Uncertainty Modeling For River Water Quality Control

Abstract

Waste Load Allocation (WLA) in rivers refers to the determination of required pollutant fractional removal levels at a set of point sources of pollution to ensure that water quality standards are maintained throughout the system. Optimal waste load allocation implies that the selected pollution treatment vector not only maintains the water quality standards, but also results in the best value for the objective function defined for the management problem. Waste load allocation problems are characterized by uncertainties due to the randomness and imprecision. Uncertainty due to randomness arises mainly due to the random nature of the variables influencing the water quality. Uncertainty due to imprecision or fuzziness is associated with setting up the water quality standards and goals of the Pollution Control Agencies (PCA), and the dischargers (e.g., industries and municipal dischargers). Many decision problems in water resources applications are dominated by natural, extreme, rarely occurring, uncertain events. However usually such events will be absent or be rarely present in the historical records. Due to the scarcity of information of these uncertain events, a realistic decision-making becomes difficult. Furthermore, water resources planners often deal with imprecision, mostly due to imperfect knowledge and insufficient or inadequate data. Therefore missing data is very common in most water resources decision problems. Missing data introduces inaccuracy in analysis and evaluation. For instance, the sample mean of the available data can be an inaccurate estimate of the mean of the complete data. Use of sample statistics estimated from inadequate samples in WLA models would lead to incorrect decisions. Therefore there is a necessity to incorporate the uncertainty due to missing data also in WLA models in addition to the uncertainties due to randomness and imprecision. The uncertainty in the input parameters due to missing or inadequate data renders the input parameters (such as mean and variance) as interval grey parameters in water quality decision-making. In a Fuzzy Waste Load Allocation Model (FWLAM), randomness and imprecision both can be addressed simultaneously by using the concept of fuzzy risk of low water quality (Mujumdar and Sasikumar, 2002). In the present work, an attempt is made to also address uncertainty due to partial ignorance due to missing data or inadequate data in the samples of input variables in FWLAM, considering the fuzzy risk approach proposed by Mujumdar and Sasikumar (2002). To address the uncertainty due to missing data or inadequate data, the input parameters (such as mean and variance) are considered as interval grey numbers. The resulting output water quality indicator (such as DO) will also, consequently, be an interval grey number. The fuzzy risk will also be interval grey number when output water quality indicator is an interval grey number. A methodology is developed for the computation of grey fuzzy risk of low water quality, when the input variables are characterized by uncertainty due to partial ignorance resulting from missing or inadequate data in the samples of input variables. To achieve this, an Imprecise Fuzzy Waste Load Allocation Model (IFWLAM) is developed for water quality management of a river system to address uncertainties due to randomness, fuzziness and also due to missing data or inadequate data. Monte Carlo Simulation (MCS) incorporating a water quality simulation model is performed two times for each set of randomly generated input variables: once for obtaining the upper bound of DO and once for the lower bound of DO, by using appropriate upper or lower bounds of interval grey input variables. These two bounds of DO are used in the estimation of grey fuzzy risk by substituting the upper and lower values of fuzzy membership functions of low water quality. A backward finite difference scheme (Chapra, 1997) is used to solve the water quality simulation model. The goal of PCA is to minimize the bounds of grey fuzzy risk, whereas the goal of dischargers is to minimize the fractional removal levels. The two sets of goals are conflicting with each other. Fuzzy multiobjective optimization technique is used to formulate the multiobjective model to provide best compromise solutions. Probabilistic Global Search Lausanne (PGSL) method is used to solve the optimization problem. Finally the results of the model are compared with the results of risk minimization model (Ghosh and Mujumdar, 2006), when the methodology is applied to the case study of the Tunga-Bhadra river system in South India. The model is capable of determining a grey fuzzy risk with the corresponding bounds of DO, at each check point, rather than specifying a single value of fuzzy risk as done in a Fuzzy Waste Load Allocation Model (FWLAM). The IFWLAM developed is based on fuzzy multiobjective optimization problem with ‘max-min’ as the operator, which usually may not result in a unique solution and there exists a possibility of obtaining multiple solutions (Karmakar and Mujumdar, 2006b). Karmakar and Mujumdar (2006b) developed a two-phase Grey Fuzzy Waste Load Allocation Model (two-phase GFWLAM), to determine the widest range of interval-valued optimal decision variables, resulting in the same value of interval-valued optimal goal fulfillment level as obtained from GFWLAM (Karmakar and Mujumdar 2006a). Following Karmakar and Mujumdar (2006b), two optimization models are developed in this study to capture all the decision alternatives or multiple solutions: one to maximize and the other to minimize the summation of membership functions of the dischargers by keeping the maximum goal fulfillment level same as that obtained in IFWLAM to obtain a lower limit and an upper limit of fractional removal levels respectively. The aim of the two optimization models is to obtain a range of fractional removal levels for the dischargers such that the resultant grey fuzzy risk will be within acceptable limits. Specification of a range for fractional removal levels enhances flexibility in decision-making. The models are applied to the case study of Tunga-Bhadra river system. A range of upper and lower limits of fractional removal levels is obtained for each discharger; within this range, the discharger can select the fractional removal level so that the resulting grey fuzzy risk will also be within specified bounds. In IFWLAM, the membership functions are subjective, and lower and upper bounds are arbitrarily fixed. Karmakar and Mujumdar (2006a) developed a Grey Fuzzy Waste Load Allocation Model (GFWLAM), in which uncertainty in the values of membership parameters is quantified by treating them as interval grey numbers. Imprecise membership functions are assigned for the goals of PCA and dischargers. Following Karmakar and Mujumdar (2006a), a Grey Optimization Model with Grey Fuzzy Risk is developed in the present study to address the uncertainty in the memebership functions of IFWLAM. The goals of PCA and dischargers are considered as grey fuzzy goals with imprecise membership functions. Imprecise membership functions are assigned to the fuzzy set of low water quality and fuzzy set of low risk. The grey fuzzy risk approach is included to account for the uncertainty due to missing data or inadequate data in the samples of input variables as done in IFWLAM. Randomness and imprecision associated with various water quality influencing variables and parameters of the river system are considered through a Monte-Carlo simulation when input parameters (such as mean and variance) are interval grey numbers. The model application is demonstrated with the case study of Tunga-Bhadra river system in South India. Finally the results of the model are compared with the results of GFWLAM (Karmakar and Mujumdar, 2006a). For the case study of Tunga Bhadra River system, it is observed that the fractional removal levels are higher for Grey Optimization Model with Grey Fuzzy Risk compared to GFWLAM (Karmakar and Mujumdar, 2006a) and therefore the resulting risk values at each check point are reduced to a significant extent. The models give a set of flexible policies (range of fractional removal levels). Corresponding optimal values of goal fulfillment level and the grey fuzzy risk are all in terms of interval grey numbers. The IFWLAM and Grey Fuzzy Optimization Model with Grey Fuzzy Risk, developed in the study do not limit their application to any particular pollutant or water quality indicator in the river system. Given appropriate transfer functions for spatial distribution of the pollutants in water body, the models can be used for water quality management of any general river system.