Multi-objective evolutionary algorithms which use non-dominated sorting and sharing have been mainly criticized for their (i) -4 computational complexity (where is the number of objectives and is the population size), (ii) non-elitism approach, and (iii) the need for specifying a sharing parameter. In this paper, we suggest a non-dominated sorting based multi-objective evolutionary algorithm (we called it the Non-dominated Sorting GA-II or NSGA-II) which alleviates all the above three difficulties. Specifically, a fast non-dominated sorting approach with computational complexity is presented. Second, a selection operator is presented which creates a mating pool by combining the parent and child populations and selecting the best (with respect to fitness and spread) solutions. Simulation results on five difficult test problems show that the proposed NSGA-II is able to find much better spread of solutions in all problems compared to PAES---another elitist multi-objective EA which pays special attention towards creating a diverse Pareto-optimal front. Because of NSGA-II's low computational requirements, elitist approach, and parameter-less sharing approach, NSGA-II should find increasing applications in the years to come.

In this paper, we provide a systematic comparison of various evolutionary approaches to multiobjective optimization using six carefully chosen test functions. Each test function involves a particular feature that is known to cause difficulty in the evolutionary optimization process, mainly in converging to the Pareto-optimal front (e.g., multimodality and deception). By investigating these different problem features separately, it is possible to predict the kind of problems to which a certain technique is or is not well suited. However, in contrast to what was suspected beforehand, the experimental results indicate a hierarchy of the algorithms under consideration. Furthermore, the emerging effects are evidence that the suggested test functions provide sufficient complexity to compare multiobjective optimizers. Finally, elitism is shown to be an important factor for improving evolutionary multiobjective search. Keywords Evolutionary algorithms, multiobjective optimization, ...

Solving optimization problems with multiple (often conflicting) objectives is, generally, a very difficult goal. Evolutionary algorithms (EAs) were initially extended and applied during the mid-eighties in an attempt to stochastically solve problems of this generic class. During the past decade, a variety of multiobjective EA (MOEA) techniques have been proposed and applied to many scientific and engineering applications. Our discussion's intent is to rigorously define multiobjective optimization problems and certain related concepts, present an MOEA classification scheme, and evaluate the variety of contemporary MOEAs. Current MOEA theoretical developments are evaluated; specific topics addressed include fitness functions, Pareto ranking, niching, fitness sharing, mating restriction, and secondary populations. Since the development and application of MOEAs is a dynamic and rapidly growing activity, we focus on key analytical insights based upon critical MOEA evaluation of c...

This paper shows how the performance of evolutionary multiobjective optimization (EMO) algorithms can be improved by the hybridization with local search. The main positive effect of the hybridization is the improvement in the convergence speed to the Paretofront. On the other hand, the main negative effect is the increase in the computation time per generation. Thus the number of generations is decreased when the available computation time is limited. As a result, the global search ability of EMO algorithms is not fully utilized. These positive and negative effects are examined by computational experiments on multiobjective permutation flowshop scheduling problems. Results of our computational experiments clearly show the importance of striking a balance between genetic search and local search. In this paper, we first modify our former multiobjective genetic local search (MOGLS) algorithm by choosing only good individuals as initial solutions for local search and assigning an appropriate local search direction to each initial solution. Next we demonstrate the importance of striking a balance between genetic search and local search through computational experiments. Then we compare the modified MOGLS with recently developed EMO algorithms: SPEA and NSGA-II. Finally, we demonstrate that local search can be easily combined with those EMO algorithms for designing multiobjective memetic algorithms.

In real world problems, one is often faced with the problem of multiple, possibly competing, goals, which should be optimized simultaneously. These competing goals give rise to a set of compromise solutions, generally denoted as Pareto-optimal. If none of the objectives have preference over the other, none of these trade-off solutions can be said to be better than any other solution in the set. Multi-objective Evolutionary Algorithms (MOEAs) can find these optimal trade-offs in order to get a set of solutions that are optimal in an overall sense.

Abstract- A large number of methods for solving multiobjective optimisation (MOO) problems have been developed. To compare these methods rigorously, or to measure the performance of a particular MOO algorithm quantitatively, a variety of performance indices (PIs) have been proposed. This paper provides an overview of the various PIs and attempts to categorise them into a certain number of classes according to their properties. Comparative studies have been conducted using a group of artificial solution sets and a group of solution sets obtained by various MOO solvers to show the advantages and disadvantages of the PIs. The comparative studies show that many PIs may be misleading in that they fail to truly reflect the quality of solution sets. Thus, it may not be a good practice to evaluate the performance of MOO solvers based on PIs only. 1

The hypervolume measure is one of the most frequently applied measures for comparing the results of evolutionary multiobjective optimization algorithms (EMOA). The idea to use this measure for selection is self-evident. A steady-state EMOA will be devised, that combines concepts of non-dominated sorting with a selection operator based on the hypervolume measure. The algorithm computes a well distributed set of solutions with bounded size thereby focussing on interesting regions of the Pareto front(s). By means of standard benchmark problems the algorithm will be compared to other well established EMOA. The results show that our new algorithm achieves good convergence to the Pareto front and outperforms standard methods in the hypervolume covered.

1 Decomposition is a basic strategy in traditional multiobjective optimization. However, this strategy has not yet widely used in multiobjective evolutionary optimization. This paper proposes a multiobjective evolutionary algorithm based on decomposition (MOEA/D). It decomposes a MOP into a number of scalar optimization subproblems and optimizes them simultaneously. Each subproblem is optimized by using information from its several neighboring subproblems, which makes MOEA/D have lower computational complexity at each generation than MOGLS and NSGA-II. Experimental results show that it outperforms or performs similarly to MOGLS and NSGA-II on multiobjective 0-1 knapsack problems and continuous multiobjective optimization problems. Index Terms multiobjective optimization, decomposition, evolutionary algorithms, memetic algorithms, Pareto optimality, computational complexity. I.

Abstract. This paper demonstrates that the self-adaptive technique of Differential Evolution (DE) can be simply used for solving a multiobjective optimization problem where parameters are interdependent. The real-coded crossover and mutation rates within the NSGA-II have been replaced with a simple Differential Evolution scheme, and results are reported on a rotated problem which has presented difficulties using existing Multi-objective Genetic Algorithms. The Differential Evolution variant of the NSGA-II has demonstrated rotational invariance and superior performance over the NSGA-II on this problem. 1

Abstract. Differential Evolution (DE) is a simple but powerful evolutionary optimization algorithm with many successful applications. In this paper we propose Differential Evolution for Multiobjective Optimization (DEMO) – a new approach to multiobjective optimization based on DE. DEMO combines the advantages of DE with the mechanisms of Paretobased ranking and crowding distance sorting, used by state-of-the-art evolutionary algorithms for multiobjective optimization. DEMO is implemented in three variants that achieve competitive results on five ZDT test problems. 1