Tanino, T., Tamaki, T., Inuiguchi, M. (eds.) Multi-Objective Programming and Goal-Programming: Theory and Applications. Advances in Soft Computing, pp.171–176. Springer, Berlin (2003)

    Chapter  Google Scholar 

  49. Katagiri, H., Sakawa, M., Kato, K., Nishizaki, I.: A fuzzy random multiobjective 0-1 programming based on the expectation optimization model using possibility and necessity measures. Math. Comput. Model. 40, 411–421 (2004)

    Article  Google Scholar 

  50. Katagiri, H., Sakawa, M., Ishii, H.: Studies of stochastic programming models using possibility and necessity measures for linear programming problems with fuzzy random variable coefficients. Electron. Commun. Jpn. Part III-Fundam. Electron. Sci. 88, 68–75 (2005)

    Article  Google Scholar 

  51. Katagiri, H., Sakawa, M., Kato, K., Ohsaki, S.: 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 III-Fundam. Electron. Sci. 88, 20–28 (2005)

    Article  Google Scholar 

  52. Katagiri, H., Hasuike, T., Ishii, H., Nishizaki, I.: Interactive multiobjective programming under random fuzzy environments. In: Proceedings of the 11th Czech-Japan Seminar on Data Analysis and Decision Making Under Uncertainty, Sendai, pp. 33–38 (2008)

    Google Scholar 

  53. Katagiri, H., Sakawa, M., Kato, K., Nishizaki, I.: Interactive multiobjective fuzzy random linear programming: maximization of possibility and probability. Eur. J. Oper. Res. 188, 530–539 (2008)

    Article  Google Scholar 

  54. Katagiri, H., Niwa, K., Kubo, D., Hasuike, T.: 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 (2010)

    Google Scholar 

  55. Katagiri, H., Kato, K., Hasuike, T.: A random fuzzy minimum spanning tree problem through a possibility-based value at risk model. Expert Syst. Appl. 39, 205–220 (2012)

    Google Scholar 

  56. Katagiri, H., Uno, T., Kato, K., Tsuda, H., Tsubaki, H.: Random fuzzy multiobjective programming: optimization of possibilistic value at risk (pVaR). Expert Syst. Appl. 40, 563–574 (2013)

    Article  Google Scholar 

  57. Klein, G., Moskowitz, H., Ravindran, A.: Interactive multiobjective optimization under uncertainty. Manag. Sci. 36, 58–75 (1990)

    Article  Google Scholar 

  58. Kwakernaak, H.: Fuzzy random variable-1. Inf. Sci. 15, 1–29 (1978)

    Article  Google Scholar 

  59. Lai, Y.-J., Hwang, C.L.: Fuzzy Mathematical Programming. Springer, Berlin (1992)

    Book  Google Scholar 

  60. Liu, B.: Fuzzy random chance-constrained programming. IEEE Trans. Fuzzy Syst. 9, 713–720 (2001)

    Article  Google Scholar 

  61. Liu, B.: Fuzzy random dependent-chance programming. IEEE Trans. Fuzzy Syst. 9, 721–726 (2001)

    Article  Google Scholar 

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

    Article  Google Scholar 

  63. Liu, B.: Uncertainty Theory. Springer, Berlin/Heidelberg/New York (2004)

    Book  Google Scholar 

  64. Luhandjula, M.K.: On possibilistic linear programming. Fuzzy Sets Syst. 18, 15–30 (1986)

    Article  Google Scholar 

  65. Luhandjula, M.K.: Multiple objective programming problems with possibilistic coefficients. Fuzzy Sets Syst. 21, 135–145 (1987)

    Article  Google Scholar 

  66. Luhandjula, M.K.: Fuzzy optimization: an appraisal. Fuzzy Sets Syst. 30, 257–282 (1989)

    Article  Google Scholar 

  67. Luhandjula, M.K.: Fuzziness and randomness in an optimization framework. Fuzzy Sets Syst. 77, 291–297 (1996)

    Article  Google Scholar 

  68. Luhandjula, M.K.: Fuzzy stochastic linear programming: survey and future research directions. Eur. J. Oper. Res. 174, 1353–1367 (2006)

    Article  Google Scholar 

  69. Luhandjula, M.K., Gupta, M.M.: On fuzzy stochastic optimization. Fuzzy Sets Syst. 81, 47–55 (1996)

    Article  Google Scholar 

  70. Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)

    Google Scholar 

  71. Mausser, H.E., Laguna, M.: A new mixed integer formulation for the maximum regret problem. Int. Trans. Oper. Res. 5, 389–403 (1998)

    Article  Google Scholar 

  72. Oliveira, C., Antunes, C.H.: Multiple objective linear programming models with interval coefficients—an illustrated overview. Eur. J. Oper. Res. 181, 1434–1463 (2007)

    Article  Google Scholar 

  73. Puri, M.L., Ralescu, D.A.: Fuzzy random variables. J. Math. Anal. Appl. 14, 409–422 (1986)

    Article  Google Scholar 

  74. Qiao, Z., Zhang, Y., Wang, G.-Y.: On fuzzy random linear programming. Fuzzy Sets Syst. 65, 31–49 (1994)

    Article  Google Scholar 

  75. Ramík, J., Římánek, J.: Inequality relation between fuzzy numbers and its use in fuzzy optimization. Fuzzy Sets Syst. 16, 123–138 (1985)

    Article  Google Scholar 

  76. Rivaz, S., Yaghoobi, M.A.: Minimax regret solution to multiobjective linear programming problems with interval objective functions coefficients. Cent. Eur. J. Oper. Res. 21, 625–649 (2013)

    Article  Google Scholar 

  77. Rommelfanger, H.: A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic variables. Fuzzy Sets Syst. 158, 1892–1904 (2007)

    Article  Google Scholar 

  78. Rommelfanger, H., Słowiński, R.: Fuzzy linear programming with single or multiple objective functions. In: Słowiński, R. (ed.) Fuzzy Sets in Decision Analysis. Operations Research and Statistics. Kluwer Academic Publishers, Boston (1998)

    Google Scholar 

  79. Rommelfanger, H., Hanuscheck, R., Wolf, J.: Linear programming with fuzzy objective. Fuzzy Sets Syst. 29, 31–48 (1989)

    Article  Google Scholar 

  80. Sakawa, M.: Fuzzy Sets and Interactive Multiobjective Optimization. Plenum Press, New York (1993)

    Book  Google Scholar 

  81. Sakawa, M., Yano, H.: Feasibility and Pareto optimality for multiobjective nonlinear programming problems with fuzzy parameters. Fuzzy Sets Syst. 43, 1–15 (1991)

    Article  Google Scholar 

  82. Sakawa, M., Takahashi, J., Yano, H.: Extended Pareto optimality concept for multiobjective linear programming problems with fuzzy parameters and its properties. Electron. Commun. Jpn. Part III-Fundam. Electron. Sci. 73, 1–9 (1990)

    Google Scholar 

  83. Słowiński, R.: Fuzzy multi-objective linear programming. Belg. J. Oper. Res. Stat. Comput. Sci. 37, 83–98 (1997)

    Google Scholar 

  84. Słowiński, R., Teghem, J.: Fuzzy vs. stochastic approaches to multicriteria linear programming under uncertainty. Nav. Res. Logist. 35, 673–695 (1988)

    Google Scholar 

  85. Słowiński, R., Teghem, J. (eds.): Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming Under Uncertainty. Kluwer Academic Publishers, Dordrecht (1990)

    Google Scholar 

  86. Stancu-Minasian, I.M.: Stochastic Programming with Multiple Objective Functions. D. Reidel Publishing Company, Dordrecht (1984)

    Google Scholar 

  87. Stancu-Minasian, I.M.: Overview of different approaches for solving stochastic programming problems with multiple objective functions. In: Słowiński, R., Teghem, J. (eds.) Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming Under Uncertainty, pp.71–101. Kluwer Academic Publishers, Dordrecht/Boston/London (1990)

    Chapter  Google Scholar 

  88. Steuer, R.E.: Algorithms for linear programming problems with interval objective function coefficients. Math. Oper. Res. 6, 333–348 (1981)

    Article  Google Scholar 

  89. Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation, and Application. Wiley, New York (1986)

    Google Scholar 

  90. Tanaka, H., Asai, K.: Fuzzy linear programming problems with fuzzy numbers. Fuzzy Sets Syst. 13, 1–10 (1984)

    Article  Google Scholar 

  91. Tanaka, H., Ichihashi, H., Asai, K.: A formulation of fuzzy linear programming problem based on comparison of fuzzy numbers. Control Cybern. 13, 185–194 (1984)

    Google Scholar 

  92. Teghem, J., Dufrane, D., Thauvoye, M., Kunsch, P.: STRANGE: an interactive method for multi-objective linear programming under uncertainty. Eur. J. Oper. Res. 26, 65–82 (1986)

    Article  Google Scholar 

  93. Urli, B., Nadeau, R.: Stochastic MOLP with incomplete information: an interactive approach with recourse. J. Oper. Res. Soc. 41, 1143–1152 (1990)

    Article  Google Scholar 

  94. Urli, B., Nadeau, R.: PROMISE/scenarios: an interactive method for multiobjective stochastic linear programming under partial uncertainty. Eur. J. Oper. Res. 155, 361–372 (2004)

    Article  Google Scholar 

  95. Wang, G.-Y., Qiao, Z.: Linear programming with fuzzy random variable coefficients. Fuzzy Sets Syst. 57, 295–311 (1993)

    Article  Google Scholar 

  96. Wang, G., Zhang, Y.: The theory of fuzzy stochastic processes. Fuzzy Sets Syst. 51, 161–178 (1992)

    Article  Google Scholar 

  97. White, D.J.: Optimality and Efficiency. Wiley, New York (1982)

    Google Scholar 

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

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, 560-8531, Japan

    Masahiro Inuiguchi

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

    Kosuke Kato

  3. Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

    Hideki Katagiri

Authors
  1. Masahiro Inuiguchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kosuke Kato
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hideki Katagiri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masahiro Inuiguchi .

Editor information

Editors and Affiliations

  1. Department of Economics and Business; Portsmouth Business School, Operations & Systems Management, University of Catania; University of Portsmouth, Catania; Portsmouth, Italy

    Salvatore Greco

  2. Department of Management Science, Lancaster University, Lancaster, United Kingdom

    Matthias Ehrgott

  3. Universidade de Lisboa, CEG-IST, Instituto Superior Técnico, Lisboa, Portugal

    José Rui Figueira

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media New York

About this chapter

Cite this chapter

Inuiguchi, M., Kato, K., Katagiri, H. (2016). Fuzzy Multi-Criteria Optimization: Possibilistic and Fuzzy/Stochastic Approaches. In: Greco, S., Ehrgott, M., Figueira, J. (eds) Multiple Criteria Decision Analysis. International Series in Operations Research & Management Science, vol 233. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3094-4_20

Download citation

Publish with us

Policies and ethics

Search

Navigation