直覺模糊環(huán)境下考慮匹配意愿的雙邊匹配決策

樂 琦

(江西財經(jīng)大學(xué)信息管理學(xué)院，江西 南昌 330013)

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直覺模糊環(huán)境下考慮匹配意愿的雙邊匹配決策

樂 琦

(江西財經(jīng)大學(xué)信息管理學(xué)院，江西 南昌 330013)

本文研究了基于直覺模糊集和匹配意愿的雙邊匹配問題。給出了直覺模糊集和雙邊匹配的概念；描述了基于直覺模糊集和匹配意愿的雙邊匹配問題。為求解該問題，首先將直覺模糊集矩陣轉(zhuǎn)化為分值矩陣；基于分值矩陣和匹配矩陣，以一對一雙邊匹配為約束，建立了考慮分值的雙邊匹配模型；依據(jù)分值矩陣，計算分值差值和倒差；進(jìn)一步地，運用倒差最大化方法計算匹配意愿矩陣；依據(jù)匹配意愿矩陣，將考慮分值的雙邊匹配模型轉(zhuǎn)化為考慮分值和匹配意愿的雙邊匹配模型；通過求解該模型獲得“最優(yōu)”雙邊匹配。最后，技術(shù)供需匹配算例說明了所提雙邊匹配決策的可行性和有效性。

雙邊匹配；直覺模糊集；匹配意愿；倒差最大化；雙邊匹配模型

1 引言

現(xiàn)實生活中存在大量的雙邊匹配問題。例如穩(wěn)定婚姻問題[1]、廣告投放中的匹配問題[2]、大學(xué)招生錄取問題[3]、服務(wù)供應(yīng)商和顧客的匹配問題[4]、人員指派問題[5]等。Gale和Shapley[6]最早針對穩(wěn)定婚姻匹配進(jìn)行了研究，提出了著名的Gale-Shapley算法。隨后，國內(nèi)外諸多學(xué)者從各種不同的視角對各種雙邊匹配問題進(jìn)行了深入研究[7-12]。由于“更優(yōu)”的雙邊匹配方案會提升雙方主體的滿意程度，提升現(xiàn)實匹配決策的效率，因此針對雙邊匹配理論與方法的研究具有重要的理論意義和實際價值。

目前，關(guān)于雙方主體偏好為序值(或稱為偏好序等)、序關(guān)系、語言、區(qū)間數(shù)等信息的雙邊匹配或多屬性雙邊匹配理論和方法已較為完善。例如，樊治平和樂琦[13]從考慮雙邊主體的最高可接受偏好序的視角給出了一種解決基于完全偏好序信息的雙邊匹配問題的嚴(yán)格雙邊匹配方法。樂琦和樊治平[14]引入了完全雙邊匹配的概念，探討了完全雙邊匹配的存在性理論，進(jìn)而從完全雙邊匹配的視角給出了一種解決基于不完全序值信息的雙邊匹配問題的方法。陳圣群等[15]針對具有語言值、精確值和區(qū)間值置信度混合信息的多屬性匹配決策問題，基于證據(jù)理論提出了一種證據(jù)融合決策方法。梁海明等[16]針對具有強(qiáng)偏好序、弱偏好序、無差異偏好序和未知偏好序信息的多滿意穩(wěn)定導(dǎo)向雙邊匹配決策問題，提出了一種新的決策分析方法。樂琦[17]針對雙方主體給出序關(guān)系信息的雙邊匹配問題，從Borda分值轉(zhuǎn)換的視角提出了一種決策方法。

2 預(yù)備知識

定義1 設(shè)E是一個非空集合，則稱I={|x∈E}為E上的直覺模糊集[18]，其中μI(x)和γI(x)分別為E中元素x屬于I的隸屬度μI:E→[0, 1]和非隸屬度γI:E→[0, 1]，且滿足0≤μI(x)+γI(x)≤1,?x∈E。

2.1直覺模糊集

定義1 設(shè)E是一個非空集合，則稱I={|x∈E}為E上的直覺模糊集[18]，其中μI(x)和γI(x)分別為E中元素x屬于I的隸屬度μI:E→[0, 1]和非隸屬度γI:E→[0, 1]，且滿足0≤μI(x)+γI(x)≤1, ?x∈E。

注1 此外，稱πI(x)=1-μI(x)-γI(x)≤1為E中元素x屬于I的猶豫度。顯然0≤πI(x)≤1, ?x∈E。特別地，若πI(x)=0，則I退化為傳統(tǒng)的模糊集。

注2 為方便起見，直覺模糊集I={|x∈E}簡記為I=<μI(x),γI(x)>。

注3 針對直覺模糊數(shù)I=<μI(x),γI(x)>，依據(jù)分值函數(shù)[20]，可計算I=<μI(x),γI(x)>的分值為sI=μI(x)-γI(x)。顯然，-1≤sI≤1，分值sI隨著差值μI(x)-γI(x)的增大而增大。因此，分值sI可作為衡量直覺模糊數(shù)I=<μI(x),γI(x)>大小的一個重要指標(biāo)[21]。

2.2雙邊匹配

定義3 設(shè)Υ為雙邊匹配，則Υ=ΥTwo∪ΥOne[25-26]，其中ΥTwo為匹配對集合，ΥOne為單身對集合。

3 基于直覺模糊集和匹配意愿的雙邊匹配決策

3.1基于直覺模糊集和匹配意愿的雙邊匹配問題描述

3.2考慮分值的雙邊匹配模型

(1)

(2)

3.3考慮分值和匹配意愿的雙邊匹配模型

模型(M-1)為多目標(biāo)優(yōu)化模型，如果進(jìn)一步考慮到雙邊匹配決策的公平性(即每個主體在匹配過程中所處地位相同)，則可以使用簡單加權(quán)方法(此時每個主體的優(yōu)先權(quán)重視為相等)將其轉(zhuǎn)化為如下單目標(biāo)雙邊匹配模型(M-2)：

(3)

(4)

(5)

(6)

于是，求解匹配意愿矩陣Ω=[ωij]p×q等價于求解如下單目標(biāo)優(yōu)化模型(M-4)：

(7)

(8)

將式(7)代入式(8)，可得：

(9)

將式(9)代入式(7)，可得：

(10)

(11)

3.4基于直覺模糊集和匹配意愿的雙邊匹配決策的步驟

基于上述分析，基于直覺模糊集和匹配意愿的雙邊匹配決策的步驟給出如下：

步驟5：求解雙邊匹配模型(M-5)，得到“最優(yōu)”雙邊匹配。

4 技術(shù)供需匹配算例

下面說明使用所提的基于直覺模糊集和匹配意愿的雙邊匹配決策的計算過程。

模型(M-1)中，P={1,…,4}，Q={1,…,6}。

表1 直覺模糊集矩陣

表2 直覺模糊集矩陣

表3 分值矩陣

表4 分值矩陣

表5 匹配意愿矩陣

表6 系數(shù)矩陣

表7 匹配矩陣

注9 需要指出的是，在文獻(xiàn)[24]中，直覺模糊偏好關(guān)系是由每個主體針對對方主體集合進(jìn)行兩兩對比的得到的，是由一個m×m方陣和一個n×n方陣構(gòu)成，而本文的直覺模糊偏好形式是由兩個m×n方陣構(gòu)成。因此，本文與林楊和王應(yīng)明[24]的研究視角是不一樣的，用林楊和王應(yīng)明[24]的方法不能解決本文所考慮的問題。

5 結(jié)語

本文針對基于直覺模糊集和匹配意愿的雙邊匹配問題，給出了一種雙邊匹配決策途徑。先將直覺模糊集矩陣轉(zhuǎn)化為分值矩陣；基于分值矩陣和匹配矩陣，建立考慮分值的雙邊匹配模型；依據(jù)分值矩陣，通過運用倒差最大化方法將考慮分值的雙邊匹配模型轉(zhuǎn)化為考慮分值和匹配意愿的雙邊匹配模型；求解該模型獲得“最優(yōu)”雙邊匹配。本文的主要創(chuàng)新點在于：(1)將直覺模糊集理論應(yīng)用于雙邊匹配決策領(lǐng)域中，(2)從主體匹配意愿的視角研究雙邊匹配決策，其匹配意愿的計算采用倒差最大化方法。本文的研究成果發(fā)展并完善了直覺模糊集信息下雙邊匹配決策方法的研究。但本文初步探討了雙方主體偏好以直覺模糊數(shù)信息給出的情形，對于以其它直覺偏好信息形式給出的雙邊匹配問題還有待于進(jìn)一步研究和探索。

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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

江西省自然科學(xué)基金資助項目(20171BAA208003, 20161BAB201025，20151BAB201026)；國家自然科學(xué)基金資助項目(71261007)

樂琦(1983-)，男(漢族)，江西東鄉(xiāng)人，江西財經(jīng)大學(xué)信息管理學(xué)院，博士，副教授，研究方向：決策理論與方法，E-mail：yueqichina@126.com.

C 934

：A