﻿ 基于双参照点的双边匹配决策方法
 控制与决策  2019, Vol. 34 Issue (6): 1286-1292 0

### 引用本文 [复制中英文]

[复制中文]
LUO Xiao, LI Wei-min, WANG Xuan-zi. Decision method for two-sided matching based on double reference points[J]. Control and Decision, 2019, 34(6): 1286-1292. DOI: 10.13195/j.kzyjc.2017.1629.
[复制英文]

### 文章历史

1. 空军工程大学 研究生院，西安 710051;
2. 西安交通大学 管理学院，西安 710049

Decision method for two-sided matching based on double reference points
LUO Xiao 1, LI Wei-min 1, WANG Xuan-zi 2
1. Graduate School, Air Force Engineering University, Xi'an 710051, China;
2. School of Management, Xi'an Jiaotong University, Xi'an 710049, China
Abstract: To solve the two-sided matching problem with bounded rationality, we propose a decision-making method based on double reference points. Firstly, according to the theory of prospect and social comparison, we use the critical value given by each agent to set up personal reference point and social reference point, calculate the gain and loss of the ordinal value relative to the double reference points, and get the comprehensive benefit of each agent. Then, considering the different risk attitude of each agent to gain and loss based on the prospect theory, the comprehensive benefit of the agent is further transformed into the perceived value that depicts the satisfaction of the agent. On this basis, an optimization model aiming at maximizing the satisfaction of two-side agents is established, and the optimal matching solution can be obtained by solving this model. Finally, an example is given to illustrate the application of the method. Compared with the existing methods, the proposed method takes into account the influence of the individual dimension and the social dimension on the psychological behavior of the agent, and can reflect the actual perceived value and the bounded rationality characteristics of the agent more comprehensively and flexibly.
Keywords: two-side matching    bounded rationality    psychological behavior    social comparison theory    prospect theory    reference point
0 引言

1 问题描述

T = [tij]m× nF = [fij]m× n分别为U方主体和V方主体给出的关于对方主体的完全序值信息.其中tij(tijN)代表uivj排在第tij位; fij(fijM)代表vjui排在第fij位.设pi (piN)和qj(qjM)分别为主体ui和主体vj提供的临界值, 临界值用于反映uivj在个人维度上的心理行为.设主体uivj的比较对象集合分别为Ui = {uk(l)}, kM, kiVj = {vh(l')}, hN, hj.其中: uk(l)表示主体ui的第l个比较对象uk, l = 1, 2, ..., li, li是主体ui的比较对象数量总和; vh(l')表示主体vj的第l'个比较对象vh, l' = 1, 2, ..., lj, lj是主体vj的比较对象数量总和; 比较对象用于反映主体uivj在社会维度上的心理行为.例如, 设U方主体集合U = {u1, u2, u3, u4}, V方主体集合V = {v1, v2, v3, v4, v5}, U方主体u1的比较对象集合可以为U1 = {u2, u4}, V方主体v3的比较对象可以为V3 = {v2, v4, v5}.

1) δ(ui)∈ V;

2) δ(vj)∈ Uvj;

3) 若δ(ui) = vj, 则δ(vj) = ui.

 图 1 考虑主体有限理性的双边匹配问题

2 基于双参照点的双边匹配方法

2.1 感知价值的计算

 (1)
 (2)
 (3)

 (4)
 (5)
 (6)

 (7)

 (8)

2.2 优化模型的构建

 (9)

2.3 优化模型的求解

 (10)

Stpe1:通过式(1) ~ (3)和(4) ~ (6), 将匹配双方偏好序值分别转化为综合益损值, 进而得到匹配双方的综合益损矩阵.

Step 2:通过式(7)和(8), 分别将匹配双方综合益损值转化成主体的感知价值, 进而得到匹配双方的感知价值矩阵.

Step 3:通过模型(9), 建立双目标规划模型, 利用线性加权法将双目标规划模型转化为单目标规划模型(10).

Step 4:求解优化模型(10), 获得匹配结果.

3 算例分析

8家企业则根据相关课题可能产生的经济效益、回报周期、社会影响和高校排名等因素给出关于高校课题组的序值矩阵

Step 1:基于序值矩阵T = [tij]6×8F = [fij]6×8, 临界值piqj, 通过式(1) ~ (3)和式(4) ~ (6), 将匹配双方的偏好序值转化为综合益损值, 其中γ1 = 0.5, γ2 = 0.5, 进而得到高校和企业双方的综合益损矩阵如下:

Step 2:文献[23]研究表明, 风险态度系数为0.88、损失规避系数为2.25时比较符合大多数决策者的实际决策行为偏好.因此, 本文设α1, α2, β1, β2的取值为0.88, λ1λ2的取值为2.25.依据式(7)和(8), 分别将高校和企业双方主体的综合益损值转化成感知价值, 进而得到感知价值矩阵如下:

Step 3:通过模型(9), 建立双目标规划模型.不失一般性, 设w1 = w2 = 0.5, 利用线性加权法将双目标规划模型进一步转化为单目标规划模型(10).

Step 4:通过Excel和Lingo11.0软件编程求解模型(10), 获得最“优”匹配结果为δ* = δE*δO*, 其中δE* = {(u1, v5), (u2, v4), (u3, v6), (u4, v2), (u5, v1), (u6, v7)}, δO* = {(v1, v1), (v3, v3)}.即u1v5匹配, u2v4匹配, u3v6匹配, u4v2匹配, u5v1匹配, u6v7匹配, v1v3未获得匹配.

4 结论

 [1] Gale D, Shapley L. College admissions and the stability of marriage[J]. American Mathematical Monthly, 1962, 69(1): 9-15. DOI:10.1080/00029890.1962.11989827 [2] Roth A E. Common and conflicting interests in two-sided matching markets[J]. European Economic Review, 1985, 27(1): 75-96. DOI:10.1016/0014-2921(85)90007-8 [3] Roth A E. New physicians: A natural experiment in market organization[J]. Science, 1990, 250(4987): 1524-1528. DOI:10.1126/science.2274783 [4] Roth A E. On the allocation of residents to rural hospital: A general property of two-sided matching markets[J]. Econometrica, 1986, 54(2): 425-427. DOI:10.2307/1913160 [5] Roth A E. A natural experiment in the organization of entry-level labor markets: Regional markets for new physicians and surgeons in the united kingdom[J]. American Economic Review, 1991, 81(3): 415-440. [6] Roth A E. The economist as engineer: Game theory, experimentation, and computation as tools for design economics[J]. Econometrica, 2002, 70(4): 1341-1378. DOI:10.1111/ecta.2002.70.issue-4 [7] Deng K H, Chiu H N, Yeh R H, et al. A fuzzy multi-criteria decision making approach for solving a bi-objective personnel assignment problem[J]. Computers & Industrial Engineering, 2009, 56(1): 1-10. [8] 梁海明, 姜艳萍. 二手房组合交易匹配决策方法[J]. 系统工程理论与实践, 2015, 35(2): 358-367. (Liang H M, Jiang Y P. Decision-making method on second-hand house combination matching[J]. Systems Engineering——Theory & Practice, 2015, 35(2): 358-367.) [9] Wong W K, Zeng X H, Au W M R, et al. A fashion mix-and-match expert system for fashion retailers using fuzzy screening approach[J]. Expert Systems With Applications, 2009, 36(2): 1750-1764. DOI:10.1016/j.eswa.2007.12.047 [10] Liu X, Ma H. A two-sided matching decision model based on uncertain preference sequences[J]. Mathematical Problems in Engineering, 2015, 2015(1): 1-10. [11] Echenique F, Pereyra J S. Strategic complementarities and unraveling in matching markets[J]. Theoretical Economics, 2016, 11(1): 1-39. [12] Erdil A, Ergin H. Two-sided matching with indifferences[J]. J of Economic Theory, 2017, 171(1): 268-292. [13] Fan Z P, Li M Y, Zhang X. Satisfied two-sided matching: A method considering elation and disappointment of agent[J]. Soft Computing, 2017, 22(21): 7227-7241. [14] 刘勇, 熊晓旋, 全冰婷. 基于灰色关联分析的双边公平匹配决策模型及应用[J]. 管理学报, 2017, 14(1): 86-92. (Liu Y, Xiong X X, Quan B T. Two-sided fair matching decision-making method and application based on grey incidence analysis[J]. Chinese J of Management, 2017, 14(1): 86-92.) [15] 乐琦. 考虑主体心理行为的双边匹配决策方法[J]. 系统工程与电子技术, 2013, 35(1): 120-125. (Yue Q. Decision method for two-sided matching considering agents' psychological behavior[J]. Systems Engineering and Electronics, 2013, 35(1): 120-125. DOI:10.3969/j.issn.1001-506X.2013.01.20) [16] Yue Q, Zhang L, Wang Z X. Matching decision considering agents' psychological behavior with incomplete ordinal number information[J]. Fuzzy Systems and Mathematics, 2014, 28(4): 90-99. [17] 陈希, 韩菁, 张晓. 考虑心理期望与感知的多属性匹配决策方法[J]. 控制与决策, 2014, 29(11): 2027-2033. (Chen X, Nan J, Zhang X. Method for multiple attribute matching decision making considering matching body's psychological aspiration and perception[J]. Control and Decision, 2014, 29(11): 2027-2033.) [18] 李铭洋, 樊治平. 考虑双方主体心理行为的稳定双边匹配方法[J]. 系统工程理论与实践, 2014, 34(10): 2591-2599. (Li M Y, Fan Z P. Method for stable two-sided matching considering psychological behavior of agents on both sides[J]. Systems Engineering——Theory & Practice, 2014, 34(10): 2591-2599. DOI:10.12011/1000-6788(2014)10-2591) [19] 吴凤平, 朱玮, 程铁军. 互联网金融背景下风险投资双边匹配选择问题研究[J]. 科技进步与对策, 2016, 33(4): 25-30. (Wu F P, Zhu W, Cheng T J. Study on the venture capital two-sided matching decision-making in internet finance[J]. Science & Technology Progress and Policy, 2016, 33(4): 25-30.) [20] 赵道致, 李锐. 考虑主体心理预期的云制造资源双边匹配机制[J]. 控制与决策, 2017, 32(5): 871-878. (Zhao D Z, Li R. Two-sided matching mechanism with agents' expectation for cloud manufacturing resource[J]. Control and Decision, 2017, 32(5): 871-878.) [21] 谢晓非, 陆静怡. 风险决策中的双参照点效应[J]. 心理科学进展, 2014, 22(4): 571-579. (Xie X F, Lu J Y. Double reference points in risky decision making[J]. Advances in Psychological Science, 2014, 22(4): 571-579.) [22] Wang L, Wang Y M, Martínez L. A group decision method based on prospect theory for emergency situations[J]. Information Sciences, 2017, 41(8): 119-135. [23] Kahneman D, Tversky A. Prospect theory: An analysis of decision under risk[J]. Econometrica, 1979, 47(2): 263-291. DOI:10.2307/1914185 [24] He X D, Zhou X Y. Portfolio choice under cumulative prospect theory: An analytical treatment[J]. Management Science, 2011, 57(2): 315-331. DOI:10.1287/mnsc.1100.1269