耦合高维排序权和对偶线性规划的综合指数算法

区域可持续与高质量发展需定期监测并科学评估。综合指数评价是应用最为广泛、信息传输效率最高的评估方法。该方法将表征区域发展质量不同侧面的系列指标加权聚合为单个指数,其核心是采用或客观或主观的方式确权。客观确权基于指标的数值统计特性实现,故权重值随指标值而改变,实非“客观”;主观确权能反映决策者智慧,但指标过多时难以实现。学界新提出的基于耦合排序权和熵权法的综合指标法允许决策者确权时仅给出权重排序,但仅支持低维情况（限制为3个指标）。本文在其基础上,借助对偶线性规划推导,使其适用性不再受限于指标数（即实现高维排序权）。同时,对指标的聚合方式进行修改,扩大了算法的适用范围;对熵权法的使用进行修正,增强了算法结果的可解译性。基于推导结果,本文进一步发展出单排序、复排序、全排序3种不同模式下的综合指数计算方法,以满足决策者主观性强、弱、无等不同情况。最后,将算法用于全球可持续发展格局的时空评价分析。本文新发展的算法同时适用于高低不同维度的指数聚合、可兼顾决策者不同层次的主观参与度,具有较强的普适性。

A method for constructing a composite index by coupling high-dimensional ranked weights and the duality of linear programming

To achieve regional sustainable development and high-quality development, regular monitoring and scientific assessment are needed. The most widespread and informative method for such an assessment is to employ a composite index. To generate a composite index, various indicators that reflect regional development quality in different aspects are weighted and then aggregated to an index. The core idea of the composite index is weighting, which can be achieved objectively or subjectively. Objective weighting is usually conducted using statistical characteristics of indicator values; thus, the resultant weights change with the values of indicators. This change questions the objectivity of the weighting. In contrast, subjective weighting well reflects the willingness of decision-makers, who feel it is difficult to determine the weights of each indicator if there are too many. Fortunately, a recently developed method eliminated this difficulty. It requires only the rank (also ranked weights) instead of the accurate weights of different indicators, but the number of indicators is restricted to only three. In this study, we improved this method by relaxing such a restriction. We employed the duality of linear programming to generalize the method for calculating a composite index based on any number of indicators. The aggregation is adjusted to expand the scope of the application, and the use of entropy weights is adjusted to improve the interpretability of results. Furthermore, we developed three different calculation modes: single, multiple, and full ranks, corresponding to three conditions of decision-makers' subjectivity: strong, weak, and non-subjective, respectively. Finally, we employed this method to examine global sustainable development patterns temporally and spatially. Since this method is suitable for high- and low-dimensional cases and can consider decision-makers' subjectivity, it has strong universality.