基于方向距离函数和信息熵的集成交叉效率模型

宁阳雪; 潘昊君; 王国强

doi:10.12299/jsues.23-0022

基于方向距离函数和信息熵的集成交叉效率模型

doi: 10.12299/jsues.23-0022

上海工程技术大学数理与统计学院, 上海 201620

基金项目: 国家自然科学基金面上项目资助（11971302）；浦东新区科技发展基金产学研专项资金（人工智能）资助（PKX2020-R02）；上海市大学生创新创业训练计划资助（CS2221001）

详细信息

作者简介:
宁阳雪（1997−），女，硕士生，研究方向为数据包络分析。E-mail：ningyx2016@163.com

通讯作者:
王国强（1977−），男，教授，博士，研究方向为最优化理论与算法、高维数据统计推断、统计优化和数据挖掘。E-mail：guoq_wang@hotmail.com

中图分类号: O221
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出版历程
- 收稿日期: 2023-02-12
- 网络出版日期: 2024-11-14
- 刊出日期: 2024-09-30

Integrated cross-efficiency model based on directional distance function and information entropy

School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China

摘要

摘要: 针对传统交叉效率模型无法处理输入输出数据中同时含有负数的问题，提出一种基于方向距离函数和信息熵的集成交叉效率模型。首先，利用方向距离函数的思想对负数进行处理。其次，结合交叉效率模型实现决策单元的完全排序。接着，借助信息熵的变异系数获取一组用于交叉效率集结的公共权重，以避免传统模型的权重偏差，且保留评价过程中的决策信息。最后，通过一个数值案例验证了本研究模型的有效性和实用性，扩展了交叉效率模型的研究范围和应用场景。
- 数据包络分析 /
- 交叉效率 /
- 方向距离函数 /
- 信息熵
Abstract: Aiming at the problem that traditional cross-efficiency model cannot handle both input and output data containing negative numbers, an integrated cross-efficiency model based on directional distance function and information entropy was proposesed. First of all, the idea of direction distance function was used to deal with negative numbers. Secondly, the complete ranking of decision units was realized by combining cross efficiency. Then, with the help of the variation coefficient of information entropy, a set of public weights for cross-efficiency integration were obtained to avoid the weight deviation of the traditional model and retain the decision information in the evaluation process. Finally, the effectiveness and practicability of the proposed model were verified by a numerical example, and the research scope and application scenarios of the cross-efficiency model were extended.
- data envelopment analysis /
- cross-efficiency /
- directional distance function /
- information entropy

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表 1 基于方向距离函数的交叉效率矩阵B

Table 1. Cross efficiency matrix B based on directional distance function

评价$\mathrm{D}\mathrm{M}{\mathrm{U} }_{d }$	被评价$\mathrm{D}\mathrm{M}{\mathrm{U} }_{j }$
评价$\mathrm{D}\mathrm{M}{\mathrm{U} }_{d }$	1	2	···	n
$ 1 $	$ {\beta }_{11} $	$ {\beta }_{12} $	···	$ {\beta }_{1n} $
$ 2 $	$ {\beta }_{21} $	$ {\beta }_{22} $	···	$ {\beta }_{2n} $
$ \vdots $	$ \vdots $	$ \vdots $		$ \vdots $
$ n $	$ {\beta }_{n1} $	$ {\beta }_{n2} $	···	$ {\beta }_{nn} $

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表 2 算例数据

Table 2. Example data

$\mathrm{D}\mathrm{M}{\mathrm{U} }_{j }$	投入		产出
$\mathrm{D}\mathrm{M}{\mathrm{U} }_{j }$	$ {x}_{1} $	$ {x}_{2} $	$ {y}_{1} $	$ {y}_{2} $	$ {y}_{3} $
1	1.03	−0.05	0.56	−0.09	−0.44
2	1.75	−0.17	0.74	−0.24	−0.31
3	1.44	−0.56	1.37	−0.35	−0.21
4	10.80	−0.22	5.61	−0.98	−3.79
5	1.30	−0.07	0.49	−1.08	−0.34
6	1.98	−0.10	1.61	−0.44	−0.34
7	0.97	−0.17	0.82	−0.08	−0.43
8	9.82	−2.32	0.48	−1.42	−1.94
9	1.59	0.00	0.52	0.00	−0.37
10	5.96	−0.15	2.14	−0.52	−0.18
11	1.29	−0.11	0.57	0.00	−0.24
12	2.38	−0.25	0.57	−0.67	−0.43
13	10.30	−0.16	9.56	−0.58	0.00

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表 3 三种模型结果比较

Table 3. Results comparison of three models

$\mathrm{D}\mathrm{M}{\mathrm{U} }_{j }$	方向距离函数模型			基于方向距离函数的交叉效率模型			本研究模型
$\mathrm{D}\mathrm{M}{\mathrm{U} }_{j }$	$ \beta $	$ 1-\beta $	排名	$ \beta $	$ 1-\beta $	排名	$ \beta $	$ 1-\beta $	排名
1	0.0351	0.9649	8	0.1494	0.8506	6	0.1332	0.8668	6
2	0.0818	0.9182	10	0.1602	0.8398	7	0.1503	0.8497	7
3	0.0000	1.0000	1	0.0510	0.9490	2	0.0456	0.9544	1
4	0.2068	0.7932	13	0.4952	0.5048	13	0.4693	0.5307	13
5	0.0757	0.9243	9	0.2927	0.7073	12	0.2878	0.7122	12
6	0.0292	0.9708	7	0.1877	0.8123	8	0.1767	0.8233	8
7	0.0000	1.0000	1	0.1120	0.8880	4	0.0948	0.9052	4
8	0.0000	1.0000	1	0.2901	0.7099	11	0.2654	0.7346	11
9	0.0055	0.9945	6	0.1387	0.8613	5	0.1253	0.8747	5
10	0.1404	0.8596	11	0.2542	0.7458	9	0.2533	0.7467	9
11	0.0000	1.0000	1	0.0667	0.9333	3	0.0604	0.9396	3
12	0.1495	0.8505	12	0.2697	0.7303	10	0.2610	0.7390	10
13	0.0000	1.0000	1	0.0455	0.9545	1	0.0464	0.9536	2

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