利用热带优化技术通过成对比较评估替代方案
Using tropical optimization techniques to evaluate alternatives via pairwise comparisons
AI总结：
本文提出基于热带优化技术的新方法，通过成对比较数据评估替代方案，采用对数切比雪夫意义下的近似方法，构建单位秩的对称矩阵，并统一处理乘法和加法比较尺度，提供紧凑向量形式的解，并通过数值示例展示新方法的优势。
摘要：
我们描述了一种基于热带优化技术的新方法，用于解决通过成对比较数据对替代方案进行评分的问题。该问题被公式化为在对数切比雪夫意义上通过单位秩的对称矩阵近似成对比较矩阵，并以热带数学的一般形式表示为热带优化问题。优化问题采用统一的形式，适用于乘法和加法比较尺度。我们应用最近的热带优化结果，为各种关于成对比较矩阵的假设下提供新的完整解。这些解以紧凑的向量形式给出，扩展了已知的解，并涉及适度的计算努力。所得结果通过数值示例进行说明。具体而言，我们通过示例展示，已知的不完整解可能遗漏由新完整解提供的更好结果。还给出了展示热带分析层次过程决策方案的热带类比示例。
英文摘要：
We describe a new approach based on tropical optimization techniques to solve the problem of rating alternatives from pairwise comparison data. The problem is formulated to approximate, in the log-Chebyshev sense, pairwise comparison matrices by reciprocal matrices of unit rank, and then represented in general terms of tropical mathematics as a tropical optimization problem. The optimization problem takes a common, unified form for both multiplicative and additive comparison scales. We apply recent results in tropical optimization to offer new complete solutions to the rating problems under various assumptions about the pairwise comparison matrices. The solutions are given in a compact vector form, which extends known solutions and involves modest computational efforts. The results obtained are illustrated with numerical examples. Specifically, we show by example that the partial solution known before may miss better results provided by the new complete solution. An example to demonstrate a tropical analogue of the analytical hierarchy process decision scheme is also given.
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