## 3 Proposed Algorithm
### Evolution of the Mainstream Market Population  
In the DILA, the mainstream market population evolves first (phase 1). The procedure for the evolution of the mainstream market population is presented in Figure 2. The members of the population will be updated by using two methods: (1) learning from the local best solution and (2) self-learning. The partially mapped crossover (PMX) operator suggested by [PERSON] and [PERSON] [13] is modified to be the learning method in the paper. For convenience, we named it MPMX. It allows an information exchange between member \(p\) (\(\Pi_{1p}\)) and the local best solution \(\Pi_{1}^{best}\) when the member's solution value (\(V_{1p}\)) is not equal to the local best solution value (\(V_{1}^{best}\)). The MPMX crossover used in this study is described as follows. Taking Figure 3 as an example, there are two members with 10 jobs between them in the diagram, namely, member \(\Pi_{1p}\) and the local best solution \(\Pi_{1}^{best}\). The member \(\Pi_{1p}\) can inherit a fixed length of subsequence from the local best solution \(\Pi_{1}^{best}\). We assumed that the fixed length is 4 and one point was selected randomly is 3, for example, positions 3 and 6, which indicated that member \(\Pi_{1p}\) would inherit the jobs in positions 3 to 6 from the local best solution \(\Pi_{1p}\). The rest of the positions in \(\Pi_{p}\), such as \(\mathrm{J}_{3}\), \(\mathrm{J}_{10}\), and \(\mathrm{J}_{8}\), were retained if they were not the same as the inherited jobs in positions 3 to 6. The order of the repetitive parts, such as \(\mathrm{J}_{5}\), \(\mathrm{J}_{4}\), and \(\mathrm{J}_{1}\), were determined by the original order of member \(\Pi_{1p}\). Then, they were filled into the blank positions 1, 2, and 9 in sequence from left to right. Then, \(\mathrm{J}_{5}\) is first placed into the blank position; when all the jobs are put into the blank positions, a new individual solution \(\Pi_{p}\) is formed. Furthermore, the DC procedure [12] is applied as the self-learning method. Figure 2: The procedure for the evolution of the mainstream market population in the DILA. When all members in population 1 have been updated, a solution \(\Pi_{1k}\) is selected, and the variable neighborhood descent (VND) algorithm is used to improve the solution. The selected solution value \(V_{1k}\) is the minimum among all the members' solution values and is different from the local best solution value \(V_{1}^{\textit{best}}\). The VND is a metaheuristic proposed by [PERSON] and [PERSON] [14] has been successfully applied to solve scheduling problems. The VND searches the solution space using a set of predefined neighborhood structures. The search escapes from local optima by systematically changing these neighborhood structures. In the paper, a series of neighborhood structures based on the insertion move of a subsequence (block) is implemented as the neighborhood structures of the VND. The variable sequence of neighborhood structures used in this paper is the same as the one in [PERSON] [15], which employs eight insertion neighborhood structures. The move of a subsequence may involve a one-job insertion move, two consecutive jobs insertion move,..., or eight consecutive jobs insertion move. For convenience, a one-job insertion move is denoted as \(\textit{INS}(1)\), a two consecutive jobs insertion move is denoted as \(\textit{INS}(2)\),..., and the eight consecutive jobs insertion move is denoted as \(\textit{INS}(8)\). Finally, two functions--the member-added function (MAF) and member-removed function (MRF)--are implemented to represent the joining and disappearance of members. A new member is added to the population, or a member of the population is eliminated according to two controlled parameters, \(c_{1}\) and \(c_{2}\), which are less than or equal to 1.0. The values \(c_{1}\) and \(c_{2}\) are set in order to generate competent probabilities that preserve population diversification. Eleven combinations of \(c_{1}\) and \(c_{2}\) shown in Table 1 are considered in this paper.