若限定，当取为矩阵的最大特征值对应的特征向量时，取最大值。习惯上，我们希望，为了得到权重系数，可求解以下规划问题，即选择，使得
w alked="862c287d-8045-4fec-a757-47944d2ccbd1" style="font-family: "Times Ne Roman";">If limited , when takingw as the matrix ’s maximum eigenvalue corresponding to the eigenvector, take the maximum value. Conventionally, we hope , in order to obtain the weight coefficients, we can solve the following planning problem, that is, to choosew, so that

=1

理解了纵横向拉开档次法的计算原理，利用该方法开展评价的完整步骤如下：
Understanding the calculation principle of the vertical and horizontal tiered method, the complete steps for conducting evaluations using this method are as follows:

（1）指标一致化处理。为了定性地判定综合评价函数的取值为越大越好，所有指标需要做类型一致化处理，将极小型指标转换为极大型指标。本文选择的转换公式为：
(1) Consistency processing of indicators. In order to qualitatively determine that a larger value of the comprehensive evaluation function is better, all indicators need to undergo type consistency processing, converting extremely small indicators into extremely large indicators. The conversion formula chosen in this paper is:

指标无量纲化处理。为了避免指标之间因为各自量纲及量级不同而存在的不可公度性，本文采用极值处理法对评价指标做无量纲化处理。对于极大型指标，处理方式为：
Dimensionality reduction of indicators. To avoid the incommensurability caused by the different dimensions and magnitudes of the indicators, this paper employs an extreme value processing method for the dimensionality reduction of evaluation indicators. For extremely large indicators, the processing method is:

式中， ，。
In the formula, , .

（3）求取权重系数。利用前文所述纵横向拉开档次法求取权重系数，并进行归一化处理。得到
(3) Calculate the weight coefficient. Use the method of expanding levels mentioned earlier to obtain the weight coefficient and perform normalization. The result is

（4）线性加权得到综合评价值。根据综合评价函数
(4) The comprehensive evaluation value is obtained through linear weighting. According to the comprehensive evaluation function

得到所有被评价对象在不同时间的评价结果。
Obtain the evaluation results of all evaluated subjects at different times.

7.2测度结果及分析
7.2Measurement Results and Analysis

本文利用Matlab软件求解纵横向拉开档次法的规划问题，得到25个指标的权重（见表7-1），根据步骤（4）计算得到重庆市农村集体经济组织发展指数（见表7-5）。
This article usesMatlabsoftware to solve the planning problem of the longitudinal and lateral spacing method, obtaining25indicator weights (see Table7-1), and calculates the development index of rural collective economic organizations in Chongqing (see Table7-5).

表7-5 重庆市农村集体经济组织发展指数（2017—2021年）
Table7-5 ChongqingAgriculturalRural Collective Economic Organization Development Index (2017—2021Year)

7.2.1重庆市农村集体经济组织发展整体水平
7.2.1Chongqing City Rural Collective Economic Organization Development Overall Level

（1）重庆市农村集体经济组织发展全市水平
(1)Chongqing CityAgriculturalDevelopment of Rural Collective Economic Organizations at the City Level

重庆市农村集体经济组织发展水平显著提升。2017—2021年，重庆市农村集体经济组织发展指数从0.509提升到0.608；发展指数增速较快，2018—2020年3年增长率分别达到0.53%、8%、11.29%，2021年发展略有减速（见表7-6）。从5年均值来看，38个涉农区县中有21个区县发展指数超过全市平均水平；有19个区县指数的年均增长率超过全市平均水平，排在前3位的依次是城口县、大渡口区、渝北区，排在后3位的依次是南川区、永川区、九龙坡区（见图7-1）。
Chongqingruralcollective economic organizations have significantly improved their development level.From 2017 to 2021in Chongqingruralcollective economic organization development index increased from0.509to0.608; the growth rate of the development index was relatively fast,with the growth rates for the years2018—2020being0.53%,8%,11.29%,and the development slightly slowed down in 2021 (see table7-6).Fromthe average over 5 years,21of the 38 agricultural-related districts and counties had development indices exceeding the city average;19counties had annual average growth rates exceeding the city average, with the top3being Chengkou County, Dadukou District, and Yubei District, and the bottom3being Nanchuan District, Yongchuan District, and Jiulongpo District (see figure7-1).

图7-1 区县农村集体经济组织发展平均水平及年均增长率
Figure7-1 CountyRuralCollective Economic Organization Development Average Level and Annual Growth Rate

重庆市各区县农村集体经济组织发展指数的整体差距在时间上表现为“先扩大、再缩小”的特点（见图7-2）。各区县发展指数在0.4—0.7之间，2019年整体差距最大，最大、最小值之差达到0.26，2017年和2021年的差值均是0.17。
The overall gap in the development index of ruralcollective economic organizations in various districts and counties of Chongqing is manifested over time as“first expanding, then narrowing”characteristics(see Figure7-2). The development index of each district and county is between0.4—0.7, with the overall gap being the largest in 2019, where the difference between the maximum and minimum values reached0.26, and the differences in 2017 and 2021 were both0.17.

图7-2 重庆市农村集体经济组织发展平均水平（2017—2021年）
Figure7-2 Average Level of Development of Rural Collective Economic Organizations in Chongqing (2017—2021 Year)

整体来看，2017年以来重庆市推进实施农村产权制度改革，逐步拓展“三变”改革试点，加速了农村集体经济组织的发展壮大。比较2017年和2021年的发展结果，重庆市在不扩大差距范围的情况下，实现了农村集体经济组织发展水平的整体提升。
Overall,since 2017Chongqing has been promoting and implementing rural property rights system reform, gradually expanding“the three transformations”reform pilot, accelerating the development and growth of rural collective economic organizations. Comparing2017and2021results, Chongqing has achieved an overall improvement in the development level of rural collective economic organizations without widening the gap.

（2）重庆市农村集体经济组织发展区域水平
(2)Chongqing CityAgriculturalRural Collective Economic Organization Development Regional Level

①区域发展指数差距
①Regional Development Index Disparity

三峡库区农村集体经济组织发展指数低于非库区。如图7-3所示，2017—2021年间，三峡库区农村集体经济组织发展指数持续低于非库区，且持续低于全市平均水平，2018年差距最大，达到0.045，之后差距逐渐缩小至2021年的0.032。但是，三峡库区农村集体经济组织发展指数的年均增长率有4.60%，高于全市平均水平（4.57%）和非库区（4.54%）。所以，可以认为三峡库区正以微弱的增速优势缓慢地实施着农村集体经济组织发展赶超战略，这个过程中不断提升了自身发展水平，但任重道远。
The Three Gorges Reservoir Area rural collective economic organization development index is lower than that of non-reservoir areas. As shown in Figure 7-3, from 2017 to 2021, the development index of rural collective economic organizations in the Three Gorges Reservoir Area has consistently been lower than that of non-reservoir areas, and has remained below the citywide average level. In 2018, the gap was the largest, reaching 0.045, and then the gap gradually narrowed to 0.032 in 2021. However, the annual average growth rate of the development index of rural collective economic organizations in the Three Gorges Reservoir Area is 4.60%, higher than the citywide average ( 4.57%) and that of non-reservoir areas ( 4.54%). Therefore, it can be said that the Three Gorges Reservoir Area is slowly implementing a rural collective economic organization development catch-up strategy with a slight growth advantage, continuously improving its own development level in the process, but there is still a long way to go.

图7-3 不同区域农村集体经济组织发展水平（2017—2021年）
Figure7-3 Development levels of rural collective economic organizations in different regions (2017—2021years)

②发展指数的区域差异及结构分解
②Regional differences and structural decomposition of the development index

为了进一步深入分析重庆市农村集体经济组织发展的区域差异，本文借鉴聂长飞、简新华（2020）^[10]的做法，运用泰尔指数将重庆市农村集体经济组织发展指数的总体差异分解为组内差异和组间差异，并计算不同差异对总体差异的贡献程度。差异计算公式如下：
In order to further analyze the regional differences in the development of rural collective economic organizations in Chongqing, this article draws on the approach of Nie Changfei and Jian Xinhua (2020）^[10] by using the Theil index to decompose the overall difference in the development index of rural collective economic organizations in Chongqing into intra-group differences and inter-group differences, and calculates the contribution of different differences to the overall difference. The formula for calculating the differences is as follows:

表示重庆市农村集体经济组织发展指数的泰尔指数，取值，有
indicates the development index of rural collective economic organizations in ChongqingTailindex, with a value of , there is

表示区县，表示区县总数，表示区县农村集体经济组织发展指数，$$表示重庆市农村集体经济组织发展指数均值。
$$ i mmersive-translate-walked="862c287d-8045-4fec-a757-47944d2ccbd1" style="font-family: "Times New Roman";">represents districts and counties, represents the total number of districts and counties, representsidistricts and countiesrural collective economicorganization development index,representsChongqingrural collective economic organization development index average.

表示不同区域农村集体经济组织发展指数的泰尔指数，有
indicates different regionsthe development index of rural collective economic organizationsThiel index, there are

表示区域内区县总数，三峡库区、非库区分别为21、17。
Indicates the total number of districts within the area for the Three Gorges Reservoir area and non-reservoir area are21 and 17.

重庆市农村集体经济组织发展指数的泰尔指数可进一步分解为区域内差异泰尔指数和区域间差异泰尔指数，有
Chongqing Cityrural collective economic organization development indexcan be further decomposed into the within-region disparity Theil index and the between-region disparity Theil index , there is

进一步定义为区域内和区域间差异对总体差异的贡献率；定义为区域内差异中各区域对总体差异的贡献率，其中分别表示区域和全市各区县农村集体经济组织发展指数之和。
Further definition as the contribution rate of intra-regional and inter-regional differences to the overall difference; definition as the contribution rate of each region to the overall difference within intra-regional differences, where represents the regions and all districts and counties in the citythe sum of the development index of rural collective economic organizations.

按照上述公式，可计算得到重庆市农村集体经济组织发展差异的分解情况，如表7-7所示。
According to the above formula, it can be calculated thatChongqingruralcollective economic organization development differences decomposition situation, as shown in table7-7.

重庆市农村集体经济组织发展指数总体差异“先扩大、再缩小”。2019年总体差异最大，泰尔指数为0.0057，这与前文的分析结论一致。2021年发展指数总体差异在5年中最小，泰尔指数为0.0019，明显低于2019年的0.0036，说明重庆各区县集体经济组织在发展的同时，差异在缩小，凸显共同进步的良性发展态势。
ChongqingAgriculturalRural Collective Economic Organization Development Index Overall Differences“First Expand, Then Contract”.In2019the overall difference was the largest, with a Theil index of0.0057, which is consistent with the previous analysis conclusions.In2021the overall difference in the development index was the smallest in5years, with a Theil index of0.0019, significantly lower than2019's0.0036, indicating that while the collective economic organizations in various districts and counties of Chongqing are developing, the differences are narrowing, highlighting a positive development trend of common progress.

表7-6 重庆市农村集体经济组织发展指数差异分解
Table7-6 ChongqingRuralCollective Economic Organization Development Index Difference Decomposition

注：括号内容为各部分差异对总体差异的贡献率。
Note: The content in parentheses indicates the contribution rate of each part to the overall difference.

重庆市农村集体经济组织区域内差异是总体差异的主要来源。从对总体差异的贡献率来看，区域间差异的贡献率从未超过20%，区域内差异的贡献率在80%以上，甚至在差异最大年份2019年超过了90%。区域内差异和区域间差异均遵循“先扩大、再缩小”的规律，其中区域内差异与总体差异的变化规律完全一致，在2019年差异最大，区域间差异则相对稳定。
Chongqing Cityruralcollective economic organization regional differences are the main source of overall differences.In terms of the contribution rate to the overall difference, the contribution rate of inter-regional differences has never exceeded20%, while the contribution rate of intra-regional differences is above80%, and even in the year with the largest difference2019it exceeded90%. Both intra-regional and inter-regional differences follow the pattern of“first expanding, then contracting”, where the variation pattern of intra-regional differences is completely consistent with that of overall differences, peaking in2019, while inter-regional differences remain relatively stable.

三峡库区农村集体经济组织发展的内部差异明显超过非库区。从结果导向来看，较2017年，2021年三峡库区农村集体经济组织发展差异维持稳定，泰尔指数均是0.0023；非库区的发展差异则明显缩小，泰尔指数从0.0044降低到0.0007。从对总体差异的贡献率来看，三峡库区的贡献率由33.85%提升到65.37%，非库区差异的贡献率由55.78%降低到16.1%。
The Three Gorges Reservoir Area rural collective economic organizations show internal differences that are significantly greater than those in non-reservoir areas. In terms of results orientation, compared to 2017, in 2021, the development differences of rural collective economic organizations in the Three Gorges Reservoir Area remain stable, with a Theil index of 0.0023; the development differences in non-reservoir areas have significantly narrowed, with the Theil index decreasing from 0.0044 to 0.0007. In terms of the contribution rate to overall differences, the contribution rate of the Three Gorges Reservoir Area increased from 33.85% to 65.37%, while the contribution rate of differences in non-reservoir areas decreased from 55.78% to 16.1%.

可以初步判断，重庆市农村集体经济组织发展存在较大的区域内部差异，三峡库区与非库区发展不平衡，尤其三峡库区内部发展极不平衡。
It can be preliminarily judged that there are significant internal regional differences in the development of rural collective economic organizations in Chongqing, with an imbalance between the Three Gorges Reservoir area and non-reservoir areas, especiallywithin the Three Gorges Reservoir areawhere development is extremely unbalanced。

（3）重庆市农村集体经济组织发展区县水平
(3)Chongqing CityAgriculturalDevelopment Level of Rural Collective Economic Organizations

①农村集体经济组织发展指数排名
①Ranking of Rural Collective Economic Organization Development Index

根据2017—2021年重庆市38个涉农区县农村集体经济组织发展指数排名来看（如表7-7所示），区县发展指数排名波动很大，排名的提升或降低与农村集体经济组织发展基础和发展速度两个因素相关，尤其是发展速度。三峡库区区县排名普遍靠后。
According tothe 2017—2021Chongqing38rural collective economic organization development index ranking of agricultural-related districts and counties (as shown in table7-7),the development index rankings of districts and counties fluctuate greatly, with the rise or fall in rankings related to two factors: the development foundation and development speed of rural collective economic organizations, especially the development speed. The rankings of districts and counties in the Three Gorges Reservoir area are generally lower.

相较于2017年，2021年各区县排名提升的区县有19个、不变的1个、降低的18个。通过分析排名提升幅度较大的区县可以发现，他们的初始排名不高，但是年均增长率都排在前列，在较短的年限内实现从中、下排位到中、上排位。比如，排名提升福幅度最大的城口县，从第34位提升到第2位，同时，城口县的年均增速达到9.67%，在各个区县中排第一；渝北区排名从35位提升至第18位，年均增速达7.88%；大足区从35位提升至17位，年均增速达到7.45%；丰都县从第23位提升至第5位，年均增速达到7.47%。通过分析排名降低幅度较大的区县可以发现，他们的年均增速大都比较低，比如奉节县从第16位降低至第32位，年均增速2.76%；秀山县从第7位降低至第22位，年均增速1.99%；永川区从第3位降低至第16位，年均增速1.67%。需要说明的是，九龙坡区降低了30位，年均增长率为负，主要是由于2019年重庆市发展西部科学城，导致九龙坡区管辖统计范围调整所致。
Compared to2017year,2021year, the districts and counties that improved their rankings are19of them, unchanged1of them, and decreased18of them.By analyzing the districts and counties with significant ranking improvements, it can be found that their initial rankings were not high, but their average annual growth rates ranked among the top, achieving a transition from mid to lower rankings to mid to upper rankings in a short period. For example, the district with the largest ranking improvement, Chengkou County, rose from34to2, while Chengkou County's average annual growth rate reached9.67%, ranking first among all districts and counties; Yubei District's ranking improved from35to18, with an average annual growth rate of7.88%; Dazhu District rose from35to17, with an average annual growth rate of7.45%; Fengdu County improved from23to5, with an average annual growth rate of7.47%. By analyzing the districts and counties with significant ranking declines, it can be found that their average annual growth rates are mostly relatively low, for example, Fengjie County dropped from16to32, with an average annual growth rate of2.76%; Xiushan County dropped from7th to22nd place, with an average annual growth rate of1.99%; Yongchuan District dropped from3rd to16th place, with an average annual growth rate of1.67%. It should be noted that Jiulongpo District dropped30places, with a negative average growth rate, mainly due to the2019development of the Western Science City in Chongqing, which led to adjustments in the statistical scope under the jurisdiction of Jiulongpo District.It seems there is no source text provided for translation. Please provide the text you would like to have translated

2021年，排名前5位的区县依次是万州区、城口县、璧山区、云阳县、丰都县，排在后5位依次的是南岸区、开州区、沙坪坝区、江北区、九龙坡区。排名在19—38位的区县中，74%的区县来自三峡库区。
2021 year, the top 5 districts are Wanzhou District, Chengkou County, Bishan District, Yunyang County, Fengdu County, while the bottom 5 districts are Nanan District, Kaizhou District, Shapingba District, Jiangbei District, Jiulongpo District. Among the districts ranked 19— 38, 74% of the districts come from the Three Gorges Reservoir area.

表7-7 重庆市农村集体经济组织发展指数排名一览表
Table7-7 Chongqing CityAgriculturalRural Collective Economic Organization Development Index Ranking Overview

注：排名增减指2021年排名减去2017年排名。
Note: The increase or decrease in ranking refers tothe year 2021minusthe year 2017ranking.

②农村集体经济组织发展指数分类
②Classification of Rural Collective Economic Organization Development Index

本文运用四分位分类法，将重庆市农村集体经济组织发展指数分为低质量水平（y<0.506）、中质量水平（0.506≤y<0.576）、中高质量水平（0.576≤y<0.612）、和高质量水平（y≥0.612），展现重庆市农村集体经济组织发展水平的时空变化特征，如表9所示。经过5年的发展，重庆市农村集体经济组织发展指数实现了由中、低水平向中高、高水平转移，尤其是实现了低水平清零。
The article uses the quartile classification method to divide the development index of rural collective economic organizations in Chongqing into low-quality level (y<0.506), medium-quality level (0.506≤y<0.576), medium-high quality level (0.576≤y<0.612), and high-quality level (y≥0.612), demonstrating the temporal and spatial variation characteristics of the development level of rural collective economic organizations in Chongqing, as shown in Table9.After5years of development, the development index of rural collective economic organizations in Chongqing has shifted from medium and low levels tomedium-high and high levels, especially achieving a complete elimination of low levels.

具体来看，2017年86.8%的区县聚集在低质量和中质量水平，高质量水平区县数量为0；2019年57.9%的区县聚集在低质量和中质量水平，低质量水平区县仅剩8个，高质量水平区县增加到4个；2021年只有18.4%的区县聚集在中质量水平，低质量水平区县为0，高质量水平区县增加到19个，中高质量和高质量水平占比达到81.6%。除了九龙坡区、奉节县保持在中质量水平，其他区县不同程度实现了跨水平转移，比如，巫山县、城口县从低质量水平跨越到高质量水平，巴南区、江津区从低质量水平跨越到中高质量水平，南岸区、开州区从低质量水平跨越到中质量水平，长寿区、合川区从中质量水平跨越到中高质量水平。
Specifically, in 2017, 86.8% of districts and counties were concentrated at low and medium quality levels, with the number of high-quality districts and counties being 0; in 2019, 57.9% of districts and counties were concentrated at low and medium quality levels, with only 8 low-quality districts remaining, and the number of high-quality districts increased to 4; in 2021, only 18.4% of districts and counties were concentrated at medium quality levels, with 0 low-quality districts, and the number of high-quality districts increased to 19; the proportion of medium-high and high-quality levels reached 81.6%. Aside from Jiulongpo District and Fengjie County, which remained at medium quality levels, other districts and counties achieved varying degrees of cross-level transfer. For example, Wushan County and Chengkou County transitioned from low-quality to high-quality levels, while Banan District and Jiangjin District moved from low-quality to medium-high quality levels, and Nanan District and Kaizhou District transitioned from low-quality to medium quality levels, while Changshou District and Hechuan District moved from medium quality to medium-high quality levels.

2021年，三峡库区21个区县均衡的分布在中、中高、高质量水平，每个水平聚集7个区县。
2021year, the Three Gorges Reservoir area21districts and counties are evenly distributed at medium, medium-high, and high quality levels, with each level aggregating7districts and counties.

表7-8 重庆市农村集体经济组织发展水平分类（2017—2021年）
Table7-8 ChongqingRuralCollective Economic Organization Development Level Classification (2017—2021Year)