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of the length of the parent roll multiplying the number of paper layers in finished products. Formulas (25) and (26) are equality constraints. 3. The Proposed Optimization Algorithm In this section, the MOEA/DTL is proposed based on two outstanding methods, namely teaching-learning-based optimization (TLBO) and MOEA/D. In order to further improve the quality of the solutions, a variable neighborhood search (VNS) algorithm is designed to be combined with the MOEA/DTL to improve the performance of the algorithm (IMOEA/DTL). 3.1. MOEA/DTL MOEA/D uses multiple weight vectors to decompose the multi-objective optimiza- tion problem into multiple single-objective sub-optimization problems. When the weight vectors are distributed uniformly, the MOEA/D assumes Pareto optimal solutions with uniform distribution generated by combining the solutions of sub-optimization prob- lems. Every individual is corresponding to a sub-optimization problem in MOEA/D. In each iteration of MOEA/D, each individual exchanges information with neighbors and coevolves [28]. The flow chart of the standard MOEA/D is shown in Figure 2.
Figure2. The flow chart of Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D).
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