Mahyar Shafii December 2007 Multi-Objective Evolutionary Optimization; Concept and Application to Calibration of Rainfall-Runoff Model Mahyar Shafii December 2007

Table of Contents Introduction Classical Methods to Solve Multi-Objective Optimization Problems Evolutionary Algorithm (EA) Terminology Multi-Objective Evolutionary Algorithms (MOEAs) MOEA Application to Calibration of Conceptual Rainfall-Runoff Models Literature Review Concluding Remarks Development of an Improved NSGA-II

Introduction Multi-Objective Optimization Optimal solution in single-objective optimization is clearly defined. In multi-objective optimization there is rather a set of alternative trade-offs, generally known as “Pareto-Optimal” solutions. Real-World Problem Several incommensurable and often competing objectives

Introduction Multi-Objective Optimization Basic Concept and Terminology Find a vector of solutions: m inequality constraints: p equality constraints: k objective functions so that,

X* Є F X Є F or, there is at least one i Є I so that Introduction Pareto Optimum concept X* Є F X Є F or, there is at least one i Є I so that Vilfredo Pareto (1896) None of solutions in Pareto optimal set can be identified as better than the others unless preference information is included (e.g. a ranking of the objectives).

Traditional Approaches Aggregating the objectives into a single and parameterized objective function and performing several runs with different parameter settings to achieve a set of solutions approximating the Pareto-optimal set. Weighting Method (Cohon, 1978) Constraint Method (Cohon, 1978) Goal Programming (Steuer, 1986) Minimax Approach (Koski, 1984)

Traditional Approaches Difficulties with classical methods: Being sensitive to the shape of the Pareto-optimal front (e.g. weighting method). Need for problem knowledge which may not be available. Restrictions on their use in some application areas (Deb, 1999). Need to several optimization runs to achieve the best parameter setting to obtain an approximation of the Pareto-optimal set.

Evolutionary Algorithms (EAs) The term evolutionary algorithm (EA) stands for a class of stochastic optimization methods that simulate the process of natural evolution. They are meta-heuristics that attempt to apply the principles of neo-Darwinian evolution to the creation of artificial intelligence (machine learning) and to optimization. Origins of EAs: Firstly proposed in the late 1950s leading to development of several EAs since the 1970s, mainly (Bäck, Hammel, and Schwefel 1997) Genetic Algorithms (GA) Evolutionary Programming (EP) Evolution Strategies (ES)

Evolutionary Algorithms (EAs) Basic Principles of EA Genotype versus Phenotype Genotype is underlying genetic coding (Genes in GA) Phenotype is expression of that coding forming a possible solution (Chromosome in GA) Mapping between G-space & P-space Selection giving a chance to each solution to reproduce a certain number of times, dependent on their quality or so-called fitness values. Variation Imitating natural capability of creating ”new” living beings by means of recombination and mutation.

Evolutionary Algorithms (EAs) Recombination involves swapping sections of two individuals’ characteristics. Note: This is not what occurs in nature ATGCCGCACC TGTCCAGTCA Parent chromosomes ATGCC AGTCA GCACC TGTCC Recombined offspring ATGCC AGTCA GCACC TGTCC A T Mutation results in a random change in one or more of an individual’s characteristics. Random mutations in genetic composition

Evolutionary Algorithms (EAs)

Multi-Objective Evolutionary Algorithms (MOEAs) Evolutionary algorithms do better than other blind search strategies in multi-objective optimization (Fonseca and Fleming (1995); Valenzuela-Rend´on and Uresti-Charre (1997)). At first, they were applied by functions aggregation. More recently, MOEAs were designed to search decision spaces for the optimal tradeoffs among a vector of objectives (Coello Coello, 2002).

Multi-Objective Evolutionary Algorithms (MOEAs) Some representatives of MOEAs in operational research through past years: Non-Dominated Sorting genetic Algorithm (NSGA), Srinivas et Deb, 1995. NSGA-II, Deb et al., 2002. Strength Pareto Evolutionary Algorithm (SPEA), Zitzler and Thiele, 1999. SPEA2, Zitzler et al., 2001. Epsilon-NSGAII, Kollat and Reed, 2005. Multi-objective Shuffled Complex Evolution Metropolis Algorithm (MOSCEM-UA), Vrugt et al., 2003.

MOEA Applications in Calibration of Conceptual Rainfall-Runoff Models Conceptual rainfall-runoff (CRR) models Calibration of RR models is a process in which parameter adjustment is made so as to match (as closely as possible) the dynamic behavior of the RR model to the observed behavior of the catchment.

MOEA Applications in Calibration of Conceptual Rainfall-Runoff Models Literature Review Gupta et al. (1998) have discussed the advantages of a multiple-objective representation of the model calibration problem and this scheme has been shown to be applicable and desirable. Purely Random Techniques Single Objective Calibration Scheme Multi-Objective Calibration Scheme Local and Global Search Algorithms Some calibration results reveal that moving from a lumped model structure to a semi-distributed model structure improves the simulation results (Ajami et al., 2004). Evolutionary-Population Based Approaches Lumped Modeling Distributed Modeling

MOEA Applications in Calibration of Conceptual Rainfall-Runoff Models Lumped Hydrological Modeling: Development an algorithm Yapo et al. (1998): MOCOM-UA Cheng et al. (2002): Fuzzy Global Optimization and GA Khu et al. (2005): NSGA-II and kNN Proposing General Framework Wagener (2001) Multi-Objective nature with aggregated function Madsen (2000) Seibert (2000) Comparison between single and multiple objective formulations Chahinian and Moussa (2007) Working on objective functions Yu and Yang (2000): Fuzzy Multi-Objective Function (FMOF)

MOEA Applications in Calibration of Conceptual Rainfall-Runoff Models Distributed Hydrological Modeling: Proposing General Framework Madsen (2003): Aggregated Objective Function Development an algorithm Cheng et al. (2006): Following previous work by the same authors. Bekele and Nicklow (2007): NSGA-II for calibration of SWAT Comparison between single and multiple objective formulations Schoups et al. (2005): Subsurface modeling Parajka et al. (2007) Sensitivity and uncertainty analysis Muleta and Nicklow (2005): SWAT, but in a single-objective scheme Although a majority of prior studies have focused on CRR applications, there are an increasing number of recent studies focusing on developing multi-objective calibration strategies for distributed hydrological models such as:

MOEA Applications in Calibration of Conceptual Rainfall-Runoff Models Remarks and recommendations Modification to the study of Madsen (2003) in order to develop an improved framework for calibration of RR process in distributed hydrological models considering: Application of MOEAs and resolving the problem in a multi-objective formulation instead of function aggregation technique to obtain Pareto optimum. Constraining input parameters.

MOEA Applications in Calibration of Conceptual Rainfall-Runoff Models Remarks and recommendations Proper study on application of hybrid EA Developing a framework to establish a criterion to choose a solution among Pareto-optimum solutions and state as the final solution of the problem As the main weakness of MOEAs is that they require a large number of function evaluations through consumption of a great deal of time, it would be promising to direct efforts towards application of meta-modeling to reduce the number of simulations.

Development of Improved NSGA-II NSGA-II, Deb et al. (2002)

Development of Improved NSGA-II Innovations Application of Heuristic Genetic Operators (Crossover and Mutation) Heuristic parent-centric recombination (PCX) operator Adaptation by Fuzzy Logic Controller (FLC) Mathematical Test Problems ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, (Deb et al., 2002)

Development of Improved NSGA-II Metrics of Performance Diversity Metrics Convergence Metrics

Development of Improved NSGA-II Results and Conclusions Diversity Metrics

Development of Improved NSGA-II Results and Conclusions Convergence Metrics

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