مقاله ترجمه شده بهبود سازی دقیق کارآمد مشکلات تخصیص افزونگی چند منظوره در سیستم های سری و موازی
قسمتی از متن انگلیسی:
Proof: Without loss of generality, we demonstrate this through an example with two subsystems and two objectives. Consider the following non-dominated solution sets for subsystem 1 and 2: ðf n o1 ð Þ x1 ,f n o2 ð ÞÞ ¼ x1 ð Þ ۳,۵ ,ð Þ ۲,۶ ,ð Þ ۴,۱ and ðf n o1 ð Þ x2 ,f n o2 ð ÞÞ ¼ x2 ð Þ ۲,۵ , ð Þ ۴,۳ ,ð Þg 1,7 . The Cartesian combination results in 9 solutions and 4 of these solutions are dominated by the remainder. For instance, the solution obtained by combining 1st solution of subsystem 1 (3,5) and 2nd solution of subsystem 2 (4,3) is (7,8), which is dominated by the solution (6,6) obtained by combining 3rd solution of subsystem 1 (4,1) and 1st solution of subsystem 2 (2,5). Hence, the Cartesian combination of the non-dominated solutions of multiple subsystems can result in dominated solutions. In order to maintain the computational efficiency, the decomposition based approach in Fig. 2 alternates between the Cartesian combining and Pareto filtering steps, e.g., filtering out dominated solutions in smaller sets rather than all at once (Phase 3, Fig. 2). The following proposition establishes that this sequential combining and filtering process does eliminate any non-dominated solution of the RAP. D
