## 3 Developed Solution Approach
### Outline of the Developed Metaheuristic  
As outlined in Figure 1, the proposed MOSS approach consists of two phases, namely initialization and improvement. In the initialization phase, the set of reference solutions, \(\mathcal{R}\), is constructed as follows. A random solution is generated using the construction heuristic described in Section 3.2. This is followed by the application of a local search heuristic based on tabu search (TS), which is outlined in Section 3.5. The new, improved solution is then checked for similarity with other solutions that have been added to the set \(\mathcal{R}\) in previous iterations. The degree of dissimilarity between two solutions, \(\mathbb{S}_{1}\) and \(\mathbb{S}_{2}\), is measured using a distance function, denoted as \(\Delta(\mathbb{S}_{1},\mathbb{S}_{2})\), which is defined in Section 3.3. The distance between the new, improved solution and every solution included in \(\mathcal{R}\) up to the current iteration is evaluated and checked against a predefined distance threshold value, \(\tilde{\Delta}\). The new solution will be added to \(\mathcal{R}\) only if \(\tilde{\Delta}\) is exceeded for all solutions currently in that set. If a number of improved solutions failed to be added to \(\mathcal{R}\) in \(n_{f}^{max}\) times, the \(\tilde{\Delta}\) threshold value is reduced to half of its value. This is done to ensure that the initialization procedure generates the required number of solutions in the reference set (\(RefSetSize\)). This construction approach tries to maintain both the quality and diversity of the initial solutions included in the reference set \(\mathcal{R}\).  
In the improvement phase, a total of \(itr^{max}\) iterations are conducted. In each iteration, a set of trial solutions, denoted as \(\mathcal{T}\), is constructed. This is done by randomly selecting two solutions from the reference set \(\mathcal{R}\) and then generating a new solution by combining some of their features, as described in Section 3.4. The combined solution is then improved via TS and used to update the non-dominated set \(\mathcal{D}\). Within the steps of TS either during the construction or the improvement phase, an incumbent solution is compared with all found non-dominated solutions that are included in a set denoted as \(\mathcal{D}\). If a non-dominated solution is found, it will be added to that set.  
Figure 1: Flow chart of the proposed multi-objective scatter-search metaheuristic.  
The newly generated solutions in set \(\mathcal{T}\) are then used to update the reference set \(\mathcal{R}\). The size of set \(\mathcal{T}\) is limited to twice the size of the set \(\mathcal{R}\), which in turn should not exceed the specified \(RefSetSize\). The main idea behind this updating mechanism is to maintain a set of diverse and high-quality solutions in the set \(\mathcal{R}\), which are used in generating new trial solutions. For the set of non-dominated solutions, no size restriction is imposed. If the generated solutions in set \(\mathcal{T}\) do not change the solutions in both sets, \(\mathcal{R}\) and \(\mathcal{D}\), a diversification procedure is applied, as explained in Section 3.6. The following subsections explain the algorithmic details of the proposed MOSS metaheuristic's main components.