a priori Methods

In this section, we describe some of the EMO algorithms which incorporate the preference information prior to the optimization process. Sincea priori methods are used in this thesis to propose new approaches, our literature has mainly focused on these methods. In this thesis, we have categorized a priori algorithms to three main categories based on the techniques that they used to incorporate the preference information: 1) Goal attainment 2) Reference Point Based 3) Light Beam Based.

Some popular approaches inGoal attainmentare described below. Goal Programming, which is proposed by Deb [1998], is one of the first attempts to apply EMO approaches to classical goal programming [Ignizio, 1976]. The ability of an EA to find multiple solutions makes it possible to simultaneously minimize the deviations from individual goals, which eliminates the need for a user-defined weight vector. The effectiveness of this evolutionary approach to goal programming has been verified empirically using several test problems as well as a real-world engineering design problem [Deb, 1998]. In this approach, the empha- sis was mainly on goal satisfaction and the algorithm does not try to find Pareto-optimal solutions close to the supplied goal. Since a DM needs to supply his/her goals before the optimization process, this approach is categorized as aa priori method.

The guided multi-objective evolutionary algorithm (G-MOEA), proposed by Branke et al. [2001], is another a priori method user needs to specify the linear trade-off between objectives before the search begin. For example, in a two-objective problem the decision maker has to specify how many units of the first objectives he/she is willing to trade for one unit of the second objective. G-MOEA then uses this trade-off information to guide the search towards the more desired regions of the Pareto-optimal front. Although G-MOEA is more flexible and intuitive than other approaches, it is not always an easy task for the decision maker to specify the trade-off between objectives, especially for many-objective problems.

Biased-Sharing, which was proposed by Deb [2003], applies the biased sharing technique to NSGA [Srinivas and Deb, 1994] where the biased Pareto-optimal solutions are generated on a desired region. To achieve this the user needs to assign weights to objectives before the optimization process. An objective with a higher priority takes a higher weight value. The main disadvantage of this technique is that it cannot find solutions on a compromise region where all objectives are of similar importance to the decision maker. Branke and Deb [2005] improved the idea of biased sharing and compared its performance with G-MOEA. They proposed a biased crowding distance in NSGA-II which has more flexibility than biased sharing in terms of finding solutions within the region of interest.

Some popular approaches inReference Point Based Algorithmsare described below. Deb et al. [2006] proposed a method that integrated use-preference information with NSGA-II [Deb et al., 2002]. The new method which was called R-NSGA-II requires the decision maker to provide one or more reference points at the beginning of the search process. In R-NSGA-II, a modified version of the crowding distance operator [Deb et al., 2002]pref- erence distance was used to favour the solutions which are closer to the reference point(s). To compute the preference distance, the Euclidean distances of all solutions to the reference

point(s) are calculated and the solutions with lower distances are ranked higher. To maintain the diversity of solutions in the desired regions, an extra parameterǫwas introduced. In this thesis, the performance of the proposed approach has been compared with R-NSGA-II. In short, the following changes have been applied to NSGA-II.

Step 1: The solution closest to the reference point should be found and assigned the rank one. To achieve this, the Euclidean distance between each solution and the reference point(s) are calculated and sorted.

Step 2: Solutions with the smaller crowding distance should be preferred. To achieve this, for each reference point the lowest rank calculated in the previous step is assigned as the crowding distance to a solution. This means that solutions closest to the reference point(s) receive lowest crowding distance values.

Step 3: To maintain diversity of solutions a ǫ parameter is used. To do this, the sum of normalized differences in objective values for all solutions is calculated. Those which have the value ofǫor less have been grouped. From each group a random solution will be picked up and the rest of the members will be assigned a large number to remain in the race.

Wickramasinghe and Li [2009] integrated reference points and light beams with particle swarm optimization (PSO). The new approach (Preference-based NSPSO), which is based on a distance metric, changes the position of particles based on the user supplied information to find the preferred regions. This distance-metric based method was compared with another user-preference based EMO (Dominance-based reference point NSPSO) [Wickramasinghe and Li, 2008] which uses dominance-based comparison. It was shown that the distance metric approach performed better than NSPSO.

The proposed approach provides different options for DM in terms of directing solutions to desired regions. For example, if the user wants to choose several non-dominated solutions in step 4, the proposed approach can display solutions which have a next best achievement function value or use clustering the current populations. The main drawback of this approach is that visualizing the Pareto-optimal front for DM beyond three objectives is difficult.

r-MOEA/D-STM [Li et al., 2014] is the user preference version of MOEA/D-STM [Li et al., 2014]. In this approach a decision maker provides his/her preferences as reference points (r). It is widely accepted that using preferences in MOEAs potentially reduces the computational cost and drives the search direction to particular areas. In this study, reference

points are used in both feasible and infeasible regions, it has been shown that r-MOEA/D- STM is able to provide solutions close to the reference points where there are two or three objectives. It should be noted that the performance of r-MOEA/D-STM has not been mea- sured on complicated Pareto-optimal shapes and many-objective problems.

Light Beam Based Algorithmsare another type of a priori user-preference methods which are very similar to reference point based algorithms. However, a light beam is passed to the optimizer rather than a reference point. One of the popular light beam algorithms is (LBS-NSGA-II) which is proposed by Deb and Kumar [2007a]. They applied light beam search to NSGA-II. The decision maker provides aspiration, reservation and a preference threshold for each objective and the procedure can be continued until a single preferred solution is obtained. To control the density of solutions, the parameterǫis used. A decision maker can choose more than one light beam so more than one set of preferred regions can be found simultaneously. It has been shown that the proposed approach can find solutions on the Pareto-optimal front up to three-objective problems, and solutions can converge satisfactorily if there are more objectives. Another light beam approach is Distance Metric which is proposed by Wickramasinghe and Li [2009]. They integrated light beams with particle swarm optimization (PSO). The new approach changes the position of particles based on the user-supplied information to find the preferred region. Distance metric was compared with another user-preference based EMO (NSPSO) [Wickramasinghe and Li, 2008] which uses a dominance-based comparison and it was shown that the distance metric approach performed better than NSPSO.