Abstract: Comprehensive evaluation is the important research director of management science and engineering, system engineering, and information science. And it is an important premise of scientific decision-making. In comprehensive evaluation, eliminating the impacts of the different dimensions corresponding to the evaluation index is a key process in a complete comprehensive evaluation activity. The existing dimensionless treatment methods are developed for traditional static evaluation problems which do not consider the factor of the number of experts or decision-makers. However, using the common linear dimensionless treatment methods to deal with the index data given by multiple decision makers directly will change the size relationship between raw data, and further use of the dimensionless data for evaluation activity will lead to unreasonable and even error evaluation results. Therefore, extending the common linear dimensionless treatment methods to the group evaluation deserves research. Aiming at this problem, we find that the index data is not the unique variable when using the common linear dimensionless treatment methods to group evaluation directly is the important reason.
To solve this issue, based on the six kinds of commonly used linear dimensionless treatment methods, we first define the problem, and further extend the six dimensionless treatment methods. So, the extended standardized treatment method, the extended extremum treatment method, the extended linear proportional method, the extended normalization method, the extended vector normalization method, and the extended efficiency coefficient method, are obtained. The significant features of the six extended methods are that only the index data is variable when using these methods for group evaluation. This solution is also suitable for dynamic evaluation which further considers the time factor in evaluation. We secondly analyze the properties of the six extended linear dimensionless treatment methods. In this process, two new properties, the lateral monotonic property, and the single variable property are proposed, which are the necessary properties the linear dimensionless treatment methods should meet when considering group evaluation. And a total of eight properties are proposed to analyze the six extended methods. We find that all these six extended methods fulfill the longitudinal monotonicity property, the lateral monotonic property, the single variable property, and the variance ratio invariance property. And there is no linear dimensionless treatment method satisfying all eight properties because they only satisfy one of the interval stability and total constancy properties. We thirdly analyze how to select the six linear dimensionless treatment methods. We use the coefficient of variation to measure the size of the original information retained by the dimensionless index information. And we find that the coefficient of variation of each index data processed by the linear proportional method, normalization method, and vector normalization method has not changed. The efficacy coefficient method can adjust the values of the translation coefficient and amplification coefficient so that the coefficient of variation of each index data processed by the efficacy coefficient method does not change. From the comparison of the coefficient of variation, the linear proportional method, normalization method, vector normalization method, and efficacy coefficient method are superior to the extreme value method and standardization method. Because only measuring the amount of information retained in the original index data after dimensionless index information from the perspective of coefficient of variation cannot compare and select the linear proportion method, normalization method, vector normalization method, and efficacy coefficient method, according to the different weighting methods, through the comparison of the variance of the dimensionless index data and the variance of the original index data, the principle of dimensionless method selection is to retain the variance of the original indicator data as much as possible after dimensionless. Based on this principle we find that among these three methods, the normalization method is the best method when the sum of all the data corresponding to the same evaluation index is more than one. If the value of the amplification coefficient is appropriate, the efficiency coefficient method is the best linear dimensionless treatment method. And when the minimum value of the special point is taken and the minimum value is less than or equal to one, the linear scaling method is the best linear dimensionless method in the linear scaling method, normalization method, and vector normalization method. We finally use a numerical example to illustrate the effectiveness of this method. And this numerical example proves that extending the common six linear dimensionless treatment methods is really necessary and will help improve the evaluation quality.
In the future, we will construct some new linear dimensionless treatment methods suitable for dynamic evaluation, and we will also research some nonlinear dimensionless treatment methods based on the regulation of the specific evaluation background.

Key words: linear dimensionless treatment method, group evaluation, lateral monotonic property, single variable property, method selection

摘要： 无量纲化处理是开展综合评价的基础,目前线性无量纲化方法很少考虑群体评价的情况。本文针对常用的6种线性无量纲化方法直接应用到群体评价中不能保证各评价者评价信息横向大小顺序的问题,首先对问题进行界定,并对6种线性无量纲化方法进行了扩展;其次进一步分析了扩展后的线性无量纲化方法的性质,并针对群体评价问题引入“横向单调性”和“变量单一性”两个性质,为线性无量纲化方法的设计研究提供重要的参考;再次以无量纲化后的数据最大程度的保留原始信息为原则,针对不同的赋权方法,给出线性无量纲化方法选择的建议;最后,用一个算例检验了方法的有效性。

关键词: 线性无量纲化方法, 群体评价, 横向单调性, 变量单一性, 方法选择