多目标最优化问题中多目标决策的最小曼哈顿距离方法
Minimum Manhattan Distance Approach to Multiple Criteria Decision Making in Multiobjective Optimization Problems
摘要：
针对多目标最优化问题的多目标决策问题，提出了一种最小曼哈顿距离（MMD）方法。该方法从归一化的理想向量中选择具有MMD的向量所对应的最终解。这个过程是等效的膝盖选择所描述的一个分而治之的方法，涉及成对比较的迭代。MMD方法能够系统地将权系数分配给多个准则，因此等效于加权和（WS）方法。由于等价性，MMD方法具有丰富的几何解释，在进化计算领域被认为是必不可少的。MMD方法是优雅的，因为所有的评估都可以通过有效的矩阵计算来执行，而无需迭代比较。虽然WS方法可能会遇到不确定的情况下，几个解决方案产生几乎相同的WS，MMD方法是能够确定最终的解决方案有区别的。由于现有的多目标进化算法的目标是后验决策，即，在得到一组Pareto最优解后再确定最终解，MMD方法可以与它们结合起来，形成一种求解多目标优化问题的强有力的求解方法。此外，该方法还支持膝关节和膝关节解决方案的可扩展定义。
英文摘要：
A minimum Manhattan distance (MMD) approach to multiple criteria decision making in multiobjective optimization problems (MOPs) is proposed. The approach selects the final solution corresponding with a vector that has the MMD from a normalized ideal vector. This procedure is equivalent to the knee selection described by a divide and conquer approach that involves iterations of pairwise comparisons. Being able to systematically assign weighting coefficients to multiple criteria, the MMD approach is equivalent to a weighted-sum (WS) approach. Because of the equivalence, the MMD approach possesses rich geometric interpretations that are considered essential in the field of evolutionary computation. The MMD approach is elegant because all evaluations can be performed by efficient matrix calculations without iterations of comparisons. While the WS approach may encounter an indeterminate situation in which a few solutions yield almost the same WS, the MMD approach is able to determine the final solution discriminately. Since existing multiobjective evolutionary algorithms aim for a posteriori decision making, i.e., determining the final solution after a set of Pareto optimal solutions is available, the proposed MMD approach can be combined with them to form a powerful solution method of solving MOPs. Furthermore, the approach enables scalable definitions of the knee and knee solutions.
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