矩形随机矩阵在方案评价和排序问题中的应用

Oleksii Oletsky
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引用次数: 0

摘要

本文研究了“状态-概率选择”模型的可能泛化问题,以便将泛化模型应用于个体或一组智能体的选择排序问题。结果表明,前人对多智能体选择和多数投票决策问题的研究结果可以很容易地推广到多智能体备选方案排序问题。在选择排序问题的重要值分布的基础上,我们可以借助众所周知的指数归一化行,转向类似的选择和投票模型。因此我们考虑两类矩阵,它们都属于平衡矩形随机矩阵。对于这样的矩阵,每行的元素和等于1,所有列的元素和相等。这两种类型都涉及到本文所考虑的两级程序。首先要构造一个表示所有可能的重要度分布的矩阵,然后在此基础上得到一个“选择状态-概率”矩阵。为了形成一个状态矩阵,建议采用两两比较和层次分析法,其中的状态矩阵属于并且其行对应于可能的重要分布。考虑了影响最佳方案和最差方案之间重要性分布的参数化传递尺度。为了进一步得到选择概率矩阵,还引入了另一个反映智能体决策程度的参数。文中讨论了这两个参数的作用,并举例说明了它们的作用。本文报道了一些数值实验的结果,这些实验说明了在层次分析法的基础上得到重要分布,并与获得备选方案的动态均衡情况有关,即当备选方案被认为是相等的情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Using of Rectangular Stochastic Matrices for the Problem of Evaluating and Ranking Alternatives
The paper investigates the issue related to a possible generalization of the “state-probability of choice” model so that the generalized model could be applied to the problem of ranking alternatives, either individual or by a group of agents. It is shown that the results obtained before for the problem of multi-agent choice and decision making by majority of votes can be easily transferred to the problem of multi-agent alternatives ranking. On the basis of distributions of importance values for the problem of ranking alternatives, we can move on to similar models for the choice and voting with the help of well-known exponential normalization of rows.So we regard two types of matrices, both of which belonging to the sort of matrices named balanced rectangular stochastic matrices. For such matrices, sums of elements in each row equal 1, and all columns have equal sums of elements. Both types are involved in a two-level procedure regarded in this paper. Firstly a matrix representing all possible distributions of importance among alternatives should be formed, and secondly a “state-probability of choice” matrix should be obtained on its base. For forming a matrix of states, which belongs and the rows of which correspond to possible distributions of importance, applying pairwise comparisons and the Analytic Hierarchy Method is suggested. Parameterized transitive scales with the parameter affecting the spread of importance between the best and the worst alternatives are regarded. For further getting the matrices of choice probabilities, another parameter which reflects the degree of the agent’s decisiveness is also introduced. The role of both parameters is discussed and illustrated with examples in the paper.The results are reported regarding some numerical experiments which illustrate getting distributions of importance on the basis of the Analytic Hierarchy Process and which are connected to gaining the situation of dynamic equilibrium of alternatives, i.e. the situation when alternatives are considered as those of equal value.
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