Water Resources Management

, Volume 4, Issue 4, pp 251–271 | Cite as

Selecting risk levels in chance-constrained reservoir operation modeling: A fuzzy set approach

  • Dragan A. Savic
  • Slobodan P. Simonovic


Many modifications, extensions, discussions, and evaluations of chance-constrained reservoir operating models have been reported in the technical literature. Lack of economic data and the fact that the establishment of acceptable risk levels in these types of models involves a human factor with all its vagueness of perception, subjectivity, and attitudes may not permit proper application of either reliability or multiobjective programming approaches. This paper presents a unique methodology for handling a practical problem of selecting risk levels in chance-constrained reservoir operation modeling. The proposed methodology is based on fuzzy set theory. Two types of fuzzy sets are used in the formulation of the reservoir long-term planning model, one for constraints and one for the objective function. An iterative solution algorithm for deriving an optimal decision using fuzzy set operations and the chance-constrained approach is developed and presented. A practical application of the approach demonstrates the feasibility and efficiency of both the proposed approach and its iterative search procedure for selecting risk levels in chance-constrained reservoir modeling.

Key words

Reservoir operation treatment of imprecision optimization satisficing fuzzy sets 


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  1. Bardossy, A., Bogardi, I., Duckstein, L., and Kelly, W. E., 1990, Fuzzy regression in hydrology, Water Resour. Res. 26(7), 1497–1508.Google Scholar
  2. Bellman, R. E. and Zadeh, L. A., 1970, Decision-making in a fuzzy environment, Management Sci. 17(4), 141–164.Google Scholar
  3. Bharathi, D. B. and Sarma, V. V. S., 1985, Estimation of fuzzy memberships for histograms, Inform. Sci. 43–59.Google Scholar
  4. Bogardi, L. Bardossy, A., and Duckstein, L., 1983, Regional management of an aquifer for mining under fuzzy environmental objectives, Water Resour. Res. 19(6), 1394–1402.Google Scholar
  5. Bras, R. L. Buchananan, R., and Curry, K. C., 1983, Real time adaptive closed loop control of reservoirs with the High Aswan dam as a case study, Water Resour. Res. 19(1), 33–52.Google Scholar
  6. Brown, C. B., 1989, Civil engineering optimizing and satisficing, Papers presented at NATO Advanced Study Institute: Optimization and Decision Support Systems in Civil Engineering, Edinburgh, UK.Google Scholar
  7. Changchit, C. and Terrell, M. P., 1989, CCGP model for multiobjective reservoir systems, J. Water Resour. Planning Management ASCE 115(5), 658–670.Google Scholar
  8. Charnes, A. and Cooper, W. W., 1959, Change-constrained programming, Management Sci. 6(1), 73–79.Google Scholar
  9. Charnes, A. and Cooper, W. W., 1983, Response to ‘Decision problems under risk and chance constrained programming: Dilemmas in the transition’, Management Sci. 29(6), 750–753.Google Scholar
  10. Chu, A. T., Kalaba, R. e., and Spingarn, K., 1979, A comparison of two methods for determining the weights of belonging to fuzzy sets, J. Optim. Theory Appl. 27(4), 531–538.Google Scholar
  11. Curry, G. L., Helm, J. C., and Clark, R. A., 1973, Chance-constrained model of system of reservoirs, J. Hydraulic Div. ASCE, 99 [HY12], 2353–2366.Google Scholar
  12. Freeling, A. N. S., 1980, Fuzzy sets and decision analysis, IEEE Trans. Systems Man Cybernet. SMC 10(7), 341–354.Google Scholar
  13. Hogan, J.A., Morris, J. G., and Thompson, H. E., 1981, Decision problems under risk and chance constrained programming: Dilemmas in the transition, Management Sci. 27(6), 698–716.Google Scholar
  14. Jaroslav Cerni Institute, 1976, Water resources analysis of the Gruza Reservoir alternatives, Internal report (in Serbian).Google Scholar
  15. Kindler, J., 1990, Water resources planning in the 90s: The case of resources allocation with imprecise demand estimates, Invited presentation at the International Symposium on Water Resources Systems Application, Winnipeg, Canada.Google Scholar
  16. Loaiciga, H. A., 1988, On the use of chance constraints in reservoir design and operation modeling, Water Resour. Res. 24(11), 1969–1975.Google Scholar
  17. Maiers, J. and Y. S. Sherif, 1985, Applications of fuzzy set theory, IEEE Trans. Systems Man Cybernet. SMC 15(1), 175–189.Google Scholar
  18. Nachtnebel, H. P., Hanisch, P., and Duckstein, L., 1986, Multicriterion analysis of small hydropower plants under fuzzy objectives, Ann. Regional Sci. 20(3), 86–100.Google Scholar
  19. Neter, J., Wasserman, W., and Kutner, M. H., 1989, Applied Linear Regression Models, Richard D. Irwin, Inc., Boston.Google Scholar
  20. Palmer, R. N. and Lund, J. R., 1985, Multi-objective analysis with subjective information, J. Water Resour. Planning and Management ASCE 111(4), 399–416.Google Scholar
  21. Pedrycz, W., 1989, Fuzzy Control and Fuzzy Systems, Research Studies Press Ltd., Somerset, England.Google Scholar
  22. Rakes, R. and Reeves, R., 1985, Selecting tolerances in chance-constrained programming: A multiple objective linear programming approach, Oper. Res. Lett. 4(2), 65–69.Google Scholar
  23. ReVelle, C., Joeres, E., and Kirby, W., 1969, The linear decision rule in reservoir management and design. 1, Development of the stochastic model, Water Resour. Res. 5(4), 767–777.Google Scholar
  24. Saaty, T. L., 1977, A scaling method for priorities in hierarchical structures, J. Math. Psych. 15, 234–281.Google Scholar
  25. Saaty, T. L., 1980, The Analytic Hierarchy Process, McGraw-Hill, New York.Google Scholar
  26. Saaty, T. L. and Vargas, L. G., 1982, The Logic of Priorities, Kluwer-Nijhoff Publishing, Boston.Google Scholar
  27. Sakawa, M., Yano, H., and Yumine, T., 1987, An interactive fuzzy satisficing method for multiobjective linear-programming problems and its application, IEEE Trans. Systems Man Cybern. SMC 17(4), 654–661.Google Scholar
  28. Sengupta, J. K., 1972, Stochastic Programming. Methods and Applications, North-Holland, Amsterdam.Google Scholar
  29. Simonovic, S. P., 1979, Two-step algorithm for design-stge long-term control of a multipurpose reservoir, Adv. Water Resour. 2, 47–49.Google Scholar
  30. Simonovic, S. P. and Marino, M. A., 1981, Reliability programming in reservoir management 2. Riskloss functions, Water Resour. Res. 17(4), 822–826.Google Scholar
  31. Simonovic, S. P., 1987, Comment on ‘Evaluation of a “Reliability Programming” reservoir model’ by J. B. Strycharczyk and J. R. Stedinger, Water Resour. Res. 23(9), 1795–1796.Google Scholar
  32. Stedinger, J. R., 1984, The performance of LDR models for preliminary design of reservoir operation, Water Resour. Res. 20(2), 215–224.Google Scholar
  33. Stedinger, J. R. and J. B. Strycharczyk, 1987, Reply, Water Resour. Res. 23(9), 1801–1802.Google Scholar
  34. Strycharczyk, J. B. and Stedinger, J. R., 1987, Evaluation of a ‘Reliability programming’ reservoir model, Water Resour. Res. 23(2), 225–229.Google Scholar
  35. Tanaka, H., Okuda, T., and Asai, K., 1974, On fuzzy-mathematical programming J. Cybern. 3(4) 37–46.Google Scholar
  36. Uan-On, T. and Helweg, O. J., 1988, Deriving the nonlinear risk-benefit algorithm for reservoirs, Water Resour. Bull. 24(2), 261–268.Google Scholar
  37. Westgate, J. T., 1980, Design of objective functions for reservoir operations, M Sc Thesis, Colorado State University, Fort Collins, Colorado.Google Scholar
  38. Zadeh, L. A., 1965, Fuzzy sets, Inform. and Control 8, 338–353.Google Scholar
  39. Zimmermann, H.-J., 1976, Description and Optimization of Fuzzy Systems, Internal. J. Gen. Systems 2, 209–215.Google Scholar
  40. Zimmermann, H.-J., 1983, Using fuzzy sets in operational research, European J. Oper. Res. 13, 201–216.Google Scholar
  41. Zimmermann, H.-J., 1985, Fuzzy Set Theory and Its Applications, Kluwer-Nijhoff Publishing, Boston.Google Scholar
  42. Zimmermann, H.-J., 1987, Fuzzy Sets, Decision Making, and Expert Systems, Kluwer Academic Publishers, Dordrecht.Google Scholar
  43. Zimmernann, H.-J., and Zysno, P., 1980, Latent connectives in human decision making, Fuzzy Sets and Systems 4, 37–51.Google Scholar

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • Dragan A. Savic
    • 1
  • Slobodan P. Simonovic
    • 1
  1. 1.Civil Engineering DepartmentUniversity of ManitobaWinnipegCanada

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