## 3 Developed Solution Approach
### Solution Recombination  
Solution recombination is an essential operator in the search procedures of modern metaheuristics. It combines some features from parent solutions to generate a new child solution. This operator helps explore more regions in the search space between or even beyond the parent solutions. This operator appears in evolutionary metaheuristics under the name of a crossover operator [47]. In swarm-intelligence metaheuristics, the mechanism of solution recombination is conducted via position-update functions [48]. Path relinking is another form of solution-recombination operators [39].  
In the current implementation of the scatter-search metaheuristic, the solution-recombination operator works as follows. First, two different parent solutions are selected randomly from the reference set, where all solutions have equal probabilities of being selected. One selected parent is labeled the leader (\(\mathcal{L}\)), and the other is labeled the follower (\(\mathcal{F}\)). These labels are used to identify the order by which genes will be inherited through the child solution (\(\mathcal{C}\)).  
Figure 6 demonstrates the gene-inheriting mechanism using solutions for the sample instance presented in Section 3.2. In this mechanism, genes are inherited for each sub-string independently for both strings constituting each chromosome. For each sub-string, a number, \(U^{X}\in[0,1]\), is randomly generated from a uniform distribution. If \(U^{X}\) is greater than a predefined recombination threshold value, denoted as \(\Theta^{X}\), the genes from the leader parent's sub-string will be copied as-is to the child solution.  
When \(U^{X}\leq\Theta^{X}\), genes in the child solution's sub-string are inherited from both parents' sub-strings as follows. First, an integer number, \(p^{X}\in\{1,2,\ldots\ell\}\), is randomly generated from a discrete uniform distribution, where \(\ell\) is the length (number of genes) of the sub-string. Figure 6 represents the randomly generated positions using a small arrow underneath the sub-strings. Then, genes from the leader solution's sub-string are copied to the child solution's sub-string, starting from the first gene up to the gene in location \(p^{X}-1\). Afterward, starting from location \(p^{X}\), the remaining genes are inherited from the follower parent using the same sequence they appear in that parent. Positions of the machine separators are inherited as-is from the parent having the larger number  
Figure 5: Illustration of the minimal moves needed to convert the second solution in Figure 4b to the first solution in Figure 4a. The numbers 1 through 5 correspond to the conducted five moves.  
of inherited genes in the corresponding workstation's sub-string, where tie-breaking is done randomly.  
Since the resultant child solution can be infeasible, the recombination operator is applied a maximum of five times on the same selected parent solutions until a feasible child solution is generated. If no feasible solution is obtained after these five iterations, either the leader or the follower solution is copied to the child solution. In this case, both parents have equal copying probabilities.