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To improve the convergence of multiple-site weather generators (SWGs) based on the brute force algorithm (MBFA), a genetic algorithm (GA) is proposed to search the overall optimal correlati...

Seeking cost-effective design of urban hydrological facilities and drainage systems is an important task for many city planners. However, such a process has always been complicated with intrinsic uncertainties. This work presented an integrated fuzzy simulation-optimization model (FSOM) for supporting Low Impact Development (LID) design under model uncertainties. Various LID implementation schemes involving green roof, bio-retention cell, and permeable pavement were simulated through an urban hydrological model. Three model parameters were assumed as fuzzy sets. In a case study, fuzzy simulation (FS) and genetic algorithm (GA) were employed to search the optimal schemes of LIDs under various confidence levels of satisfying flood control constraints. Comparison of FSOM to traditional deterministic and stochastic models were also carried out. It was shown that FSOM could offer a flexible way of defining and assessing uncertainties associated with hydrological modeling and generate solutions that were comparable to those from either deterministic or stochastic models. However, FSOM also showed limitation of high computational requirement.

A Gini-coefficient based stochastic optimization (GBSO) model was developed by integrating the hydrological model, water balance model, Gini coefficient and chance-constrained programming (CCP) into a general multi-objective optimization modeling framework for supporting water resources allocation at a watershed scale. The framework was advantageous in reflecting the conflicting equity and benefit objectives for water allocation, maintaining the water balance of watershed, and dealing with system uncertainties. GBSO was solved by the non-dominated sorting Genetic Algorithms-II (NSGA-II), after the parameter uncertainties of the hydrological model have been quantified into the probability distribution of runoff as the inputs of CCP model, and the chance constraints were converted to the corresponding deterministic versions. The proposed model was applied to identify the Pareto optimal water allocation schemes in the Lake Dianchi watershed, China. The optimal Pareto-front results reflected the tradeoff between system benefit ( α SB ) and Gini coefficient ( α G ) under different significance levels (i.e. q ) and different drought scenarios, which reveals the conflicting nature of equity and efficiency in water allocation problems. A lower q generally implies a lower risk of violating the system constraints and a worse drought intensity scenario corresponds to less available water resources, both of which would lead to a decreased system benefit and a less equitable water allocation scheme. Thus, the proposed modeling framework could help obtain the Pareto optimal schemes under complexity and ensure that the proposed water allocation solutions are effective for coping with drought conditions, with a proper tradeoff between system benefit and water allocation equity.

Optimization techniques have emerged as robust tools to aid the planning and design of urban drainage facilities in cost-effective ways. Such an effort was traditionally aided by heuristic methods (like genetic algorithm), which was generally time-consuming and also challenging in reaching convergence for large-scale problems with wide decision spaces. This study proposed a novel optimization method, denoted as two-level optimization (TO) scheme, for supporting rainwater storage pond design in an urban drainage system. Polynomial regression models were established as surrogate models to facilitate the solution of the optimization framework using traditional iteration algorithm. The TO scheme firstly sought the optimal layout of storage ponds on tributary sub-watersheds, and then proceeded to that of the mainstream one to yield the final solution. Through a case study, the TO scheme was compared with the traditional global optimization (GO) scheme where the physical simulation model was dynamically linked with genetic algorithm (GA) to seek the global optimal solution. The performance of two schemes under different constraint settings was analyzed. Effects of related issues such as start-point selection and mainstream design on tributary sub-watersheds were also discussed. The results showed that the proposed TO scheme is a prominent alternative to the traditional GO scheme to support urban water managers for a more science-based decision making towards storage pond implementation in large-scale practical problems. Ministry of Education (MOE) The research work was supported by Singapore’s Ministry of Education (MOE) AcRF Tier 1 Project (Ref No. RG170/16; WBS No.: 4011766.030).

A generalized likelihood uncertainty estimation (GLUE) framework coupling with artificial neural network (ANN) models in two surrogate schemes (i.e. GAE-S1 and GAE-S2) was proposed to improve the efficiency of uncertainty assessment in flood inundation modelling. The GAE-S1 scheme was to construct an ANN to approximate the relationship between model likelihoods and uncertain parameters for facilitating sample acceptance/rejection instead of running the numerical model directly; thus, it could speed up the Monte Carlo simulation in stochastic sampling. The GAE-S2 scheme was to establish independent ANN models for water depth predictions to emulate the numerical models; it could facilitate efficient uncertainty analysis without additional model runs for locations concerned under various scenarios. The results from a study case showed that both GAE-S1 and GAE-S2 had comparable performances to GLUE in terms of estimation of posterior parameters, prediction intervals of water depth, and probabilistic i...

Fuzzy programming model has been widely used in water resources management, but its applicability has been significantly restricted in dealing with triangular or trapezoidal shaped fuzzy sets due to intrinsic complexity in converting fuzzy constraints into their deterministic forms. In this study, a novel superiority-inferiority-based sequential fuzzy programming (SISFP) model was proposed for supporting water supply–demand analysis under uncertainty. The SISFP method could transform fuzzy objective function and constraints with general-shaped fuzzy coefficients into their crisp equivalent by using fuzzy superiority and inferiority measures. The water supply–demand management system in Tianjin Binhai New Area, China, consisting of five sources of water, five water users at three districts (i.e. Tanggu, Hanggu, and Dagang), was used for methodology demonstration. The proposed model could effectively address the complex nature of fuzzy characterization of water-transfer safety factor, wastewater reclamation rate, the net benefits derived from water, and water-saving rate of the system; and also take demand management measures into consideration. The obtained solutions have sought a well balance among the water availability, water demand, adoption of water-saving measures, and benefit/cost of each water users. The advantage and necessity of SISFP in dealing with general-shaped fuzzy parameters were further verified by comparing to fuzzy models with both deterministic and specially-shaped fuzzy parameters.

A stochastic optimization model for urban drainage design (SODD) was proposed in this study to help analyze the tradeoff between investment and acceptable flood damage in urban drainage designs considering effects of both uncertainty and climate change. The simulation model (i.e. SWMM), driven by designed rainfall either from existing Intensity-Duration-Frequency (IDF) curves or future ones subjected to climate change conditions, was used to simulate flooding scenarios. The generalized uncertainty analysis estimation (GLUE) and Monte Carlo simulation methods were employed to quantify the system reliability which was adopted in the constraints of the optimization model. The results from a case study showed that the deterministic optimization was computationally efficient with no randomness encountered in hydrological simulation, although its solution was hardly reliable in achieving the target for flood mitigation. The stochastic version, on the other hand, could offer richer information on system reliability and help managers make a more robust decision. The results also revealed that the rainfall extremes under the impact of climate change could significantly affect system investment. The proposed method is advantageous in facilitating cost-effective planning towards a risk-based drainage design in light of various complexities.

An uncertainty assessment framework based on Karhunen–Loevè expansion (KLE) and probabilistic collocation method (PCM) was introduced to deal with flood inundation modelling under uncertainty. The Manning's roughness for channel and floodplain were treated as 1D and 2D, respectively, and decomposed by KLE. The maximum flow depths were decomposed by the 2nd-order PCM. Through a flood modelling case with steady inflow hydrographs based on five designed testing scenarios, the applicability of KLE-PCM was demonstrated. The study results showed that the Manning's roughness assumed as a 1D/2D random field could efficiently alleviate the burden of random dimensionality within the analysis framework, and the introduced method could significantly reduce repetitive runs of the physical model as required in the traditional Monte Carlo simulation (MCS). The study sheds some light on reducing the computational burden associated with flood modelling under uncertainty which is useful for the related damage quantification and risk management.

This paper presents the application of an improved particle swarm optimization (PSO) technique for training an artificial neural network (ANN) to predict water levels for the Heshui Watershed, China. Daily values of rainfall and water levels from 1988 to 2000 were first analyzed using ANNs trained with the conjugate gradient, gradient descent, and Levenberg-Marquardt neural network (LM-NN) algorithms. The best results were obtained from the LM-NN, and these results were then compared with those from PSO-based ANNs, including the conventional PSO neural network (CPSONN) and the improved PSO neural network (IPSONN) with passive congregation. The IPSONN algorithm improves PSO convergence by using the selfish herd concept in swarm behavior. The results show that the PSO-based ANNs performed better than the LM-NN. For models run using a single parameter (rainfall) as input, the root mean square error (RMSE) of the testing data set for IPSONN was the lowest (0.152 m) compared to those for CPSONN (0.161 ...

A generalized likelihood uncertainty estimation (GLUE) method incorporating moving least squares (MLS) with entropy for stochastic sampling (denoted as GLUE-MLS-E) was proposed for uncertainty analysis of flood inundation modelling. The MLS with entropy (MLS-E) was established according to the pairs of parameters/likelihoods generated from a limited number of direct model executions. It was then applied to approximate the model evaluation to facilitate the target sample acceptance of GLUE during the Monte-Carlo-based stochastic simulation process. The results from a case study showed that the proposed GLUE-MLS-E method had a comparable performance as GLUE in terms of posterior parameter estimation and predicted confidence intervals; however, it could significantly reduce the computational cost. A comparison to other surrogate models, including MLS, quadratic response surface and artificial neural networks (ANN), revealed that the MLS-E outperformed others in light of both the predicted confidence interval and the most likely value of water depths. ANN was shown to be a viable alternative, which performed slightly poorer than MLS-E. The proposed surrogate method in stochastic sampling is of practical significance in computationally expensive problems like flood risk analysis, real-time forecasting, and simulation-based engineering design, and has a general applicability in many other numerical simulation fields that requires extensive efforts in uncertainty assessment. Copyright © 2014 John Wiley & Sons, Ltd.

Generalized Likelihood Uncertainty Estimation (GLUE), a simplified Bayesian method, was adopted to determine the parametric uncertainty in hydrological modeling. A preliminary analysis of the summer flows of the Kootenay Watershed, Canada, was modeled to portray a typical uncertainty analysis procedure. SLURP, a robust hydrologic model was chosen for this procedure. The results demonstrated the viability of applying the GLUE method in conjunction with the SLURP hydrological model, following which the posterior probability distributions of the parameters was analyzed. The performance of this technique was verified by examining the flows’ prediction intervals for a period of 2 years, enabling valid future hydrological forecasting for the watershed.

A simulation-based fuzzy optimization method (SFOM) was proposed for regional groundwater pumping management in considering uncertainties. SFOM enhanced the traditional groundwater management models by incorporating a response matrix model (RMM) into a fuzzy chance-constrained programming (FCCP) framework. RMM was used to approximate the input–output relationship between pumping actions and subsurface hydrologic responses. Due to its explicit expression, RMM could be easily embedded into an optimization model to help seek cost-effective pumping solutions. A groundwater management case in Pinggu District of Beijing, China, was used to demonstrate the method's applicability. The study results showed that the obtained system cost and pumping rates would vary significantly under different confidence levels of constraints satisfaction. The decision-makers could identify the best groundwater pumping strategy through analyzing the tradeoff between the risk of violating the related water resources conservation target and the economic benefit. Compared with traditional approaches, SFOM was particularly advantageous in linking simulation and optimization models together, and tackling uncertainties using fuzzy chance constraints.

A joint Monte Carlo and fuzzy possibilistic simulation (MC-FPS) approach was proposed for flood risk assessment. Monte Carlo simulation was used to evaluate parameter uncertainties associated with inundation modeling, and fuzzy vertex analysis was applied for promulgating human-induced uncertainty in flood damage estimation. A study case was selected to show how to apply the proposed method. The results indicate that the outputs from MC-FPS would present as fuzzy flood damage estimate and probabilistic-possibilistic damage contour maps. The stochastic uncertainty in the flood inundation model and fuzziness in the depth-damage functions derivation would cause similar levels of influence on the final flood damage estimate. Under the worst scenario (i.e. a combined probabilistic and possibilistic uncertainty), the estimated flood damage could be 2.4 times higher than that computed from conventional deterministic approach; considering only the pure stochastic effect, the flood loss would be 1.4 times higher. It was also indicated that uncertainty in the flood inundation modeling has a major influence on the standard deviation of the simulated damage, and that in the damage-depth function has more notable impact on the mean of the fitted distributions. Through applying MC-FPS, rich information could be derived under various α-cut levels and cumulative probabilities, and it forms an important basis for supporting rational decision making for flood risk management under complex uncertainties.

A modified trapezoidal-shaped fuzzy chance-constrained mixed-integer programming (TFCMP) model was advanced for municipal solid waste (MSW) management. Compared with conventional methods, TFCMP was advantageous in handling fuzzy-type uncertainties in both the left- and right-hand sides of model constraints and could be used to reflect the possibility of constraints violation at predefined confidence levels. Mixed-integer programming (MIP) was embedded into the general framework of TFCMP for handling capacity-expansion issues. The solid waste management system in a typical Chinese city was used to demonstrate the applicability of TFCMP. Study results indicated that a variety of cost-effective MSW-flow allocation solutions could be obtained from TFCMP under various scenarios of system reliability. A trade-off between the total system cost and the reliability of satisfying model constraints can be analyzed to gain an in-depth insight into the characteristics of MSW systems. Generally, decision alter...

An integrated simulation-assessment approach (ISAA) was developed in this study to systematically tackle multiple uncertainties associated with hydrocarbon contaminant transport in subsurface and assessment of carcinogenic health risk. The fuzzy vertex analysis technique and the Latin hypercube sampling (LHS) based stochastic simulation approach were combined into a fuzzy-Latin hypercube sampling (FLHS) simulation model and was used for predicting contaminant transport in subsurface under coupled fuzzy and stochastic uncertainties. The fuzzy-rule-based risk assessment (FRRA) was used for interpreting the general risk level through fuzzy inference to deal with the possibilistic uncertainties associated with both FLHS simulations and health-risk criteria. A study case involving health risk assessment for a benzene-contaminated site was examined. The study results demonstrated the proposed ISAA was useful for evaluating risks within a system containing complicated uncertainties and interactions and providing supports for identifying cost-effective site management strategies.

A stochastic robust interval linear programming model (IPRO) was developed for supporting municipal solid waste management under uncertainty. The model improves upon the existing stochastic robust optimization (SRO) and interval linear programming (ILP) methods by allowing evaluations of trade-offs among expected costs, cost variability, and risk of violating relax constraints simultaneously, as well as reflections of complex uncertainties through both interval and stochastic theories. A long-term waste management problem was used to demonstrate the applicability of IPRO model. The results indicated that IPRO normally led to interval solutions, where waste-management alternatives could be generated by adjusting the decision-variable values within their intervals. The model could also help waste managers to identify desired policies that under various environmental, economic, system-feasibility and system-reliability constraints.

A stochastic robust fuzzy interval linear programming (SRFILP) model was proposed for supporting municipal solid waste (MSW) management under multiple uncertainties. The method integrated stochastic robust optimization (SRO), interval linear programming (ILP) and fuzzy possibilistic programming (FPP) methods into a general framework and could simultaneously deal with uncertainties expressed as fuzzy sets, stochastic variables and discrete intervals. The SRFILP model was applied to a hypothetical problem of municipal solid waste management. The results demonstrated that flexible interval solutions under different I±-cut levels could be generated, which could help decision makers gain an in-depth insight into system complexities associated with solid waste management. The waste-management alternatives could be generated by adjusting the decision-variable values within their solution intervals. In addition, the proposed method could be used to help evaluate the trade-off between solution robustness and model robustness, and help waste managers identify desired cost-effective policies considering environmental, economic, system-feasibility and system-reliability constraints.

An inexact two-stage stochastic robust programming model (ITSRP) was developed in this study for dealing with water resources allocation problems under uncertainty. ITSRP was formulated ba...

A hybrid stochastic robust chance-constraint programming (SRCCP) model was developed in this study for supporting municipal solid waste management under uncertainty. The method improves upon the existing robust-optimization (RO) and chance-constraint programming (CCP) approaches by allowing analysis on trade-offs among expected value of the objective function, variation in the value of the objective function and the risk of violating constraints that contain uncertain parameters. SRCCP could be used to examine the balance between solution robustness and model robustness, and was especially useful for analyzing the reliability of satisfying (or risk of violating) system constraints under complex uncertainties. A long-term municipal solid waste management problem was used to demonstrate the applicability of SRCCP, with violations for capacity constraints being assumed under various significance levels. The study results demonstrated that a higher system cost may guarantee that waste-management requirements and environmental criteria be met, and a lower cost may lead to a higher risk of violating the related regulations. The proposed SRCCP model could be used by waste managers for identifying desired waste-management policies under various environmental, economic, and system-reliability constraints and complex uncertainties.

The development of optimization models for energy systems planning has attracted considerable interest over the past decades. However, the uncertainties that are inherent in the planning process and the complex interactions among various uncertain parameters are challenging the capabilities of these developed tools. Therefore, the objective of this study is to develop a hybrid interval-fuzzy two-stage stochastic energy systems planning model (IFTEM) to deal with various uncertainties that can be expressed as fuzzy numbers, probability distributions and discrete intervals. The developed IFTEM is then applied to a hypothetical regional energy system. The results indicate that the IFTEM has advantages in reflecting complexities of various system uncertainties as well as dealing with two-stage stochastic decision problems within energy systems.

Water quality management is complicated with a variety of uncertainties and nonlinearities. This leads to difficulties in formulating and solving the resulting inexact nonlinear optimization problems. In this study, an inexact chance-constrained quadratic programming (ICCQP) model was developed for stream water quality management. A multi-segment stream water quality (MSWQ) simulation model was provided for establishing the relationship between environmental responses and pollution-control actions. The relationship was described by transformation matrices and vectors that could be used directly in a multi-point-source waste reduction (MWR) optimization model as water-quality constraints. The interval quadratic polynomials were employed to reflect the nonlinearities and uncertainties associated with wastewater treatment costs. Uncertainties associated with the water-quality parameters were projected into the transformation matrices and vectors through Monte Carlo simulation. Uncertainties derived from water quality standards were characterized as random variables with normal probability distributions. The proposed ICCQP model was applied to a water quality management problem in the Changsha section of the Xiangjiang River in China. The results demonstrated that the proposed optimization model could effectively communicate uncertainties into the optimization process, and generate inexact solutions containing a spectrum of wastewater treatment options. Decision alternatives could then be obtained by adjusting different combinations of the decision variables within their solution intervals. Solutions from the ICCQP model could be used to analyze tradeoffs between the wastewater treatment cost and system-failure risk due to inherent uncertainties. The results are valuable for supporting decision makers in seeking cost-effective water management strategies.

A factorial-design-based stochastic modeling system (FSMS) was developed in this study to systematically investigate impacts of uncertainties associated with hydrocarbon-contaminant transport in subsurface. FSMS integrated a solute transport model, factorial analysis, and Monte Carlo technique into a general framework, and effectively analyze the individual and joint effects of input parameters' uncertainties that are associated with hydrogeological conditions. Four input parameters (i.e. the mean and the variance of permeability as well as the mean and the variance of porosity) were assumed to be of uncertain nature, and the factorial design and Monte Carlo simulation algorithm were incorporated into a groundwater flow and solute transport model developed in this study. Under each factorial experiment, a number of Monte Carlo simulations were implemented. A pilot-scale physical modeling system was used to illustrate the applicability of the proposed methodology. The simulation results reveal that the uncertainties in input parameters pose considerable influences on the predicted output; especially, variations in the mean of porosity will have significant impacts on the modeling output. The results obtained from the systematic uncertainty analysis methods proposed in this study, such as mean, standard deviation, and percentile can provide useful information for further decision-making regarding the petroleum contamination problem.

An inexact two-stage stochastic quadratic programming (ITQP) model is developed for water resources management under uncertainty. The model is a hybrid of inexact quadratic programming and two-stage stochastic programming. It can deal with the uncertainties presented as both probabilities and intervals. Moreover, it can deal with nonlinearities in objective function to reflect the effects of marginal utility on the benefit and cost components. Using quadratic form in the objective function rather than linear one, the ITQP can minimize the unfair competition of water resources among multiple users under uncertain water conditions. In the modeling formulation, penalties are imposed when policies expressed as the promised water supply targets are violated. In its solution process, the ITQP model is transformed into two deterministic submodels based on an interactive algorithm and a derivative algorithm, which correspond to the lower and upper bounds of the desired objective. Interval solutions, which are feasible and stable in the given decision space, can then be obtained by solving the two submodels sequentially. The developed method is then applied to a case study of water resources management planning. The results indicate that reasonable solutions have been obtained. They can help provide bases for identifying desired water-allocation plans with maximized system benefit and minimized system-failure risk.

Planning for water quality management systems is complicated by a variety of uncertainties and nonlinearities, where difficulties in formulating and solving the resulting inexact nonlinear optimization problems exist. With the purpose of tackling such difficulties, this paper presents the development of an interval-fuzzy nonlinear programming (IFNP) model for water quality management under uncertainty. Methods of interval and fuzzy programming were integrated within a general framework to address uncertainties in the left- and right-hand sides of the nonlinear constraints. Uncertainties in water quality, pollutant loading, and the system objective were reflected through the developed IFNP model. The method of piecewise linearization was developed for dealing with the nonlinearity of the objective function. A case study for water quality management planning in the Changsha section of the Xiangjiang River was then conducted for demonstrating applicability of the developed IFNP model. The results demonstrated that the accuracy of solutions through linearized method normally rises positively with the increase of linearization levels. It was also indicated that the proposed linearization method was effective in dealing with IFNP problems; uncertainties can be communicated into optimization process and generate reliable solutions for decision variables and objectives; the decision alternatives can be obtained by adjusting different combinations of the decision variables within their solution intervals. It also suggested that the linearized method should be used under detailed error analysis in tackling IFNP problems.

An extended fuzzy parametric programming (EFPP) model was proposed for supporting water resources allocation problems under uncertainty. EFPP deals with flexible constraints (i.e., fuzzy relationships) by allowing violation of constraints at certain satisfaction degrees (i.e.,αlevels) and employs fuzzy ranking method to handle trapezoidal-shaped fuzzy coefficients. The objective function is defuzzified by usingβcuts and weighting factors. The applicability of EFPP was demonstrated by a numerical example and a water resources allocation case. A series of decision alternatives at various satisfaction degrees were obtained. Generally, the higher theαlevel, the lower the system benefit. In comparison, theβlevel in the objective function posed less sensitive impacts on both objective function and model solutions. The reliability of EFPP was tested by comparing its solutions with those from fuzzy chance constrained programming (FCCP). The results indicated that EFPP performed equally well with FCCP in addressing parameter uncertainties, but it demonstrated a wider applicability due to its extended capacity of handling fuzzy relationships in the model constraints.

A genetic-algorithm-aided stochastic optimization (GASO) model was developed in this study for supporting regional air quality management under uncertainty. The model incorporated genetic algorithm (GA) and Monte Carlo simulation techniques into a general stochastic chance-constrained programming (CCP) framework and allowed uncertainties in simulation and optimization model parameters to be considered explicitly in the design of least-cost strategies. GA was used to seek the optimal solution of the management model by progressively evaluating the performances of individual solutions. Monte Carlo simulation was used to check the feasibility of each solution. A management problem in terms of regional air pollution control was studied to demonstrate the applicability of the proposed method. Results of the case study indicated the proposed model could effectively communicate uncertainties into the optimization process and generate solutions that contained a spectrum of potential air pollutant treatment options with risk and cost information. Decision alternatives could be obtained by analyzing tradeoffs between the overall pollutant treatment cost and the system-failure risk due to inherent uncertainties.

This study investigated the applicability of using genetic algorithm for tackling chance-constrained programming models utilized for solving environmental pollution-control management problems. Compared with conventional chance-constrained methods, the proposed one could deal with stochastic models with both the left- and right-hand-side constraints being involved with random variables. Two study cases which were related to air quality management and river water pollution control were applied to illustrate the applicability of the proposed method. Both study cases had needs of seeking cost-effective management schemes under uncertainty. The study results indicated that the GA-based CCP models could effectively communicate uncertainties into optimization process, and generate solutions that contain a spectrum of potential waste treatment options with both risk and cost information. Decision alternatives could be obtained by analyzing tradeoffs between the waste handling cost and the system-failure risk due to inherent uncertainties. Compared with the linearization solution method of traditional CCP models, the introduction a GA algorithm could facilitate the solution of more general stochastic models. The proposed method is not restricted to environmental problems and can also be applied to many other engineering management systems.

A simulation-based interval quadratic waste load allocation (IQWLA) model was developed for supporting river water quality management. A multi-segment simulation model was developed to generate water-quality transformation matrices and vectors under steady-state river flow conditions. The established matrices and vectors were then used to establish the water-quality constraints that were included in a water quality management model. Uncertainties associated with water quality parameters, cost functions, and environmental guidelines were described as intervals. The cost functions of wastewater treatment units were expressed in quadratic forms. A water-quality planning problem in the Changsha section of Xiangjiang River in China was used as a study case to demonstrate applicability of the proposed method. The study results demonstrated that IQWLA model could effectively communicate the interval-format uncertainties into optimization process, and generate inexact solutions that contain a spectrum of potential wastewater treatment options. Decision alternatives can be generated by adjusting different combinations of the decision variables within their solution intervals. The results are valuable for supporting local decision makers in generating cost-effective water quality management strategies.