基于對(duì)稱模糊數(shù)和最優(yōu)h值線性回歸模型的質(zhì)量功能展開研究

朱　倩,　陸秋君

(上海理工大學(xué) 理學(xué)院,上海　200093)

注:*表示兩者之間的關(guān)系顯著.

?

基于對(duì)稱模糊數(shù)和最優(yōu)h值線性回歸模型的質(zhì)量功能展開研究

朱倩,陸秋君

(上海理工大學(xué) 理學(xué)院,上海200093)

模糊線性回歸參數(shù)h決定模糊系數(shù)的可能性分布,影響模型的系統(tǒng)可靠性.分析了含有參數(shù)p的對(duì)稱模糊數(shù)的模型,得出質(zhì)量功能展開中具有最優(yōu)h值的函數(shù)關(guān)系.當(dāng)參數(shù)p給定時(shí),考慮系統(tǒng)模糊性和系統(tǒng)隸屬度兩個(gè)因素,依據(jù)模糊線性回歸模型系統(tǒng)可靠性最大的原則,得到最優(yōu)h值;并將該理論應(yīng)用于質(zhì)量功能展開的實(shí)例,選取有代表性的3個(gè)p值,分別對(duì)應(yīng)二次根型、三角型和拋物型對(duì)稱模糊數(shù)進(jìn)行討論,得到對(duì)應(yīng)的最優(yōu)h值.在最優(yōu)h值的前提下,分析對(duì)應(yīng)的系統(tǒng)模糊性和可靠性的變化規(guī)律,通過對(duì)比分析,得到系統(tǒng)可靠性最大的函數(shù)關(guān)系.

模糊線性回歸; 對(duì)稱模糊數(shù); 最優(yōu)h值; 質(zhì)量功能展開; 系統(tǒng)可靠性

質(zhì)量功能展開(QFD)起源于20世紀(jì)末的日本[1],它將客戶的需求通過不同階段的產(chǎn)品規(guī)劃轉(zhuǎn)變?yōu)楫a(chǎn)品功能,已被用來作為顧客導(dǎo)向的方法,分析客戶對(duì)不同屬性的產(chǎn)品的需求,從而促進(jìn)產(chǎn)品的設(shè)計(jì)[2].QFD的方法已成功應(yīng)用在許多行業(yè),包括軟件開發(fā)過程[3]、供應(yīng)鏈管理[4]、項(xiàng)目選擇和分配[5]、預(yù)測分析[6]及投資[7]等.

到目前為止,在質(zhì)量功能展開中確定函數(shù)關(guān)系有兩大分支.一是有經(jīng)驗(yàn)的產(chǎn)品專家根據(jù)清晰輸入或者模糊輸入決定函數(shù)關(guān)系.在傳統(tǒng)的質(zhì)量功能展開中,客戶要求傾向于主觀性和定量性,得到的結(jié)論往往是不精確的.另一分支是模糊回歸的方法.Tanaka等[8]根據(jù)Zadeh[9]的擴(kuò)展原理首次將模糊回歸的方法引入QFD中.Kim等[10]用模糊線性回歸(FLR)模型來估計(jì)QFD中的函數(shù)關(guān)系,并提出了一種模糊多準(zhǔn)則的方法進(jìn)行產(chǎn)品設(shè)計(jì).Moskowitz等[11]、Kim等[12]研究了具有對(duì)稱三角模糊數(shù)的模糊線性回歸模型中，模糊線性回歸參數(shù)h與隸屬函數(shù)的形狀和模糊參數(shù)的展形之間的關(guān)系.Liu等[13]在綜合考慮了系統(tǒng)模糊性和系統(tǒng)隸屬度的因素下,提出系統(tǒng)可靠性最大的準(zhǔn)則,選擇具有最優(yōu)h值的模糊線性回歸模型.Liu等[14]研究了具有對(duì)稱三角模糊數(shù)的模糊線性回歸模型在質(zhì)量功能展開中的應(yīng)用,Chen等[15]將其推廣到具有非對(duì)稱模糊數(shù)的線性回歸模型.

本文分析含有參數(shù)p的對(duì)稱模糊數(shù),推導(dǎo)最優(yōu)h值模糊線性回歸模型的求解過程;并將該理論模型應(yīng)用于質(zhì)量功能展開,在不同p值前提下,對(duì)應(yīng)得到最優(yōu)h值、系統(tǒng)模糊性和系統(tǒng)可靠性進(jìn)行相關(guān)分析,得到最優(yōu)模糊函數(shù)關(guān)系.

1　理論知識(shí)

1.1對(duì)稱模糊數(shù)的線性回歸模型

傳統(tǒng)的線性回歸模型,假設(shè)被解釋變量是解釋變量的線性組合.有m對(duì)樣本觀察數(shù)據(jù){(y1,x1),(y2,x2),…,(yi,xi),…,(ym,xm)},其中,yi是第i個(gè)清晰觀察值,xi=(xi0,xi1,…,xij,…,xin)是第i個(gè)清晰輸入向量,并對(duì)任意的i,有xi0=1成立,xij是第i個(gè)樣本的第j個(gè)變量的觀測值.一般來說,被解釋變量和解釋變量之間的模糊函數(shù)關(guān)系為

(2)

(3)

式中,[f(xi)]h是h水平下通過輸入向量xi,根據(jù)式(1)得到的模糊線性回歸函數(shù)的模糊輸出,是一個(gè)取值區(qū)間.

圖1　不同p值時(shí)具有對(duì)稱模糊數(shù)的隸屬函數(shù)

通過中心值fc(xi)和展形fs(xi),得到預(yù)測模糊值f(xi)=(fc(xi),fs(xi))L,即

(4)

(5)

同時(shí),可以得到h(0≤h<1)水平下的f(xi)=(fc(xi),fs(xi))L的區(qū)間:

(6)

當(dāng)h=0時(shí),f(xi)取0水平截集為模糊集合支集的閉包.

此時(shí),式(3)可以對(duì)應(yīng)寫成

(7)

Tanaka等[8]提出模糊線性回歸模型的系統(tǒng)模糊性,記為Δ.

(8)

(9)

(10)

1.2模糊線性回歸模型的可靠性

(11)

(12)

(13)

由式(13)可得式(1)中所有數(shù)據(jù)集的系統(tǒng)可靠性,記為z.

(14)

z越大,表明得到的模糊線性回歸模型效果越好.

1.3對(duì)稱模糊系數(shù)的最優(yōu)h值

在給定樣本數(shù)據(jù)對(duì)情況下,對(duì)具有對(duì)稱模糊系數(shù)的模糊線性回歸模型,考慮模型的系統(tǒng)模糊性最小和系統(tǒng)可靠性最大兩項(xiàng)準(zhǔn)則,為回歸模型選擇最優(yōu)h值.選擇最優(yōu)h值的具體步驟如下:

(15)

(16)

(17)

(18)

若定義

(19)

則式(18)可以寫成

(20)

(21)

因此,式(1)中對(duì)于h1的模糊線性回歸模型的系統(tǒng)可靠性zh1為

(22)

對(duì)于h2,由式(15)～(17),(19)可得

(23)

(26)

(27)

(28)

若設(shè)h1=0,h2=h,則式(15)～(17),(23)～(25),(27)整理為

(29)

(30)

(31)

(32)

(33)

(34)

(35)

在式(1)中,基于系統(tǒng)可靠性最大的準(zhǔn)則,為了推導(dǎo)出求解最優(yōu)h值的公式,需對(duì)式(35)關(guān)于h求導(dǎo).

(37)

其中,z0≥0,w0≥0.若pw0≥z0成立,則0≤h*<1.一般情況下,很難保證h*≥0,當(dāng)pw0

(38)

根據(jù)式(36)～(38),可以推導(dǎo)出

依據(jù)式(39)和極值條件可以得出h*是式(35)的最大值點(diǎn).

根據(jù)上述分析可以得出:在p>0的條件下,式(35)的最優(yōu)解為

(40)

其系統(tǒng)可靠性

(41)

2　實(shí)例分析

將理論應(yīng)用于質(zhì)量功能展開的案例,討論在不同p值下,依據(jù)系統(tǒng)可靠性最大的準(zhǔn)則,獲得顧客需求(CRs)和工程特性(ECs)最優(yōu)的函數(shù)關(guān)系.

2.1相關(guān)說明

表1顯示包裝機(jī)的相關(guān)數(shù)據(jù),其中,CR1-CR4分別表示“提高炸藥包裝的質(zhì)量”、“提高炸藥包裝的效率”、“減少包裝的噪音”和“增加機(jī)器的硬度”;EC1-EC7分別表示“提高模型修剪的精度”、“提高炸藥包裝的精密”、“增加炸藥包裝的控制力”、“增加炸藥包裝的效率”、“增加壓錘的硬度”、“減少凸輪電力傳輸?shù)脑胍簟焙汀皽p少機(jī)器床的高度”;Proc表示第c種產(chǎn)品,c=1,2,3,4,5;CR1,CR2,…,CR7的單位分別為m-2,m-2,N,ns-1,HRC,dB,m.

表1包裝機(jī)的相關(guān)數(shù)據(jù)

Tab.1Corresponding datas of the packing machine

變量EC1EC2EC3EC4EC5EC6EC7Pro1Pro2Pro3Pro4Pro5MinMaxCR1···3.14.02.53.32.015CR2···3.13.01.82.93.915CR3·2.23.74.31.83.515CR4·1.63.73.33.74.015Pro1117589055751.9Pro286658550681.7Pro3129607050551.8Pro498628045801.7Pro51010657555701.6Min64556040501.5Max15127010060902

注:*表示兩者之間的關(guān)系顯著.

2.2識(shí)別函數(shù)關(guān)系

其系統(tǒng)可靠性

表2顧客需求(CRs)和工程特性(ECs)之間的函數(shù)關(guān)系(h=0)

Tab.2Functional relationships between the customer requirements (CRs) and the engineering characteristics (ECs) (h=0)

系數(shù)y1y2y3y4X0(0.3219,0.0000)L(0.0000,0.2314)LX1(1.7444,0.0000)L(0.0000,0.1587)L(1.4727,0.0000)LX2(3.0748,0.0000)LX3(0.0000,0.2752)L(1.6207,0.0000)LX4(1.9966,0.0000)LX5(1.3352,0.0000)L(0.5565,0.0000)LX6(0.3543,0.0000)L(0.0000,0.2245)L(4.0219,0.0000)L(1.3010,0.0000)LX7(0.0000,0.3057)L(2.3040,0.0000)L(3.5198,0.0745)L

表3f1,f2,f3和f4的對(duì)應(yīng)結(jié)果(h=0,p=1)

Tab.3Corresponding results of f1,f2,f3and f4(h=0,p=1)

Tab.4Functional relationships between the customer requirements (CRs) and the engineering

系數(shù)y1y2y3y4X0(0.3219,0.0000)L(0.0000,0.4628)LX1(1.7444,0.0000)L(0.0000,0.2691)L(1.4727,0.0000)LX2(3.0748,0.0000)LX3(0.0000,0.5504)L(1.6207,0.0000)LX4(1.9966,0.0000)LX5(1.3352,0.0000)L(0.5565,0.0000)LX6(0.3543,0.0000)L(0.0000,0.4490)L(4.0219,0.0000)L(1.3010,0.0000)LX7(0.0000,0.6114)L(2.3040,0.0000)L(3.5198,0.1490)L

2.3h值變化的相關(guān)分析

(44)

(45)

圖的參數(shù)隨h的變化圖像

(46)

(47)

圖的參數(shù)隨h的變化圖像

圖的參數(shù)隨h的變化圖像

圖的參數(shù)隨h的變化圖像

圖6　不同p值時(shí)參數(shù)的變化

2.4對(duì)比分析

3　結(jié)　論

通過分析對(duì)稱模糊數(shù)的線性回歸模型,當(dāng)隸屬函數(shù)含有參數(shù)p時(shí),考慮系統(tǒng)模糊性和系統(tǒng)隸屬度兩個(gè)因素,根據(jù)系統(tǒng)可靠性最大的準(zhǔn)則,推導(dǎo)選擇最優(yōu)h值的方法.將其理論應(yīng)用到QFD的實(shí)例中,判斷顧客需求與工程特性之間的函數(shù)關(guān)系,同時(shí)討論不同p值下相關(guān)參數(shù)的變化規(guī)律.當(dāng)p=2時(shí),即含有拋物型模糊數(shù)的模糊線性回歸模型的系統(tǒng)可靠性最大,此時(shí)得到的模型更合理,有助于管理者作出正確高效的決策.

[1]AKAO Y.Quality function deployment:integrating customer requirements into product design[M].Cambridge,MA:Productivity Press,1990.

[2]FUNG R Y K,REN S J,XIE J X.The prioritisation of attributes in customer requirement management[C]∥Proceedings of IEEE International Conference on Systems,Man,and Cybernetics.Beijing:IEEE,1996,2:953-958.

[3]BARNETT W D,RAJA M K.Application of QFD to the software development process[J].International Journal of Quality & Reliability Management,1995,12(6):24-42.

[4]ONUT S,TOSUN S.An integrated methodology for supplier selection under the presence of vagueness:a case in banking sector,Turkey[J].Journal of Applied Mathematics,2014,8(10):1-14.

[5]CHEN C C,ZHANG Q.Applying quality function deployment techniques in lead production project selection and assignment[J].Advanced Materials Research,2014,945:2954-2959.

[6]靳忠偉,葉舟,陳康民,等.α-加權(quán)模糊線性回歸模型及其在電力需求預(yù)測中的應(yīng)用[J].上海理工大學(xué)學(xué)報(bào),2005,27(4):319-322.

[7]CELIK M,CEBI S,KAHRAMAN C,et al.An integrated fuzzy QFD model proposal on routing of shipping investment decisions in crude oil tankermarket[J].Expert Systems with Applications,2009,36(3):6227-6235.

[8]TANAKA H,UEJIMA S,ASAI K.Linear regression analysis with fuzzy model[J].IEEE Transactions on Systems,Man,and Cybernetics,1982,12(6):903-907.

[9]ZADEH L A.The concept of a linguistic variable and its application to approximate reasoning-Ⅱ[J].Information Sciences,1975,8(4):301-357.

[10]KIM K J,MOSKOWITZ H,DHINGRA A,et al.Fuzzy multicriteria models for quality function deployment[J].European Journal of Operational Research,2000,121(3):504-518.

[11]MOSKOWITZ H,KIM K.On assessing thehvalue in fuzzy linear regression[J].Fuzzy Sets and Systems,1993,58(3):303-327.

[12]KIM K J,MOSKOWITZ H,KOKSALAN M.Fuzzy versus statistical linear regression[J].European Journal of Operational Research,1996,92(2):417-434.

[13]LIU X L,CHEN Y Z.A systematic approach to optimizinghvalue for fuzzy linear regression with symmetric triangular fuzzy numbers[J].Mathematical Problems in Engineering,2013,9(6):1-9.

[14]LIU Y Y,CHEN Y Z,ZHOU J,et al.Fuzzy linear regression models for QFD using optimizedhvalues[J].Engineering Applications Artificial Intelligence,2015,39:45-54.[15]CHEN F N,CHEN Y Z,ZHOU J,et al.Optimizinghvalue for fuzzy linear regression with asymmetric triangular fuzzy coefficients[J].Engineering Applications of Artificial Intelligence,2016,47:16-24.

[16]任麗麗,陸秋君.基于模糊線性回歸的電子商務(wù)交易額預(yù)測[J].統(tǒng)計(jì)與決策,2013(3):31-34.

(編輯:石瑛)

Quality Function Deployment Based on a Kind of Linear Regression Model with Symmetric Fuzzy Numbers and Optimized h Values

ZHU Qian,LU Qiujun

(College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China)

The parameterh,in a fuzzy linear regression(FLR) model,determines the probability distribution of the fuzzy coefficients and influences the system credibility of the FLR model.To obtain the functional relationships with optimizedhvalues in quality function deployment(QFD),the FLR model was analyzed,where the coefficients were assumed as symmetric fuzzy numbers containing parameterp.When the parameterpwas given,considering both the system fuzziness and membership,thehvalues were determined according to the criterion of maximum credibility of the system.An illustrative example was provided to demonstrate the performance of the proposed approach.Meanwhile,the optimizedhvalues were obtained under different representatives ofpvalues,namely,the secondary root,triangle and parabolic symmetric fuzzy numbers.The changing disciplines of the system fuzziness and reliability were analyzed with regard to the optimizedhvalues.The maximum system credibility of the functional relationship can be obtained by comparing the corresponding results.

fuzzy linear regression; symmetric fuzzy number; optimized h value; quality function deployment; system credibility

1007-6735(2016)04-0318-11

10.13255/j.cnki.jusst.2016.04.003

2016-01-30

朱倩(1992-),女,碩士研究生.研究方向:應(yīng)用數(shù)理統(tǒng)計(jì).E-mail:zhuqian_1992@126.com

陸秋君(1974-),女,副教授.研究方向:應(yīng)用數(shù)理統(tǒng)計(jì)、經(jīng)濟(jì)計(jì)量學(xué)和模糊數(shù)學(xué).E-mail:fuzzy_lu@163.com

O 212

A