直觉模糊环境下考虑匹配意愿的双边匹配决策
2017-07-01乐琦
乐 琦
(江西财经大学信息管理学院，江西 南昌 330013)
直觉模糊环境下考虑匹配意愿的双边匹配决策
乐 琦
(江西财经大学信息管理学院，江西 南昌 330013)
本文研究了基于直觉模糊集和匹配意愿的双边匹配问题。给出了直觉模糊集和双边匹配的概念；描述了基于直觉模糊集和匹配意愿的双边匹配问题。为求解该问题，首先将直觉模糊集矩阵转化为分值矩阵；基于分值矩阵和匹配矩阵，以一对一双边匹配为约束，建立了考虑分值的双边匹配模型；依据分值矩阵，计算分值差值和倒差；进一步地，运用倒差最大化方法计算匹配意愿矩阵；依据匹配意愿矩阵，将考虑分值的双边匹配模型转化为考虑分值和匹配意愿的双边匹配模型；通过求解该模型获得“最优”双边匹配。最后，技术供需匹配算例说明了所提双边匹配决策的可行性和有效性。
双边匹配；直觉模糊集；匹配意愿；倒差最大化；双边匹配模型
1 引言
现实生活中存在大量的双边匹配问题。例如稳定婚姻问题[1]、广告投放中的匹配问题[2]、大学招生录取问题[3]、服务供应商和顾客的匹配问题[4]、人员指派问题[5]等。Gale和Shapley[6]最早针对稳定婚姻匹配进行了研究，提出了著名的Gale-Shapley算法。随后，国内外诸多学者从各种不同的视角对各种双边匹配问题进行了深入研究[7-12]。由于“更优”的双边匹配方案会提升双方主体的满意程度，提升现实匹配决策的效率，因此针对双边匹配理论与方法的研究具有重要的理论意义和实际价值。
目前，关于双方主体偏好为序值(或称为偏好序等)、序关系、语言、区间数等信息的双边匹配或多属性双边匹配理论和方法已较为完善。例如，樊治平和乐琦[13]从考虑双边主体的最高可接受偏好序的视角给出了一种解决基于完全偏好序信息的双边匹配问题的严格双边匹配方法。乐琦和樊治平[14]引入了完全双边匹配的概念，探讨了完全双边匹配的存在性理论，进而从完全双边匹配的视角给出了一种解决基于不完全序值信息的双边匹配问题的方法。陈圣群等[15]针对具有语言值、精确值和区间值置信度混合信息的多属性匹配决策问题，基于证据理论提出了一种证据融合决策方法。梁海明等[16]针对具有强偏好序、弱偏好序、无差异偏好序和未知偏好序信息的多满意稳定导向双边匹配决策问题，提出了一种新的决策分析方法。乐琦[17]针对双方主体给出序关系信息的双边匹配问题，从Borda分值转换的视角提出了一种决策方法。
2 预备知识
定义1 设E是一个非空集合，则称I={
2.1直觉模糊集
定义1 设E是一个非空集合，则称I={
注1 此外，称πI(x)=1-μI(x)-γI(x)≤1为E中元素x属于I的犹豫度。显然0≤πI(x)≤1, ∀x∈E。特别地，若πI(x)=0，则I退化为传统的模糊集。
注2 为方便起见，直觉模糊集I={
注3 针对直觉模糊数I=<μI(x),γI(x)>，依据分值函数[20]，可计算I=<μI(x),γI(x)>的分值为sI=μI(x)-γI(x)。显然，-1≤sI≤1，分值sI随着差值μI(x)-γI(x)的增大而增大。因此，分值sI可作为衡量直觉模糊数I=<μI(x),γI(x)>大小的一个重要指标[21]。
2.2双边匹配
定义3 设Υ为双边匹配，则Υ=ΥTwo∪ΥOne[25-26]，其中ΥTwo为匹配对集合，ΥOne为单身对集合。
3 基于直觉模糊集和匹配意愿的双边匹配决策
3.1基于直觉模糊集和匹配意愿的双边匹配问题描述
3.2考虑分值的双边匹配模型
(1)
(2)
3.3考虑分值和匹配意愿的双边匹配模型
模型(M-1)为多目标优化模型，如果进一步考虑到双边匹配决策的公平性(即每个主体在匹配过程中所处地位相同)，则可以使用简单加权方法(此时每个主体的优先权重视为相等)将其转化为如下单目标双边匹配模型(M-2)：
(3)
(4)
(5)
(6)
于是，求解匹配意愿矩阵Ω=[ωij]p×q等价于求解如下单目标优化模型(M-4)：
(7)
(8)
将式(7)代入式(8)，可得：
(9)
将式(9)代入式(7)，可得：
(10)
(11)
3.4基于直觉模糊集和匹配意愿的双边匹配决策的步骤
基于上述分析，基于直觉模糊集和匹配意愿的双边匹配决策的步骤给出如下：
步骤5：求解双边匹配模型(M-5)，得到“最优”双边匹配。
4 技术供需匹配算例
下面说明使用所提的基于直觉模糊集和匹配意愿的双边匹配决策的计算过程。
模型(M-1)中，P={1,…,4}，Q={1,…,6}。
表1 直觉模糊集矩阵
表2 直觉模糊集矩阵
表3 分值矩阵
表4 分值矩阵
表5 匹配意愿矩阵
表6 系数矩阵
表7 匹配矩阵
注9 需要指出的是，在文献[24]中，直觉模糊偏好关系是由每个主体针对对方主体集合进行两两对比的得到的，是由一个m×m方阵和一个n×n方阵构成，而本文的直觉模糊偏好形式是由两个m×n方阵构成。因此，本文与林杨和王应明[24]的研究视角是不一样的，用林杨和王应明[24]的方法不能解决本文所考虑的问题。
5 结语
本文针对基于直觉模糊集和匹配意愿的双边匹配问题，给出了一种双边匹配决策途径。先将直觉模糊集矩阵转化为分值矩阵；基于分值矩阵和匹配矩阵，建立考虑分值的双边匹配模型；依据分值矩阵，通过运用倒差最大化方法将考虑分值的双边匹配模型转化为考虑分值和匹配意愿的双边匹配模型；求解该模型获得“最优”双边匹配。本文的主要创新点在于：(1)将直觉模糊集理论应用于双边匹配决策领域中，(2)从主体匹配意愿的视角研究双边匹配决策，其匹配意愿的计算采用倒差最大化方法。本文的研究成果发展并完善了直觉模糊集信息下双边匹配决策方法的研究。但本文初步探讨了双方主体偏好以直觉模糊数信息给出的情形，对于以其它直觉偏好信息形式给出的双边匹配问题还有待于进一步研究和探索。
[1] Ishida Y, Hashimoto S. Asymmetric characterization of diversity in symmetric stable marriage problems: An example of agent evacuation [J]. Procedia Computer Science, 2015, 60: 1472-1481.
[2] Byun J, Jang S. Effective destination advertising: Matching effect between advertising language and destination type [J]. Tourism Management, 2015, 50：31-40.
[3] Liu Qijun, Peng Yaping. Corruption in college admissions examinations in China [J]. International Journal of Educational Development, 2015, 41: 104-111.
[4] Xu Xinkui, Wang Chun, Zeng Yong, et al.Matching service providers and customers in two-sided dynamic markets[J].IFAC-PapersOnLine, 2015, 48(3): 2208-2213.
[5] Gharote M, Patil R, Lodha S, et al. Assignment of trainees to software project requirements: A stable matching based approach [J]. Computers & Industrial Engineering, 2015, 87(9): 228-237.
[6] Gale D, Shapley L S. College admissions and the stability of marriage [J]. American Mathematical Monthly, 1962(1): 9-15.
[7] Zhang Xi, Tan C L. Handwritten word image matching based on Heat Kernel Signature [J]. Pattern Recognition, 2015, 48(11): 3346-3356.
[8] Trégou⊇t T. Gender-based price discrimination in matching markets [J]. International Journal of Industrial Organization, 2015, 42(9): 34-45.
[9] Tewari G, Youll J, Maes P. Personalized location based brokering using an agent-based intermediary architecture [J]. Decision Support Systems, 2003, 34(2): 127-137.
[10] Yashiv E. Labor search and matching in macroeconomics [J]. European Economic Review, 2007, 51(8): 1859-1895.
[11] Ehlers L. Truncation strategies in matching markets [J]. Mathematics of Operations Research, 2008, 33(2): 327-335.
[12] 樊治平, 李铭洋, 乐琦. 考虑稳定匹配条件的双边满意匹配决策方法[J]. 中国管理科学, 2014, 22(4): 112-118.
[13] 樊治平, 乐琦. 基于完全偏好序信息的严格双边匹配方法[J]. 管理科学学报, 2014, 17(1): 21-34.
[14] 乐琦, 樊治平. 基于不完全序值信息的双边匹配决策方法[J]. 管理科学学报, 2015, 18(2): 23-35.
[15] 陈圣群, 王应明, 施海柳. 多属性匹配决策的等级置信度融合法[J]. 系统工程学报, 2015, 30(1): 25-33.
[16] 梁海明, 姜艳萍, 孔德财. 考虑偏好序的多满意稳定导向双边匹配决策方法[J]. 系统工程理论与实践, 2015, 35(6): 1535-1546.
[17] 乐琦. 基于序关系信息的双边匹配决策方法[J]. 系统工程学报, 2015, 30(5): 601-606.
[18] Atanassov K. Intuitionistic fuzzy sets [J].Fuzzy Sets and Systems, 1986, 20(1): 87-96.
[19] Zadeh L A. Fuzzy sets [J].Information and Control, 1965, 8(3): 338-356.
[20] Chen S M, Tan J M. Handling multicriteria fuzzy decision-making problems based on vague set theory [J]. Fuzzy Sets and Systems, 1994, 67(2): 163-172.
[21] 徐泽水.直觉模糊偏好信息下的多属性决策途径[J]. 系统工程理论与实践, 2007, 27(11), 62-71.
[22] 胡浩, 徐少华, 宋继冉, 等. 基于直觉模糊动态信息交互的多属性群决策模型[J]. 中南大学学报(自然科学版), 2015, 46(8)：2923-2929.
[23] 蔡久顺, 张执国, 师鹏, 等. 基于直觉模糊多属性群决策的风险排序方法[J]. 工程数学学报, 2015, 32(5): 650-658.
[24] 林杨, 王应明. 考虑直觉模糊偏好关系的双边稳定匹配及应用[J].控制与决策, 2015, 30(12): 2212-2218.
[25] Gale D. The two-sided matching problem：Origin, development and current issues [J]. International Game Theory Review, 2001, 3(2-3): 237-252.
[26] Echenique F. What matchings can be stable? The testable implications of matching theory [J]. Mathematics of Operations Research, 2008, 33(3): 757-768.
[27] 王应明. 运用离差最大化方法进行多指标决策与排序[J]. 系统工程与电子技术, 1998, 20(7): 24-26.
Two-sided Matching Decision Considering Matching Aspiration under the Intuitionistic Fuzzy Circumstance
YUEQi
(School of Information Management, Jiangxi University of Finance and Economics, Nanchang 330013, China)
The two-sided matching problem has always been concerned by the scholars in the fields of economic management and so on. Due to the complexity and fuzzy uncertainty of objective things, the preferences given by two-sided agents are in the format of intuitionistic fuzzy sets sometimes. The two-sided matching decision problem based on intuitionistic fuzzy sets and matching aspirations is an urgent need research new topic in psychology and decision science with rich actual backgrounds, and still has forward position and exploration. The theory of intuitionistic fuzzy set has been widely applied in the field of decision, but the application in the field of two-sided matching decision are relatively rare. Therefore, how to introduce the related theories of intuitionistic fuzzy set and matching aspiration into the two-sided matching decision problem and develop scientific and effective decision method have important theoretical significance and practical application value with respect to the research on two-sided matching decision. In this paper, the two-sided matching problem is investigated based on intuitionistic fuzzy sets and matching aspirations. The concepts of intuitionistic fuzzy set and two-sided matching are firstly introduced. Then, the two-sided matching problem based on intuitionistic fuzzy sets and matching aspirations is described. In order to solve this problem, the intuitionistic fuzzy set matrixes are transformed into score matrixes. Based on score matrixes and matching matrixes, a two-sided matching model considering scores under the constraint conditions of one-to-one two-sided matching is developed. Moreover, the score deviations and the score reciprocal-deviations are calculated based on score matrixes. Then the matching aspiration matrix can be calculated by using the maximum score reciprocal-deviation principle. The two-sided matching model considering scores is converted into a two-sided matching model considering scores and matching aspirations according to the matching aspiration matrix. The “optimal” two-sided matching can be obtained by solving the model. Lastly, the feasibility and effectiveness of the proposed two-sided matching decision is illustrated with an example of technology supply-demand matching. The research achievements of this paper develop and prefect the decision theories and methods for two-sided matching based on intuitionistic fuzzy sets and matching aspiration. But this paper discussed preliminarily this case that the preferences of two-sided agents are intuitionistic fuzzy sets. When the preferences of two-sided agents are in the format of interval-valued intuitionistic fuzzy sets, triangular intuitionistic fuzzy numbers, or trapezoidal intuitionistic fuzzy numbers in the two-sided matching problem, the above problem has yet to be further researched and explored.
two-sided matching; intuitionistic fuzzy set; matching aspiration; maximum reciprocal-deviation; two-sided matching model
1003-207(2017)06-0161-08
10.16381/j.cnki.issn1003-207x.2017.06.017
2015-12-15；
：2016-04-07
江西省自然科学基金资助项目(20171BAA208003, 20161BAB201025，20151BAB201026)；国家自然科学基金资助项目(71261007)
乐琦(1983-)，男(汉族)，江西东乡人，江西财经大学信息管理学院，博士，副教授，研究方向：决策理论与方法，E-mail：yueqichina@126.com.
C 934
：A
