# An Empirical Study of Cluster-Based MOEA/D Bare Bones PSO for Data Clustering +
## 3 Materials and Methods
### Decomposition-Based Multiobjective Evolutionary Algorithm  
The decomposition-based MOEA (MOEA/D) uses three approaches--the weighted sum, Tchebycheff and penalty-based boundary intersection (PBI)--to convert MOP into scalar optimization subproblems, simultaneously solved by evolving a population ofsolutions [29]. Unlike previous MOO techniques, such as dominance-based MOEA, there is no clear ranking of Pareto solutions, while with MOEA/D, such a ranking is possible, as the \(g^{pbi}\) function is expressed as a scalar value as defined in the objective below. MOEA/D with the PBI approach for an \(m\)-objective problem [29] is defined as follows:  
\[\text{Minimize:}g^{pbi}(x|\lambda,z^{*})=d_{1}+\theta d_{2}\] \[\text{subject to:}\ x\in\Omega \tag{3}\]  
where  
\[d_{1} =\frac{||(z^{*}-F(x))^{T}\lambda||}{||\lambda||}\] \[d_{2} =||F(x)-(z^{*}-d_{1}\lambda)||, \tag{4}\]  
where \(x\) is a candidate solution, \(\theta>0\) is a predefined penalty parameter, \(z^{*}=(z^{*}_{1},\dots,z^{*}_{m})^{T}\) is the reference point such that \(z^{*}_{i}=min\{f_{i}(x)|x\in\Omega\}\), \(\tilde{~{}}=(\lambda_{1},\dots,\lambda_{m})^{T}\) is a weight vector such that \(\lambda_{i}\geq 0\) for \(\forall i=1,\dots,m\) and \(\sum_{i=1}^{m}\lambda_{i}=1\). In this study, \(x\) is a candidate solution that undergoes the evaluation process of cluster-based MOEA/D to be selected for next generation, as detailed in Algorithm 2, line 15.  
```
1:Initialization phase:
2: Set external population EP = \(\emptyset\)
3: Initialize \(N\), number of subproblems
4: Generate a uniform spread of \(N\) weight vectors \(\lambda^{1},\dots,\lambda^{N}\)
5: For each \(i=1,\dots,N\), generate \(B(i)=\{i_{1},\dots,i_{T}\}\) where \(\lambda^{i_{1}},\dots,\lambda^{i_{T}}\) are the \(T\) closest weight vectors to \(\lambda^{i}\) using Euclidean distance
6: Using solutions in \(\mathit{pbestList}\) to generate initial population \(x^{1},\dots,x^{N}\in\Omega\) where \(x^{i}\) is current solution to \(i\)th subproblem and set fitness value of all individuals \(FV^{i}=F(x^{i})\) for \(i=1,\dots,N\)
7: Initialize reference point \(z=(z_{1},\dots,z_{m})^{T}\) where \(z_{i}\) is the best value so far for objective \(f_{i}\)
8: Set number of clusters \(k\) as 2
9: Cluster individuals \(x^{1},\dots,x^{N}\) based on their similarity in the \(FV\) matrix and find which \(x^{ic}\) is most representative for each cluster \(c\).
10:Update phase:
11:for\(c=1,\dots,k\)do
12: Reproduction: Generate new solution \(y\) using genetic operators on two solutions \(x^{k}\) and \(x^{l}\) where indexes \(k\) and \(l\) are randomly chosen from \(B(i)\)
13: Improvement: Apply any improvement heuristic on \(y\) to produce \(y^{\prime}\)
14: Update \(z\): For each \(j=1,\dots,m\), if \(z_{j}<f_{j}(y^{\prime})\), set \(z_{j}=f_{j}(y^{\prime})\)
15: Update neighbors: For each \(j\in B(i)\) belonging to cluster \(c\), if \(g^{pbi}(y^{\prime}|\lambda^{j},z)\leq g^{pbi}(x^{i}|\lambda^{j},z)\), \(g^{pbi}\) values calculated using (3), set \(x^{j}=y^{\prime}\) and \(FV^{j}=F(y^{\prime})\)
16: Update EP: Remove solutions in EP dominated by \(y^{\prime}\) and add \(y^{\prime}\) to EP if no vectors in EP dominate \(y^{\prime}\).
17:endfor
18: Termination Condition: Stop and output EP if termination condition is met.
```  
**Algorithm 2** Pseudocode for Cluster-based MOEA/D