基于比较标准的多准则问题求解新原理

Using tropical optimization techniques in bi-criteria decision problems

We consider decision problems of rating alternatives based on their pairwise comparisons according to two criteria. Given pairwise comparison matrices for each criterion, the problem is to find the overall scores of the alternatives. We offer a solution that involves the minimax approximation of the comparison matrices by a common consistent matrix of unit rank in terms of the Chebyshev metric in logarithmic scale. The approximation problem reduces to a bi-objective optimization problem to minimize the approximation errors simultaneously for both comparison matrices. We formulate the problem i…

Algebraic solution to constrained bi-criteria decision problem of rating alternatives through pairwise comparisons

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (…

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

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。

使用成对比较法确定消费者在酒店选择中的偏好

Using pairwise comparisons to determine consumer preferences in hotel selection

本文通过202名学生的调查数据，结合直接评分与三种成对比较方法评估酒店选择准则的偏好，并利用统计技术比较不同评估结果以做出准确判断。

Algebraic solution to box-constrained bi-criteria problem of rating alternatives through pairwise comparisons

We consider a decision-making problem to evaluate absolute ratings of alternatives that are compared in pairs according to two criteria, subject to box constraints on the ratings. The problem is formulated as the log-Chebyshev approximation of two pairwise comparison matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank), to minimize the approximation errors for both matrices simultaneously. We rearrange the approximation problem as a constrained bi-objective optimization problem of finding a vector that determines the approximating consistent matrix, and then …

一种变分问题比较原理的合成方法，及其在最优传输中的应用

A synthetic approach to comparison principles for variational problems, with applications to optimal transport

本文提出了一种基于子模性和替代性的合成方法，用于在无限维空间中推导最优传输问题的比较原理，无需依赖极小解的正则性假设。