In the dissertation the solution approaches for different fuzzy optimization problems are presented. The single-level optimization problem with fuzzy objective is solved by its reformulation into a biobjective optimization problem. A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Necessary and sufficient optimality conditions of the the convex nonlinear fuzzy optimization problem are derived in differentiable and nondifferentiable cases. A fuzzy optimization problem with both fuzzy objectives and constraints is also investigated in the thesis in the linear case. These solution approaches are applied to fuzzy bilevel optimization problems. In the case of bilevel optimization problem with fuzzy objective functions, two algorithms are presented and compared using an illustrative example. For the case of fuzzy linear bilevel optimization problem with both fuzzy objectives and constraints k-th best algorithm is adopted.

La modélisation de phénomènes complexes rencontrés dans les problématiques industrielles peut conduire à l'étude de codes de simulation numérique. Ces simulateurs peuvent être très coûteux en temps d'exécution (de quelques heures à plusieurs jours), mettre en jeu des paramètres incertains et même être intrinsèquement stochastiques. Fait d'importance en optimisation basée sur de tels simulateurs, les dérivées des sorties en fonction des entrées peuvent être inexistantes, inaccessibles ou trop coûteuses à approximer correctement. Ce mémoire est organisé en quatre chapitres. Le premier chapitre traite de l'état de l'art en optimisation sans dérivées et en modélisation d'incertitudes. Les trois chapitres suivants présentent trois contributions indépendantes --- bien que liées --- au champ de l'optimisation sans dérivées en présence d'incertitudes. Le deuxième chapitre est consacré à l'émulation de codes de simulation stochastiques coûteux --- stochastiques au sens où l'exécution de simulations avec les mêmes paramètres en entrée peut donner lieu à des sorties distinctes. Tel était le sujet du projet CODESTOCH mené au Centre d'été de mathématiques et de recherche avancée en calcul scientifique (CEMRACS) au cours de l'été 2013 avec deux doctorants de Électricité de France (EDF) et du Commissariat à l'énergie atomique et aux énergies alternatives (CEA). Nous avons conçu quatre méthodes de construction d'émulateurs pour des fonctions dont les valeurs sont des densités de probabilité. Ces méthodes ont été testées sur deux exemples-jouets et appliquées à des codes de simulation industriels concernés par trois phénomènes complexes: la distribution spatiale de molécules dans un système d'hydrocarbures (IFPEN), le cycle de vie de grands transformateurs électriques (EDF) et les répercussions d'un hypothétique accident dans une centrale nucléaire (CEA). Dans les deux premiers cas l'émulation est une étape préalable à la résolution d'un problème d'optimisation. Le troisième chapitre traite de l'influence de l'inexactitude des évaluations de la fonction objectif sur la recherche directe directionnelle --- un algorithme classique d'optimisation sans dérivées. Dans les problèmes réels, l'imprécision est sans doute toujours présente. Pourtant les utilisateurs appliquent généralement les algorithmes de recherche directe sans prendre cette imprécision en compte. Nous posons trois questions. Quelle précision peut-on espérer obtenir, étant donnée l'inexactitude ? À quel prix cette précision peut-elle être atteinte ? Quels critères d'arrêt permettent de garantir cette précision ? Nous répondons à ces trois questions pour l'algorithme de recherche directe directionnelle appliqué à des fonctions dont l'imprécision sur les valeurs --- stochastique ou non --- est uniformément bornée. Nous déduisons de nos résultats un algorithme adaptatif pour utiliser efficacement des oracles de niveaux de précision distincts. Les résultats théoriques et l'algorithme sont validés avec des tests numériques et deux applications réelles: la minimisation de surface en conception mécanique et le placement de puits pétroliers en ingénierie de réservoir. Le quatrième chapitre est dédié aux problèmes d'optimisation affectés par des paramètres imprécis, dont l'imprécision est modélisée grâce à la théorie des ensembles flous. Plusieurs méthodes ont déjà été publiées pour résoudre les programmes linéaires où apparaissent des coefficients flous, mais très peu pour traiter les problèmes non linéaires. Nous proposons un algorithme pour répondre à une large classe de problèmes par tri non-dominé itératif. / The modeling of complex phenomena encountered in industrial issues can lead to the study of numerical simulation codes. These simulators may require extensive execution time (from hours to days), involve uncertain parameters and even be intrinsically stochastic. Importantly within the context of simulation-based optimization, the derivatives of the outputs with respect to the inputs may be inexistent, inaccessible or too costly to approximate reasonably. This thesis is organized in four chapters. The first chapter discusses the state of the art in derivative-free optimization and uncertainty modeling. The next three chapters introduce three independent---although connected---contributions to the field of derivative-free optimization in the presence of uncertainty. The second chapter addresses the emulation of costly stochastic simulation codes---stochastic in the sense simulations run with the same input parameters may lead to distinct outputs. Such was the matter of the CODESTOCH project carried out at the Summer mathematical research center on scientific computing and its applications (CEMRACS) during the summer of 2013, together with two Ph.D. students from Electricity of France (EDF) and the Atomic Energy and Alternative Energies Commission (CEA). We designed four methods to build emulators for functions whose values are probability density functions. These methods were tested on two toy functions and applied to industrial simulation codes concerned with three complex phenomena: the spatial distribution of molecules in a hydrocarbon system (IFPEN), the life cycle of large electric transformers (EDF) and the repercussions of a hypothetical accidental in a nuclear plant (CEA). Emulation was a preliminary process towards optimization in the first two cases. In the third chapter we consider the influence of inaccurate objective function evaluations on direct search---a classical derivative-free optimization method. In real settings inaccuracy may never vanish, however users usually apply direct search algorithms disregarding inaccuracy. We raise three questions. What precision can we hope to achieve, given the inaccuracy? How fast can this precision be attained? What stopping criteria can guarantee this precision? We answer these three questions for directional direct search applied to objective functions whose evaluation inaccuracy stochastic or not is uniformly bounded. We also derive from our results an adaptive algorithm for dealing efficiently with several oracles having different levels of accuracy. The theory and algorithm are validated with numerical tests and two industrial applications: surface minimization in mechanical design and oil well placement in reservoir engineering. The fourth chapter considers optimization problems with imprecise parameters, whose imprecision is modeled with fuzzy sets theory. A number of methods have been published to solve linear programs involving fuzzy parameters, but only a few as for nonlinear programs. We propose an algorithm to address a large class of fuzzy optimization problems by iterative non-dominated sorting. The distributions of the fuzzy parameters are assumed only partially known. We also provide a criterion to assess the precision of the solutions and make comparisons with other methods found in the literature. We show that our algorithm guarantees solutions whose level of precision at least equals the precision on the available data.

Im Beitrag werden drei Methoden für den Entwurf und die Bemessung von Textilbetonverstärkungen vorgestellt. Für eine Vorbemessung wird die Variantenuntersuchung angewendet, z.B. für die Bestimmung der Anzahl an Textillagen. Für die Festlegung von Realisierungen mehrerer kontinuierlicher Entwurfsvariablen unter Berücksichtigung unterschiedlicher Entwurfsziele und Entwurfsnebenbedingungen werden die Fuzzy-Optimierung und die direkte Lösung der Entwurfsaufgabe skizziert. Mit der Fuzzy-Optimierung werden Kompromisslösungen für die multikriterielle Entwurfsaufgabe ermittelt. Die direkte Lösung basiert auf der explorativen Datenanalyse von Punktmengen, die als Ergebnis einer unscharfen Tragwerksanalyse vorliegen, und liefert Bereiche – sog. Entwurfsteilräume – als Grundlage für die Auswahl des Entwurfs.

<p>У докторској дисертацији су предложена два модела: модел за планирање испорука и агрегационо планирање производње и модел за прогнозу купчеве потражње у ланцу снабдевања у аутомобилској индустрији. Оба модела примењена су на студији случаја у предузећима два добављача из области аутомобилске индустрије у Републици Србији. Истраживање је показало применљивост предложених модела на практичним проблемима у присуству неизвесности и употребљивост њихових резултата у аутомобилској индустрији. Модели су показали боље резултате у односу на практичне податке у предузећима и у односу на основне стратегије за планирање производње и залиха које се користе у аутомобилској индустрији.</p> / <p>U doktorskoj disertaciji su predložena dva modela: model za planiranje isporuka i agregaciono planiranje proizvodnje i model za prognozu kupčeve potražnje u lancu snabdevanja u automobilskoj industriji. Oba modela primenjena su na studiji slučaja u preduzećima dva dobavljača iz oblasti automobilske industrije u Republici Srbiji. Istraživanje je pokazalo primenljivost predloženih modela na praktičnim problemima u prisustvu neizvesnosti i upotrebljivost njihovih rezultata u automobilskoj industriji. Modeli su pokazali bolje rezultate u odnosu na praktične podatke u preduzećima i u odnosu na osnovne strategije za planiranje proizvodnje i zaliha koje se koriste u automobilskoj industriji.</p> / <p>In doctoral dissertation are proposed two models: model for delivery planning and aggregate production planning and model for customer demand forecasting in supply chain in automotive industry. Both models are applied on the case study in enterprises of two suppliers from area of automotive industry in Republic of Serbia. The research shows applicability of proposed models on practital problems in the presence of uncertanty and usability of their results in automotive industry. Models have showed better results in regard to both the practical data in enterprises and a basic strategies for production planning and inventory planning which are used in automotive industry.</p>

In the dissertation the solution approaches for different fuzzy optimization problems are presented. The single-level optimization problem with fuzzy objective is solved by its reformulation into a biobjective optimization problem. A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Necessary and sufficient optimality conditions of the the convex nonlinear fuzzy optimization problem are derived in differentiable and nondifferentiable cases. A fuzzy optimization problem with both fuzzy objectives and constraints is also investigated in the thesis in the linear case. These solution approaches are applied to fuzzy bilevel optimization problems. In the case of bilevel optimization problem with fuzzy objective functions, two algorithms are presented and compared using an illustrative example. For the case of fuzzy linear bilevel optimization problem with both fuzzy objectives and constraints k-th best algorithm is adopted.:1 Introduction 1 1.1 Why optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fuzziness as a concept . . . . . . . . . . . . . . . . . . . . .. . . . . . . 2 1.3 Bilevel problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Preliminaries 11 2.1 Fuzzy sets and fuzzy numbers . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Fuzzy order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Fuzzy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 3 Optimization problem with fuzzy objective 19 3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Local optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Existence of an optimal solution . . . . . . . . . . . . . . . . . . . . 25 4 Linear optimization with fuzzy objective 27 4.1 Main approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Membership function value . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4.1 Special case of triangular fuzzy numbers . . . . . . . . . . . . 36 4.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 5 Optimality conditions 47 5.1 Differentiable fuzzy optimization problem . . . . . . . . . . .. . . . 48 5.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . .. 49 5.1.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Nondifferentiable fuzzy optimization problem . . . . . . . . . . . . 51 5.2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . . 52 5.2.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 54 5.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 Fuzzy linear optimization problem over fuzzy polytope 59 6.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 The fuzzy polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 6.3 Formulation and solution method . . . . . . . . . . . . . . . . . . .. . 65 6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7 Bilevel optimization with fuzzy objectives 73 7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.2 Solution approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 7.3 Yager index approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4 Algorithm I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.5 Membership function approach . . . . . . . . . . . . . . . . . . . . . . .78 7.6 Algorithm II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 7.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8 Linear fuzzy bilevel optimization (with fuzzy objectives and constraints) 87 8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.2 Solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9 Conclusions 95 Bibliography 97

Im Beitrag werden drei Methoden für den Entwurf und die Bemessung von Textilbetonverstärkungen vorgestellt. Für eine Vorbemessung wird die Variantenuntersuchung angewendet, z.B. für die Bestimmung der Anzahl an Textillagen. Für die Festlegung von Realisierungen mehrerer kontinuierlicher Entwurfsvariablen unter Berücksichtigung unterschiedlicher Entwurfsziele und Entwurfsnebenbedingungen werden die Fuzzy-Optimierung und die direkte Lösung der Entwurfsaufgabe skizziert. Mit der Fuzzy-Optimierung werden Kompromisslösungen für die multikriterielle Entwurfsaufgabe ermittelt. Die direkte Lösung basiert auf der explorativen Datenanalyse von Punktmengen, die als Ergebnis einer unscharfen Tragwerksanalyse vorliegen, und liefert Bereiche – sog. Entwurfsteilräume – als Grundlage für die Auswahl des Entwurfs.

In this study, methodologies for modeling grey uncertainty in water resources systems are developed, specifically for the problems in two identified areas in water resources: waste load allocation in streams and floodplain planning. A water resources system is associated with some degree of uncertainty, due to randomness of hydrologic and hydraulic parameters, imprecision and subjectivity in management goals, inappropriateness in model selection, inexactness of different input parameters for inadequacy of data, etc. Uncertainty due to randomness of input parameters could be modeled by the probabilistic models, when probability distributions of the parameters may be estimated. Uncertainties due to imprecision in the management problem may be addressed by the fuzzy decision models. In addition, some parameters in any water resources problems need to be addressed as grey parameters, due to inadequate data for an accurate estimation but with known extreme bounds of the parameter values. Such inexactness or grey uncertainty in the model parameters can be addressed by the inexact or grey optimization models, representing the parameters as interval grey numbers. The research study presented in this thesis deals with the development of grey and fuzzy optimization models, and the combination of the two for water resources systems decision-making. Three grey fuzzy optimization models for waste load allocation, namely (i) Grey Fuzzy Waste Load Allocation Model (GFWLAM), (ii) two-phase GFWLAM and (iii) multiobjective GFWLAM, and a Grey Integer Programming (GIP) model for floodplain planning, are developed in this study. The Grey Fuzzy Waste Load Allocation Model (GFWLAM) for water quality management of river system addresses uncertainty in the membership functions for imprecisely stated management goals of the Pollution Control Agency (PCA) and dischargers. To address the imprecision in fixing the boundaries of membership functions (also known as membership parameters), the membership functions themselves are treated as imprecise in the model and the membership parameters are expressed as interval grey numbers. The conflict between the fuzzy goals of PCA and dischargers is modeled using the concept of fuzzy decision, but because of treating the membership parameters as interval grey numbers, in the present study, the notion of ‘fuzzy decision’ is extended to the notion of ‘grey fuzzy decision’. A terminology ‘grey fuzzy decision’ is used to represent the fuzzy decision resulting from the imprecise membership functions. The model provides flexibility for PCA and dischargers to specify their aspirations independently, as the membership parameters for membership functions are interval grey numbers in place of a deterministic real number. In the 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. The methodology is demonstrated with the case studies of a hypothetical river system and the Tunga-Bhadra river system in Karnataka, India. Formulation of GFWLAM is based on the approach for solving fuzzy multiple objective optimization problem using max-min as the operator, which usually may not result in a unique solution. The two-phase GFWLAM captures all the alternative optimal solutions of the GFWLAM. The solution technique in the Phase 1 of two-phase GFWLAM is the same as that of GFWLAM. The Phase 2 maximizes upper bounds and minimizes lower bounds of decision variables, keeping the optimal value of goal fulfillment level same as obtained in the Phase 1. The two-phase GFWLAM gives the unique, widest, intervals of the optimal fractional removal levels of pollutant corresponding to the optimal value of goal fulfillment level. The solution increases the widths of interval-valued fractional removal levels of pollutants by capturing all the alternative optimal solutions and thus enhances the flexibility and applicability in decision-making. The model is applied to the case study of Tunga-Bhadra river system, which shows the existence of multiple solutions when the GFWLAM is applied to the same case study. The width of the interval of optimal fractional removal level plays an important role in the GFWLAM, as more width in the fractional removals implies a wider choice to the decision-makers and more applicability in decision-making. The multiobjective GFWLAM maximizes the width of the interval-valued fractional removal levels for providing a latitude in decision-making and minimizes the width of goal fulfillment level for reducing the system uncertainty. The multiobjective GFWLAM gives a new methodology to get a satisfactory deterministic equivalent of a grey fuzzy optimization problem, using the concept of acceptability index for a meaningful ranking between two partially or fully overlapping intervals. The resulting multiobjective optimization model is solved by fuzzy multiobjective optimization technique. The consistency of the solution is verified by solving the problem with fuzzy goal programming technique. The multiobjective GFWLAM avoids intermediate submodels unlike GFWLAM, so that the solution from a single deterministic equivalent of the GFWLAM adequately covers all possible situations. Although the solutions obtained from multiobjective GFWLAM provide more flexibility than those of the GFWLAM, its application is limited to grey fuzzy goals expressed by linear imprecise membership functions only, whereas GFWLAM has the capability to solve the model with any monotonic nonlinear imprecise membership functions also. The methodology is demonstrated with the case studies of a hypothetical river system and the Tunga-Bhadra river system in Karnataka, India. The Grey Integer Programming (GIP) model for floodplain planning is based on the floodplain planning model developed by Lund (2002), to identify an optimal mix of flood damage reduction options with probabilistic flood descriptions. The model demonstrates how the uncertainty of various input parameters in a floodplain planning problem can be modeled using interval grey numbers in the optimization model. The GIP model for floodplain planning does not replace a post-optimality analysis (e.g., sensitivity analysis, dual theory, parametric programming, etc.), but it provides additional information for interpretation of the optimal solutions. The results obtained from GIP model confirm that the GIP is a useful technique for interpretation of the solutions particularly when a number of potential feasible measures are available in a large scale floodplain planning problem. Though the present study does not directly compare the GIP technique with sensitivity analysis, the results indicate that the rigor and extent of post-optimality analyses may be reduced with the use of GIP for a large scale floodplain planning problem. Application of the GIP model is demonstrated with the hypothetical example as presented in Lund (2002).

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.