ANÁLISIS DEL PROBLEMA

Although the EA is known to be inherently robust to low-level of noise due to its distributed nature of individuals and its non-reliance on gradient information, such a property may not extend well into EMOO that requires the evolutionary search to maintain a diverse set of

CHAPTER 2. 37

non-dominated solutions uniformly distributing along the tradeoff. A few existing noise- handling techniques in EMOO include the approaches of periodic re-evaluation of archived solutions [21], probabilistic Pareto ranking [94], and extended averaging scheme [186] etc.

According to Jin and Branke [107], the different approaches for handling noise can be classified as 1) explicit averaging, 2) implicit averaging, and 3) selection modification. In explicit averaging, the objective values are averaged over a number of samples, H to compute the expected values. Increasing the number of samples reduces the degree uncertainty by a factor of

√

H at the expense of increasing computational cost. In implicit averaging, a large population is used instead of re-evaluating and averaging the objective values over a number of samples. When population size is large, there are many similar solutions and the solutions are implicitly averaged as the MOEA revisit promising regions repeatedly. In selection modification, the ranking and selection procedures are modified such that a solution is judged better than another solution only if it satisfies certain conditions. However, the two noise-handling heuristics, namely the experiential learning directed perturbation operator and the gene adaptation selection strategy, that will be presented in Chapter 3 do not fall under any of the three categories. Therefore, it would be appropriate to define an additional class of “heuristical” techniques for improving MOEA performance in noisy environments.

Explicit averaging: Existing EMOO approaches that applies explicit averaging include [186] and [22]. Using NSGAII [43] as the baseline algorithm, Singh extended the re-sampling method and probabilistic selection scheme [94] to solve a noisy groundwater remediation design problem. In this work, the technique of extended averaging is proposed to reduce the bias introduced by small sample size used in simple averaging. The extended averaging approach performs the averaging over all samples of identical individuals, which can be easily extended over different generations.

Similar to Singh, Bui et al [22] applied NSGAII as the baseline algorithm as three dif- ferent approaches are investigated: NSGAII with probabilistic Pareto ranking and NSGAII with two variants of explicit averaging based NSGAII. In order to reduce the number of

evaluations required, the mechanisms of fitness inheritance is also extended from noisy SO optimization in this work. In particular, a threshold that is calculated from the offspring’s objective values and estimated variance is used to determine if the offspring will undergo multiple re-evaluation or adopt the mean fitness of the parents. The investigation concludes that the probabilistic approach will yield better results initially but explicit averaging will provide better results eventually.

In [11], Basseur and Zitzler studied the impact of noise on indicator-based MOEAs. A significant difference between this work and the previous two approaches is that, instead of expected objectives values, the expected -indicator values are sought. As the computation of the expected -indicator values is very intensive, three different approaches of estimating the expected indicator values are compared and analyzed.

Implicit averaging: The periodic re-evaluation and reinsertion of archived solutions can be classified under implicit averaging. Adapting from SPEA [230], Buche et al [21] proposed the noise tolerant strength Pareto evolutionary algorithm (NTSPEA) with an improved ro- bust performance against noise. In particular, the elite preservation scheme is modified to reduce the detrimental effect of outliers, and every solution is assigned a lifetime that is dependent on the fraction of the archive it dominates. Any archive solutions with expiring lifetime is re-evaluated and added to the evolving population. In the subsequent archive up- dating, expired archive solutions will not be considered. However, the re-evaluated solutions will participate in the archiving process.

Selection modification: Currently, noise-handling schemes of the third category, par- ticularly the use of probabilistic Pareto ranking scheme, is the most popular approach. Hughes [94] introduced a probabilistic approach for Pareto ranking scheme to account for the presence of uncertainties, and demonstrated the possible deficiencies of layered ranking approach adopted in NSGA [188]. In a similar vein, Hughes [95] extended the probabilistic ranking to handle constraints in noisy environments. In the proposed multi-objective prob- abilistic selection evolutionary algorithm (MOPSEA), elitism is implemented by replacing

CHAPTER 2. 39

Table 2.2: Parameter settings of the simulation study

Parameter Settings

Chromosome Binary coding; 15 bits per decision variable.

Population Population size 100; Archive (or secondary population) size 100. Selection Binary tournament selection

Crossover operator Uniform crossover Crossover rate

0.8

Mutation operator Bit-flip mutation

Mutation rate _{chromosome length}1 for ZDT1, ZDT4 and ZDT6;

1

bit number per variable for FON and KUR;

Ranking scheme Pareto ranking.

Diversity operator Niche count with radius 0.01 in the normalized objective space. Evaluation number 50,000

part of the evolving population with the best individuals.

While Hughes assumes that noise is normal distributed, Teich [203] considers a uniform noise distribution. Teich extended the SPEA algorithm in two ways 1) a probabilistic strength fitness is used and 2) the update of the external set is based on the a percentage of the best solutions and solutions with a fitness that is above a certain threshold. Building upon these works, Fieldsend and Everson [57] considered the computation of Probabilistic ranking under different conditions such as unknown noise properties, independent noise for each objectives and etc. Based on preliminary theoretical analysis, an online variance learning scheme is presented and validated empirically.