[1] A fuzzy optimization model is developed for the seasonal water quality management of river systems. The model addresses the uncertainty in a water quality system in a fuzzy probability framework. The occurrence of low water quality is treated as a fuzzy event. Randomness associated with the water quality indicator is linked to this fuzzy event using the concept of probability of a fuzzy event. In most water quality management models the risk level for violation of a water quality standard is constrained by a preassigned value through a chance constraint. In the fuzzy risk approach a range of risk levels is specified by considering a fuzzy set of low risk, instead of using a chance constraint. Thus the two levels of uncertainty, one associated with low water quality and the other with low risk, are quantified and incorporated in the management model. The model takes into account the seasonal variations of river flow to specify seasonal fraction removal levels for the pollutants. Fuzzy sets of low water quality are considered in each season. The membership functions of these fuzzy sets represent the degree of low water quality associated with the discrete states of water quality in a season. The fuzzy risk of low water quality in a season at a checkpoint in the river system is expressed in terms of the degree of low water quality and the steady state probabilities associated with discrete river flows. Considering the fuzzy goals of the pollution control agency and dischargers, and the crisp constraints, the water quality management problem is formulated as a fuzzy optimization model. The model solution gives seasonal fraction removal levels for the pollutants. Application of the fuzzy optimization model is illustrated with a hypothetical river system.

ABSTRACT:A fuzzy waste-load allocation model, FWLAM, is developed for water quality management of a river system using fuzzy multiple-objective optimization. An important feature of this model is its capability to incorporate the aspirations and conflicting objectives of the pollution control agency and dischargers. The vagueness associated with specifying the water quality criteria and fraction removal levels is modeled in a fuzzy framework. The goals related to the pollution control agency and dischargers are expressed as fuzzy sets. The membership functions of these fuzzy sets are considered to represent the variation of satisfaction levels of the pollution control agency and dischargers in attaining their respective goals. TWO formulations-namely, the MAX-MIN and MAX-BIAS formulations-are proposed for FWLAM. The MAX-MIN formulation maximizes the minimum satisfaction level in the system. The MAX-BIAS formulation maximizes a bias measure, giving a solution that favors the dischargers. Maximization of the bias measure attempts to keep the satisfaction levels of the dischargers away from the minimum satisfaction level and that of the pollution control agency close to the minimum satisfaction level. Most of the conventional water quality management models use waste treatment cost curves that are uncertain and nonlinear. Unlike such models, FWLAM avoids the use of cost curves. Further, the model provides the flexibility for the pollution control agency and dischargers to specify their aspirations independently.

Wafer quolify munagemenfof0 river sysrem is addressed i n n fizzy andprobobilisric framework. Two ryper of uneerfoinry, namely randomness and vagueness. ore rreared simrrlfonmii.dy in the monogemmf problem. A.fuzzy-ser-bo.~ed dqfinifion rhaf is a more general case offhe existing crisp-set-based dqfmirion of low w l e r qualify is infroduced. The evenr of low wafer qu01it)i af r? check-poinr in rhe river syrrsm i.s considered as n fuzzy even?. The rirk of low ~, a f e r qualify ir rhen definedm rhepobabiliry of rhe fuzzy even1 s f l m wafer qu01if.y. Insreadof consfrainiag fhe risk ofviolafim of warw quality srmdord by nfinile value of risk throuplr a chance con.~framr. LI furry sef sf low risk rhaf con.sidws a range ofri.yk levels with approprioc membership values i$ inrroduced. Diffkrmr goalr o.wocioird with rhr manopemenf problem m e expressed as ,fuzzy iei-i. The rcirrlring manapmenf prohlem ir furmuluiedas a,fuzzy mslriobjperive oprimimtinn nrohlem I

Reservoir operation modeling and optimization are inevitable components of water resources planning and management. Determination of reservoir operating policy is a multi-stage decision-making problem characterized by uncertainty. Uncertainty in inflows and power demands lead to varying degrees of the working of a reservoir from one period to another. This transition, being ambiguous in nature, can be addressed in a fuzzy framework. The different working states of the reservoir are described as fuzzy states. Based on the degree of success in meeting the power demand and randomness associated with inflows, hydropower production is considered as a random fuzzy event. This paper examines the scope of profust reliability theory, a theory used in the reliability analysis of manufactured systems, in the performance optimization of a hydropower reservoir system. The operating policy derived from a profust reliability-based optimization model is compared with a simulation model. The model is then used to derive the optimal operation policy for a hypothetical reservoir fed by normally distributed inflow, for a period of five years. The results show that the model is useful in deriving optimal operating policies with improved reliabilities in hydropower production.

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