Abstract
A standard lower-side attainment values based inexact fuzzy two-stage programming (SLA-IFTSP) approach is proposed for supporting multi-water resources management under multi-uncertainties. The method improves upon the existing inexact two-stage stochastic programming by the introduction of a standard average lower-side attainment values based fuzzy linear programming. Multi-uncertainties such as intervals, probabilistic and/or possibilistic distributions and their combinations in water resources management can be directly communicated into the water allocation process. The risk of infeasibility caused by the random water availabilities can be analyzed by imposing economic penalties when the designed water allocations would not be satisfied after the occurrence of random seasonal flows. Based on the standard average lower-side attainment index, the fuzzy random relationships representing various subjective judgments in the model can be transformed into corresponding deterministic ones without additional constraints, and thus guarantee a higher computational efficiency. A hypothetical case regarding two-source water resources management is adopted for demonstrating its applicability. Reasonable solutions have been generated. They provide desired water allocations with maximized system benefit under different water availability levels. The solutions of intervals with different probabilities can be used for generating decision alternatives. Comparisons between the solutions from SLA-IFTSP and those from ITSP are also undertaken. They show that SLA-IFTSP can generate more reasonable water allocation patterns with higher net system benefits than ITSP.
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Abbreviations
- f :
-
Net system benefit ($)
- B j :
-
Benefit for allocating per unit of water to user j
- T mj :
-
Allocation target promised to user j by surface reservoir m (the first-stage decision variable)
- T nj :
-
Allocation target promised to user j by local river n (the first-stage variable)
- C j :
-
Penalty cost for per unit of water not delivered to user j
- q m :
-
Fuzzy random variable being equal to available water amount of reservoir m
- q n :
-
Fuzzy random variable being equal to available water amount of local river n
- D mjkh :
-
Amount by which water-allocation target T mj is not met when the inflow of reservoir m is h level
- D njkh :
-
Amount by which water-allocation target T nj is not met when the inflow of local river n is h level
- θ mj :
-
Variation factor representing water losses in the allocating stage from reservoir m to user j
- θ nj :
-
Variation factor representing water losses in the allocating stage from local river n to user j
- J :
-
Index for users
- M :
-
Index for reservoir
- n :
-
Index for local river
- k :
-
Index for planning periods
- H j :
-
Maximum total water shortage for user j over the planning horizon
- G k :
-
Maximum total water shortage for all users during each planning period
- BR minj :
-
Minimum water demand of user j
- BR maxj :
-
Maximum water demand of user j
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Acknowledgments
This research has been supported by the Major Science and Technology Program for Water Pollution Control and Treatment (2009ZX07104-004). The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions.
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Hu, Q., Huang, G., Liu, Z. et al. Inexact fuzzy two-stage programming for water resources management in an environment of fuzziness and randomness. Stoch Environ Res Risk Assess 26, 261–280 (2012). https://doi.org/10.1007/s00477-011-0503-7
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DOI: https://doi.org/10.1007/s00477-011-0503-7