Multi‐objective particle swarm optimization for generating optimal trade‐offs in reservoir operation

A multi‐objective particle swarm optimization (MOPSO) approach is presented for generating Pareto‐optimal solutions for reservoir operation problems. This method is developed by integrating Pareto dominance principles into particle swarm optimization (PSO) algorithm. In addition, a variable size external repository and an efficient elitist‐mutation (EM) operator are introduced. The proposed EM‐MOPSO approach is first tested for few test problems taken from the literature and evaluated with standard performance measures. It is found that the EM‐MOPSO yields efficient solutions in terms of giving a wide spread of solutions with good convergence to true Pareto optimal solutions. On achieving good results for test cases, the approach was applied to a case study of multi‐objective reservoir operation problem, namely the Bhadra reservoir system in India. The solutions of EM‐MOPSOs yield a trade‐off curve/surface, identifying a set of alternatives that define optimal solutions to the problem. Finally, to facilitate easy implementation for the reservoir operator, a simple but effective decision‐making approach was presented. The results obtained show that the proposed approach is a viable alternative to solve multi‐objective water resources and hydrology problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Multi‐objective framework for structural model identification

Structural identification based on measured dynamic data is formulated in a multi‐objective context that allows the simultaneous minimization of the various objectives related to the fit between measured and model predicted data. Thus, the need for using arbitrary weighting factors for weighting the relative importance of each objective is eliminated. For conflicting objectives there is no longer one solution but rather a whole set of acceptable compromise solutions, known as Pareto solutions, which are optimal in the sense that they cannot be improved in any objective without causing degradation in at least one other objective. The strength Pareto evolutionary algorithm is used to estimate the set of Pareto optimal structural models and the corresponding Pareto front. The multi‐objective structural identification framework is presented for linear models and measured data consisting of modal frequencies and modeshapes. The applicability of the framework to non‐linear model identification is also addressed. The framework is illustrated by identifying the Pareto optimal models for a scaled laboratory building structure using experimentally obtained modal data. A large variability in the Pareto optimal structural models is observed. It is demonstrated that the structural reliability predictions computed from the identified Pareto optimal models may vary considerably. The proposed methodology can be used to explore the variability in such predictions and provide updated structural safety assessments, taking into consideration all Pareto structural models that are consistent with the measured data. Copyright © 2005 John Wiley & Sons, Ltd.     

Fuzzy optimum design of aseismic structures

It was pointed out in our previous paper that there exists a vast amount of fuzzy information in both objective and constraint functions of optimum design of structures. Then the idea of fuzzy optimum design of structures was first proposed and the problem with fuzzy allowable intervals of the physical variables (structural responses and sizes) could be solved via the α-level cut approach. In this paper, the procedure of fuzzy optimum design of aseismic structures is further developed. For this purpose, the concepts and definitions of fuzzy predictive earthquake intensity, fuzzy response spectrum and fuzzy structural response are given. A solution of the programming problems with generalized fuzzy constraints (including both fuzzy constraint functions and their fuzzy allowable intervals) is put forward; one of its special cases is the χ-level cut solution in Reference 1. The satisfaction degree of a fuzzy constraint function to its fuzzy allowable interval is defined, by which the problem considered is transformed into a series of non-fuzzy programmings. Then, a series of optimum points can be obtained which make up the solution of the fuzzy programming. As the solution of fuzzy programming contains not one but a series of optimum points, a two-step approach is presented to the fuzzy optimum design of aseismic structures. The first step is to find out the set of minimum cost design points corresponding to different design levels. In the second step, both construction cost and earthquake-caused loss expectation in the service life of the structure are traded off to find the optimum design level as well as a corresponding optimum design scheme.     

Automatic calibration of HEC‐HMS using single‐objective and multi‐objective PSO algorithms

This study presents single‐objective and multi‐objective particle swarm optimization (PSO) algorithms for automatic calibration of Hydrologic Engineering Center‐ Hydrologic Modeling Systems rainfall‐runoff model of Tamar Sub‐basin of Gorganroud River Basin in north of Iran. Three flood events were used for calibration and one for verification. Four performance criteria (objective functions) were considered in multi‐objective calibration where different combinations of objective functions were examined. For comparison purposes, a fuzzy set‐based approach was used to determine the best compromise solutions from the Pareto fronts obtained by multi‐objective PSO. The candidate parameter sets determined from different single‐objective and multi‐objective calibration scenarios were tested against the fourth event in the verification stage, where the initial abstraction parameters were recalibrated. A step‐by‐step screening procedure was used in this stage while evaluating and comparing the candidate parameter sets, which resulted in a few promising sets that performed well with respect to at least three of four performance criteria. The promising sets were all from the multi‐objective calibration scenarios which revealed the outperformance of the multi‐objective calibration on the single‐objective one. However, the results indicated that an increase of the number of objective functions did not necessarily lead to a better performance as the results of bi‐objective function calibration with a proper combination of objective functions performed as satisfactorily as those of triple‐objective function calibration. This is important because handling multi‐objective optimization with an increased number of objective functions is challenging especially from a computational point of view. Copyright © 2012 John Wiley & Sons, Ltd.     

A diversified multiobjective GA for optimizing reservoir rule curves

The paper develops an efficient macro-evolutionary multiobjective genetic algorithm (MMGA) for optimizing the rule curves of a multi-purpose reservoir system in Taiwan. Macro-evolution is a new kind of high-level species evolution that can avoid premature convergence that may arise during the selection process of conventional GAs. MMGA enriches the capabilities of GA to handle multiobjective problems by diversifying the solution set. Simulation results using a benchmark test problem indicate that the proposed MMGA yields better-spread solutions and converges closer to the true Pareto frontier than the nondominated sorting genetic algorithm-II (NSGA-II). When applied to a real case study, MMGA is able to generate uniformly spread solutions for a two-objective problem involving water supply and hydropower generation. Results of this work indicate that the proposed MMGA is highly competitive and provides a viable alternative to solve multiobjective optimization problems for water resources planning and management.     

Sparse calibration of subsurface flow models using nonlinear orthogonal matching pursuit and an iterative stochastic ensemble method

We introduce a nonlinear orthogonal matching pursuit (NOMP) for sparse calibration of subsurface flow models. Sparse calibration is a challenging problem as the unknowns are both the non-zero components of the solution and their associated weights. NOMP is a greedy algorithm that discovers at each iteration the most correlated basis function with the residual from a large pool of basis functions. The discovered basis (aka support) is augmented across the nonlinear iterations. Once a set of basis functions are selected, the solution is obtained by applying Tikhonov regularization. The proposed algorithm relies on stochastically approximated gradient using an iterative stochastic ensemble method (ISEM). In the current study, the search space is parameterized using an overcomplete dictionary of basis functions built using the K-SVD algorithm. The proposed algorithm is the first ensemble based algorithm that tackels the sparse nonlinear parameter estimation problem.     

Grey fuzzy optimization model for water quality management of a river system

A grey fuzzy optimization model is developed for water quality management of river system to address uncertainty involved in fixing the membership functions for different goals of Pollution Control Agency (PCA) and dischargers. The present model, Grey Fuzzy Waste Load Allocation Model (GFWLAM), has the capability to incorporate the conflicting goals of PCA and dischargers in a deterministic framework. The imprecision associated with specifying the water quality criteria and fractional removal levels are modeled in a fuzzy mathematical framework. To address the imprecision in fixing the lower and upper bounds of membership functions, the membership functions themselves are treated as fuzzy in the model and the membership parameters are expressed as interval grey numbers, a closed and bounded interval with known lower and upper bounds but unknown distribution information. The model provides flexibility for PCA and dischargers to specify their aspirations independently, as the membership parameters for different membership functions, specified for different imprecise goals are interval grey numbers in place of a deterministic real number. In the final solution optimal fractional removal levels of the pollutants are obtained in the form of interval grey numbers. This enhances the flexibility and applicability in decision-making, as the decision-maker gets a range of optimal solutions for fixing the final decision scheme considering technical and economic feasibility of the pollutant treatment levels. Application of the GFWLAM is illustrated with case study of the Tunga–Bhadra river system in India.     

A two-phase grey fuzzy optimization approach for water quality management of a river system

An extension of the Grey Fuzzy Waste Load Allocation Model (GFWLAM) developed in an earlier work is presented here to address the problem of multiple solutions. Formulation of GFWLAM is based on the approach for solving fuzzy multiple objective optimization problems with max–min as the operator, which usually may not result in a unique solution. The multiple solutions of fuzzy multiobjective optimization model should be obtained as parametric equations or equations that represent a subspace. A two-phase optimization technique, two-phase GFWLAM, is developed to capture all alternative or multiple solutions of GFWLAM. The optimization model in Phase 1 is exactly same as the optimization model described in GFWLAM. The optimization model in Phase 2 maximizes the upper bounds of fractional removal levels of pollutants and minimizes the lower bounds of fractional removal levels of pollutants keeping the value of goal fulfillment level same as obtained from Phase 1. The widths of the interval-valued fractional removal levels play an important role in decision-making as these can be adjusted within their intervals by the decision-maker considering technical and economic feasibility in the final decision scheme. Two-phase GFWLAM widens the widths of interval-valued removal levels of pollutants, thus enhancing the flexibility in decision-making. The methodology is demonstrated with a case study of the Tunga-Bhadra river system in India.     

A macro-evolutionary multi-objective immune algorithm with application to optimal allocation of water resources in Dongjiang River basins, South China

Macro-evolution is a new kind of high-level species evolution inspired by the dynamics of species extinction and diversification at large time scales. Immune algorithms are a set of computational systems inspired by the defense process of the biological immune system. By taking advantage of the macro-evolutionary algorithm and immune learning of artificial immune systems, this article proposes a macro-evolutionary multi-objective immune algorithm (MEMOIA) for optimizing multi-objective allocation of water resources in river basins. A benchmark test problem, namely the Viennet problem, is utilized to evaluate the performance of the proposed new algorithm. The study indicates that the proposed algorithm yields a much better spread of solutions and converges closer to the true Pareto frontier compared with The Non-dominated Sorting Genetic Algorithm and Improving the Strength Pareto Evolutionary Algorithm. MEMOIA is applied to a water allocation problem in the Dongjiang River basin in southern China, with three objectives named economic interests (OF ₁), water shortages (OF ₂) and the amount of organic pollutants in water (OF ₃). The results demonstrate the capabilities of MEMOIA as well as its suitability as a viable alternative for enhanced water allocation and management in a river basin.     

Agricultural effluent control under uncertainty: An inexact double-sided fuzzy chance-constrained model

An inexact double-sided fuzzy chance-constrained programming (IDFCCP) method was developed in this study and applied to an agricultural effluent control management problem. IDFCCP was formulated through incorporating interval linear programming (ILP) into a double-sided fuzzy chance-constrained programming (DFCCP) framework, and could be used to deal with uncertainties expressed as not only possibility distributions associated with both left- and right-hand-side components of constraints but also discrete intervals in the objective function. The study results indicated that IDFCCP allowed violation of system constraints at specified confidence levels, where each confidence level consisted of two reliability scenarios. This could lead to model solutions with high system benefits under acceptable risk magnitudes. Furthermore, the introduction of ILP allowed uncertain information presented as discrete intervals to be communicated into the optimization process, such that a variety of decision alternatives can be generated by adjusting the decision-variable values within their intervals. The proposed model could help decision makers establish various production patterns with cost-effective water quality management schemes under complex uncertainties, and gain in-depth insights into the trade-offs between system economy and reliability.     

Evaluating water reuse applications under uncertainty: generalized intuitionistic fuzzy-based approach

Water reuse is a viable option to increase urban water supply, especially under new realities of climate change and increasing anthropogenic activities. A sustainable water reuse application should be cost-effective and have acceptable health risk to consumers. Water reuse application evaluation is complex because data acquisitions are usually associated with the problems of uncertainty, hesitancy, and parameterization. In this paper, a generalized intuitionistic fuzzy soft set (GIFSS)-based decision support framework is proposed to provide an effective approach to describe uncertainty and hesitancy in an intuitionistic fuzzy number. In addition, the modified measures of comparison and similarity are proposed to compare water reuse applications. Then, the proposed framework is applied to the City of Penticton (British Columbia, Canada) to evaluate seven water reuse applications. The evaluation results show that the applications of garden flower watering and public parks watering are the most preferred alternatives, which are consistent with the existing practice in the city. Furthermore, the results are highly affected by the generalized parameter and the weights of evaluation criteria. Both the comparison measure-based and similarity measure-based evaluations within the same GIFSS-based framework produce consistent results, indicating an applicable and efficient methodology.     

An inexact fuzzy-random-chance-constrained air quality management model

An inexact fuzzy-random-chance-constrained programming model (IFRCCMM) was developed for supporting regional air quality management under uncertainty. IFRCCMM was formulated through integrating interval linear programming within fuzzy-random-chance-constrained programming framework. It could deal with parameter uncertainties expressed as not only fuzzy random variables but also discrete intervals. Based on the stochastic and fuzzy chance-constrained programming algorithms, IFRCCMM was solved when constraints was satisfied under different satisfaction and violation levels of constraints, leading to interval solutions with different risk and cost implications. The proposed model was applied to a regional air quality management problem for demonstration. The obtained results indicated that the proposed model could effectively reflect uncertain components within air quality management system through employing multiple uncertainty-characterization techniques (in random, fuzzy and interval forms), and help decision makers analyze trade-offs between system economy and reliability. In fact, many types of solutions (i.e. conservative solutions with lower risks and optimistic solutions with higher risks) provided by IFRCCMM were suitable for local decision makers to make more applicable decision schemes according to their understanding and preference about the risk and economy. In addition, the modeling philosophy is general and applicable to many other environmental problems that may be complicated with multiple forms of uncertainties.     

An Elitist Multiobjective Tabu Search for Optimal Design of Groundwater Remediation Systems

This study presents a new multiobjective evolutionary algorithm (MOEA), the elitist multiobjective tabu search (EMOTS), and incorporates it with MODFLOW/MT3DMS to develop a groundwater simulation‐optimization (SO) framework based on modular design for optimal design of groundwater remediation systems using pump‐and‐treat (PAT) technique. The most notable improvement of EMOTS over the original multiple objective tabu search (MOTS) lies in the elitist strategy, selection strategy, and neighborhood move rule. The elitist strategy is to maintain all nondominated solutions within later search process for better converging to the true Pareto front. The elitism‐based selection operator is modified to choose two most remote solutions from current candidate list as seed solutions to increase the diversity of searching space. Moreover, neighborhood solutions are uniformly generated using the Latin hypercube sampling (LHS) in the bounded neighborhood space around each seed solution. To demonstrate the performance of the EMOTS, we consider a synthetic groundwater remediation example. Problem formulations consist of two objective functions with continuous decision variables of pumping rates while meeting water quality requirements. Especially, sensitivity analysis is evaluated through the synthetic case for determination of optimal combination of the heuristic parameters. Furthermore, the EMOTS is successfully applied to evaluate remediation options at the field site of the Massachusetts Military Reservation (MMR) in Cape Cod, Massachusetts. With both the hypothetical and the large‐scale field remediation sites, the EMOTS‐based SO framework is demonstrated to outperform the original MOTS in achieving the performance metrics of optimality and diversity of nondominated frontiers with desirable stability and robustness.     

Development of a fuzzy-queue-based stochastic quadratic program for water resources planning

In this study, a fuzzy-queue (FQ)-based inexact stochastic quadratic programming (SQP) method is developed through coupling FQ technique with inexact SQP. FQ-SQP improves upon the existing stochastic programming methods by considering the effects of queuing phenomenon during the water resources allocation process. FQ-SQP cannot only handle uncertainties expressed as interval values, random variables, and fuzzy sets, but also tackle nonlinearity in the objective function; more importantly, it can reflect the effects of FQ on water resources allocation and system benefit. The FQ-SQP model is applied to a case study of planning water resources management, where FM/FM/1 (fuzzy exponential interarrival time, fuzzy exponential service time, and one server) queue is incorporated within the SQP modeling framework. Based on α-cut analysis technique, interval solutions with fuzzy arrival and service rates have been generated, which result in different water resources allocation patterns as well as changed waiting water amounts and system benefits. Results indicate that consideration of queuing problem impacts on water resources allocation can provide more useful information for decision makers and gain in-depth insights into the effects of queuing problems for water resources allocation.     

A stochastic rough-approximation water management model for supporting sustainable water-environment strategies in an irrigation district of arid region

In this study, a stochastic rough-approximation water management model (SRAWM) associated with optimistic and pessimistic options is proposed for supporting regional sustainability in an irrigation system (IS) of an arid region with uncertain information. SRAWM can not only handle conventional stochastic variations in objective functions or constraints, but also tackle objective and subjective (i.e., risk performance of the decision maker) fuzziness through rough-approximation model based on measure Me. The developed model would be applied to a real case study of an irrigation district (ID) in Kaidu-kongque River Basin, China, which is encountering challenges in economic development and a serious environmental crisis (e.g., drought, water deficit, land deterioration, stalinization, soil erosion and water pollution) synchronously. Simulation technical (i.e., support vector regression) is put into SRAWM framework to reflect dynamic prediction of water demand in the future. Results of optimized irrigation area, water allocation, water deficit, pollution reduction, water and soil erosion and system benefit under various water-environmental policies (corresponding to various ecological effects) are obtained. Tradeoffs between ecological and irrigative water usages can facilitate the local decision makers rectifying the current irrigation patterns and ecological protection polices. Moreover, compromises between systemic benefit and failure risk can help policymakers to generate a robust risk-control plan under uncertainties. These detections are beneficial to achieve conjunctive goals of socio-economic development and eco-environmental sustainability in such an arid IS.     

Two-stage fuzzy chance-constrained programming: application to water resources management under dual uncertainties

In this study, a two-stage fuzzy chance-constrained programming (TFCCP) approach is developed for water resources management under dual uncertainties. The concept of distribution with fuzzy probability (DFP) is presented as an extended form for expressing uncertainties. It is expressed as dual uncertainties with both stochastic and fuzzy characteristics. As an improvement upon the conventional inexact linear programming for handling uncertainties in the objective function and constraints, TFCCP has advantages in uncertainty reflection and policy analysis, especially when the input parameters are provided as fuzzy sets, probability distributions and DFPs. TFCCP integrates the two-stage stochastic programming (TSP) and fuzzy chance-constrained programming within a general optimization framework. TFCCP incorporates the pre-regulated water resources management policies directly into its optimization process to analyze various policy scenarios; each scenario has different economic penalty when the promised amounts are not delivered. TFCCP is applied to a water resources management system with three users. Solutions from TFCCP provide desired water allocation patterns, which maximize both the system’s benefits and feasibility. The results indicate that reasonable solutions were generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of stream flows, α-cut levels and fuzzy dominance indices.     

Planning water resources management systems using a fuzzy-boundary interval-stochastic programming method

In this study, a fuzzy-boundary interval-stochastic programming (FBISP) method is developed for planning water resources management systems under uncertainty. The developed FBISP method can deal with uncertainties expressed as probability distributions and fuzzy-boundary intervals. With the aid of an interactive algorithm woven with a vertex analysis, solutions for FBISP model under associated α-cut levels can be generated by solving a set of deterministic submodels. The related probability and possibility information can also be reflected in the solutions for the objective function value and decision variables. The developed FBISP is also applied to water resources management and planning within a multi-reservoir system. Various policy scenarios that are associated with different levels of economic consequences when the pre-regulated water-allocation targets are violated are analyzed. The results obtained are useful for generating a range of decision alternatives under various system conditions, and thus helping decision makers to identify desired water resources management policies under uncertainty.     

Multi-objective optimal design for flood risk management with resilience objectives

In flood risk management, the divergent concept of resilience of a flood defense system cannot be fully defined quantitatively by one indicator and multiple indicators need to be considered simultaneously. In this paper, a multi-objective optimization (MOO) design framework is developed to determine the optimal protection level of a levee system based on different resilience indicators that depend on the probabilistic features of the flood damage cost arising under the uncertain nature of rainfalls. An evolutionary-based MOO algorithm is used to find a set of non-dominated solutions, known as Pareto optimal solutions for the optimal protection level. The objective functions, specifically resilience indicators of severity, variability and graduality, that account for the uncertainty of rainfall can be evaluated by stochastic sampling of rainfall amount together with the model simulations of incurred flood damage estimation for the levee system. However, these model simulations which usually require detailed flood inundation simulation are computationally demanding. This hinders the wide application of MOO in flood risk management and is circumvented here via a surrogate flood damage modeling technique that is integrated into the MOO algorithm. The proposed optimal design framework is applied to a levee system in a central basin of flood-prone Jakarta, Indonesia. The results suggest that the proposed framework enables the application of MOO with resilience objectives for flood defense system design under uncertainty and solves the decision making problems efficiently by drastically reducing the required computational time.     

Incorporating spatial and temporal information for urban drainage model calibration: An approach using preference ordering genetic algorithm

In order to successfully calibrate an urban drainage model, multiple criteria should be considered, which raises the issue of adopting a method for comparing different parameter sets according to a set of objectives. Multi-objective genetic algorithms (MOGA) have proved effective in numerous such applications, where most of the techniques relying on the condition of Pareto efficiency to compare different solutions. However, as the number of criteria increases, the ratio of Pareto optimal to feasible solutions increases as well, worsening the efficiency of the genetic algorithm search. In this paper, firstly the drawbacks of single objective calibration approach are highlighted. Then, a new MOGA, the preference ordering genetic algorithm, is proposed, that alleviates the drawbacks of conventional Pareto-based methods. The efficacy of this algorithm is demonstrated on the calibration of a physically-based, distributed sewer network model, and the comparison is made with a known MOGA NSGA-II. The results are very encouraging because the obtained parameter sets closely resembled both calibration and validation events. The identifiability of 10 model parameters were analysed, showing significantly smaller ranges of optimal values for parameters related to impervious areas compared to those related to pervious areas, which is reasonable considering relatively low rainfall intensities. In addition to standard ways of presenting calibration results, “radar” plots were also used to present information on trade-off for eight objective functions for four rainfall-runoff events.     

Rule reduction in fuzzy logic for better interpretability in reservoir operation

Decision‐making in reservoir operation has become easy and understandable with the use of fuzzy logic models, which represent the knowledge in terms of interpretable linguistic rules. However, the improvement in interpretability with increase in number of fuzzy sets (‘low’, ‘high’, etc) comes with the disadvantage of increase in number of rules that are difficult to comprehend by decision makers. In this study, a clustering‐based novel approach is suggested to provide the operators with a limited number of most meaningful operating rules. A single triangular fuzzy set is adopted for different variables in each cluster, which are fine‐tuned with genetic algorithm (GA) to meet the desired objective. The results are compared with the multi fuzzy set fuzzy logic model through a case study in the Pilavakkal reservoir system in Tamilnadu State, India. The results obtained are highly encouraging with a smaller set of rules representing the actual fuzzy logic system. Copyright © 2007 John Wiley & Sons, Ltd.