# An Empirical Study of Cluster-Based MOEA/D Bare Bones PSO for Data Clustering +
## 3 Materials and Methods
### Cluster-Based MOEA/D BBPSO  
Each solution in a multiobjective clustering (MOC) problem is regarded as a set of cluster centers \(\mathbf{X}_{s}\) or \(\mathbf{v}\) in (1). Cluster connectedness (Conn) and within-sum-of-squares (WSS) [3] are used as objective functions in this work to evaluate the best solution candidates. The final solution is then selected using crowding distance on the external population. In a MOC framework, the population \(P\) contains \(S\) solutions with \(\mathbf{X}=\{\mathbf{X}_{1},\dots,\mathbf{X}_{s}\}\) where the matrix of each solution \(\mathbf{X}_{s}\) represents \(C\) cluster centers of a dataset containing \(N\) objects and \(D\) dimensions as follows:\[\mathbf{X}_{s}=\begin{bmatrix}x_{11}&x_{12}&\ldots&x_{1D}\\ x_{21}&x_{22}&\ldots&x_{2D}\\ \vdots&\vdots&\ddots&\vdots\\ x_{C1}&x_{C2}&\ldots&x_{CD}\end{bmatrix} \tag{5}\]  
In the CM-BBPSO clustering framework, there are two parts. The BBPSO clustering algorithm is in charge of updating the particles in the swarm. The _pBest_s from BBPSO are selected based on the quantization error-based fitness function from [23]. BBPSO is used because it requires lesser parameters than PSO and performs well in clustering, based on previous work [30]. Next, we subject the _pbest_ solutions found by BBPSO to the cluster-based MOEA/D algorithm for solution evolving and selection to find the _gbest_ for the BBPSO framework. The population size for the cluster-based MOEA/D is the same as the swarm size for the BBPSO. The objective functions Conn and WSS are used to update \(FV\) matrix and then evaluate solutions in lines 6, 15, 16 in Algorithm 2. Newly evolved solutions are compared with existing ones and the better one is selected to be updated. The fitness matrix \(FV\) containing WSS and Connectedness fitness values of the better solution are stored.  
The cluster-based approach used in this study is adapted from [13] and applied at line 9 in Algorithm 2 in the MOEA/D PBI framework from [29] to find the _gBest_ candidates, given all _pBest_. Here, we apply k-means clustering with \(k=2\) to guide the solution selection. Based on the fitness values in the \(FV\) matrix of \(N\) subproblems, we cluster them. The improved potential individual \(y^{\prime}\) is evaluated against neighbors belonging to the group and added to \(EP\) if it has a more favorable evaluation value. In doing so, the application of the cluster-based approach is to improve the local search within the objective space for new candidates, while the genetic operators maintain a random global search. We will test on higher \(k\) values to explore deeper in future work. In this work, we apply a small \(k\) value and simple conventional algorithms and distance measures, such as k-means and Euclidean distance, to study the effects of clustering-based MOEA/D as a preliminary study. Indeed, more investigation is required to understand deeply the Pareto solutions found in the objective space and the more sophisticated methods to deal with challenging Pareto solutions as proposed in [18].  
To investigate its performance, we tested on 13 public datasets and compared cluster-based MOEA/D BBPSO (CM-BBPSO) with the following MOPSO clustering frameworks: IBEA QPSO (IB-QPSO), SPEA2 QPSO (SP-QPSO), MOEA/D QPSO (M-QPSO)and BBPSO (M-BBPSO), and MOEA/GLU QPSO (MG-QPSO) and BBPSO (MG-BBPSO), where the respective MOEAs are individually incorporated into a fuzzy, QPSO or BBPSO clustering framework as described in Algorithm 1 in [4]. An abbreviation list is given before the References section.