论文标题
使用log-chebyshev近似的热带优化用于求解成对比较的多标准问题
Application of tropical optimization for solving multicriteria problems of pairwise comparisons using log-Chebyshev approximation
论文作者
论文摘要
我们考虑一个决策问题，以找到在多个标准下成对比较的替代方案的绝对等级，但要以两侧边界对等级之间比率的形式受到限制。给定根据标准进行的成对比较的矩阵，该问题通过共同的一致矩阵（单位等级的对称相互矩阵）的对数 - chebyshev近似，以同时使所有矩阵的近似错误最小化。我们将近似问题重新排列作为确定近似一致矩阵的向量的约束多目标优化问题。然后，该问题在热带代数的框架中表示，该框架涉及基于能力的半决赛的理论和应用，并为模糊和间隔算术提供了正式的基础。我们采用热带优化的方法和结果来开发一种新方法，以根据各种最佳原则来处理多目标优化问题。从最大排序，词典订购和词素最大订购最优性的意义上讲，新的完整解决方案是以紧凑的矢量形式给出的，可以进行正式分析和有效的计算。我们提出了解决对成对比较的四个替代方案的多准则问题的数值示例，以说明该技术并将其与他人进行比较。
We consider a decision-making problem to find absolute ratings of alternatives that are compared in pairs under multiple criteria, subject to constraints in the form of two-sided bounds on ratios between the ratings. Given matrices of pairwise comparisons made according to the criteria, the problem is formulated as the log-Chebyshev approximation of these matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank) to minimize the approximation errors for all matrices simultaneously. We rearrange the approximation problem as a constrained multiobjective optimization problem of finding a vector that determines the approximating consistent matrix. The problem is then represented in the framework of tropical algebra, which deals with the theory and applications of idempotent semirings and provides a formal basis for fuzzy and interval arithmetic. We apply methods and results of tropical optimization to develop a new approach for handling the multiobjective optimization problem according to various principles of optimality. New complete solutions in the sense of the max-ordering, lexicographic ordering and lexicographic max-ordering optimality are obtained, which are given in a compact vector form ready for formal analysis and efficient computation. We present numerical examples of solving multicriteria problems of rating four alternatives from pairwise comparisons to illustrate the technique and compare it with others.
