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

Evolutionary Algorithms are a standard tool for multi-objective optimization that are able to approximate the Pareto front in a single optimization run. However, for some selection operators, the algorithm stagnates at a certain distance from the Pareto front without convergence for further iterations.

Abstract. In this paper, we propose an algorithm based on the clonal selection principle to solve multiobjective optimization problems (either constrained or unconstrained). The proposed approach uses Pareto dominance and feasibility to identify solutions that deserve to be cloned, and uses two types of mutation: uniform mutation is applied to the clones produced and non-uniform mutation is applied to the “not so good ” antibodies (which are represented by binary strings that encode the decision variables of the problem to be solved). We also use a secondary (or external) population that stores the nondominated solutions found along the search process. Such secondary population constitutes the elitist mechanism of our approach and it allows it to move towards the true Pareto front of a problem over time. Our approach is compared with three other algorithms that are representative of the state-of-the-art in evolutionary multiobjective optimization. For our comparative study, three metrics are adopted and graphical comparisons with respect to the true Pareto front of each problem are also included. Results indicate that the proposed approach is a viable alternative to solve multiobjective optimization problems.

In this paper, we propose a new mechanism to maintain diversity in multi-objective optimization problems. The proposed mechanism is based on the use of stripes that are applied on objective function space and that is independent of the search engine adopted to solve the multi-objective optimization problem. In order to validate the proposed approach, we included it in a multi-objective particle swarm optimizer. Our approach was compared with respect to two multi-objective evolutionary algorithms which are representative of the state-of-the-art in the area. The results obtained indicate that our proposed mechanism is a viable alternative to maintain diversity in the context of multi-objective optimization. 1.

The overall aim in multi-objective optimization is to aid the decisionmaking process when tackling multi-criteria optimization problems. In an a posteriori approach, the strategy is to produce a set of nondominated solutions that represent a good approximation to the Pareto optimal front so that the decision-makers can select the most appropriate solution. In this paper we propose the use of diversity measures to guide the search and hence, to enhance the performance of the multi-objective search algorithm. We propose the use of diversity measures to guide the search in two different ways. First, the diversity in the objective space is used as a helper objective when evaluating candidate solutions. Secondly, the diversity in the solution space is used to choose the most promising strategy to approximate the Pareto optimal front. If the diversity is low, the emphasis is on exploration. If the diversity is high, the emphasis is on exploitation. We carry out our experiments on a two-objective optimization problem, namely space allocation in academic institutions. This is a real-world problem in which the decision-makers want to see a set of alternative diverse solutions in order to compare them and select the most appropriate allocation. 1.

Abstract. The aim of this work is to propose and validate a new multiobjective optimization algorithm based on the emulation of the immune system behavior. The rationale of this work is that the artificial immune system has, in its elementary structure, the main features required by other multiobjective evolutionary algorithms described in literature. The proposed approach is compared with the NSGA2 algorithm, that is representative of the state-of-the-art in multiobjective optimization. Algorithms are tested versus three standard problems (unconstrained and constrained), and comparisons are carried out using three different metrics. Results show that the proposed approach have performances similar or better than those produced by NSGA2, and it can become a valid alternative to standard algorithms. 1

Optimisation (MOO) has been focused on achieving the Pareto optimal front by explicitly analysing the dominance level of individual solutions. While such approaches have produced good results for a variety of problems, they are computationally expensive due to the complexities of deriving the dominance level for each solution against the entire population. TB_MOO (Threshold Based Multiobjective Optimisation) is a new artificial life approach to MOO problems that does not analyse dominance, nor perform any agent-agent comparisons. This reduction in complexity results in a significant decrease in processing overhead. Results show that TB_MOO performs comparably, and often better, than its more complicated counter-parts with respect to distance from the Pareto optimal front, but is slightly weaker in terms of distribution and extent. 1

Nondominated Neighbor Immune Algorithm (NNIA) is proposed for multiobjective optimization by using a novel nondominated neighbor-based selection technique, an immune inspired operator, two heuristic search operators, and elitism. The unique selection technique of NNIA only selects minority isolated nondominated individuals in the population. The selected individuals are then cloned proportionally to their crowding-distance values before heuristic search. By using the nondominated neighbor-based selection and proportional cloning, NNIA pays more attention to the less-crowded regions of the current trade-off front. We compare NNIA with NSGA-II, SPEA2, PESA-II, and MISA in solving five DTLZ problems, five ZDT problems and three low-dimensional problems. The statistical analysis based on three performance metrics including the Coverage of two sets, the Convergence metric, and the Spacing, show that the unique selection method is effective, and NNIA is an effective algorithm for solving multiobjective optimization problems. The empirical study on NNIA’s scalability with respect to the number of objectives shows that the new algorithm scales well along the number of objectives.

In this paper, we introduce two measurements for computing the diversity and spread of non-dominated solutions in the objective space. These measurements compute the angular positions of solutions in the objective space and are able to find a percentage which indicates the distribution of solutions in the space. Also, because we are able to compute the positions of the solutions, the spread of solutions along the non-dominated front can also be measured. This is more important when we evaluate solutions of a problem with a large number of objectives, the objective space of which cannot be illustrated graphically. These measurements are being examined to measure distribution of several sets of non-dominated solutions in the objective space.

The research area of evolutionary multiobjective optimization (EMO) is reaching better understandings of the properties and capabilities of EMO algorithms, and accumulating much evidence of their worth in practical scenarios. An urgent emerging issue is that the favoured EMO algorithms scale poorly when problems have ‘many ’ (e.g. five or more) objectives. One of the chief reasons for this is believed to be that, in many-objective EMO search, populations are likely to be largely composed of nondominated solutions. In turn, this means that the commonly-used algorithms cannot distinguish between these for selective purposes. However, there are methods that can be used validly to rank points in a nondominated set, and may therefore usefully underpin selection in EMO search. Here we discuss and compare several such methods. Our main finding is that simple variants of the often-overlooked ‘Average Ranking ’ strategy usually outperform other methods tested, covering problems with 5—20 objectives and differing amounts of inter-objective correlation. Categories and Subject Descriptors I.2.8 [Problem solving, control methods and search]: Heuristic