Research Article A Systematic Approach to Optimizing Value ...
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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 210164, 9 pages http://dx.doi.org/10.1155/2013/210164 Research Article A Systematic Approach to Optimizing ℎ Value for Fuzzy Linear Regression with Symmetric Triangular Fuzzy Numbers Xilong Liu and Yizeng Chen School of Management, Shanghai University, Shanghai 200444, China Correspondence should be addressed to Yizeng Chen; [email protected] Received 17 July 2013; Revised 12 October 2013; Accepted 14 October 2013 Academic Editor: T. Warren Liao Copyright © 2013 X. Liu and Y. Chen. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A systematic approach is proposed to optimize ℎ value for fuzzy linear regression (FLR) analysis using minimum fuzziness criteria with symmetric triangular fuzzy numbers (TFNs). Firstly, a new concept of credibility is defined to evaluate the performance of FLR models with different ℎ values when a set of sample data pairs is given. Secondly, based on the defined concept of credibility, a programming model is formulated to optimize the value of ℎ. Finally, both the numerical study and the real application show that the approach proposed in this paper is effective and efficient; that is, optimal value for ℎ can be determined definitely with respect to a set of given sample data pairs. 1. Introduction Statistic regression analysis and fuzzy regression analysis are two types of methods underlying different philosophies to assess the functional relationship between the dependent and independent variables and determine the best-fit model for describing the relationship, by exploiting the knowledge from the given input-output data pairs. In statistical regression analysis, deviations between the observed values and the estimates are assumed to be random errors disturbed by a probabilistic distribution. Different from statistic regression analysis, in fuzzy regression analysis, the deviations are attributed to the imprecision of the observed values and/or the indefiniteness of model structure. Tanaka et al. [1] firstly proposed fuzzy linear regression (FLR) analysis using the fuzzy functions defined by Zadeh’s extension principle [2], in which the observed values can differ from the estimated values to a certain degree of belief [3]. us, the uncertainty in this type of regression model becomes fuzziness, not randomness, and the disturbance is incorporated into the fuzzy coefficients, and the final objective is to adjust the fuzzy coefficients from the available sample data pairs. According to [3], the existing FLR methods can be roughly classified into the following two categories based on criterion function, that is, FLR methods using minimum fuzziness criteria and FLR methods using fuzzy least-squares criteria. By using the first category of FLR methods, FLR model can be built by minimizing the system vagueness. e first FLR method in [1] was extended by using other types of fuzzy coefficients, including general LP-type fuzzy coefficients [4], exponential fuzzy coefficients [5], and tri- angular fuzzy coefficients [6, 7], Chen et al. depended on symmetric triangular fuzzy numbers to study determination Method for Parameters of Rock’s shear strength through least absolute linear regression, and the analysis of practical engineering computation, and comparison to other methods shows that the method is reasonable [8], trapezoidal fuzzy coefficients [9], and Kheirfam and Verdegay extend the dual simplex method to a type of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. e results obtained lead to a solution for fuzzy linear programming problems and the optimal value function with fuzzy coefficients [10]. And interval regression, where a model with interval coefficients is assumed, is regarded as the simplest version of fuzzy regression analysis [11–13]. Some fuzzy nonlinear regression approaches also were proposed [3, 7]; a research indicated that prediction performance of the nonlinear multiple regression model is higher than that of the fuzzy inference system model [14]. Based on the other direction of FLR methods, FLR model will be built
Transcript of Research Article A Systematic Approach to Optimizing Value ...
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 210164 9 pageshttpdxdoiorg1011552013210164
Research ArticleA Systematic Approach to Optimizing ℎ Value for Fuzzy LinearRegression with Symmetric Triangular Fuzzy Numbers
Xilong Liu and Yizeng Chen
School of Management Shanghai University Shanghai 200444 China
Correspondence should be addressed to Yizeng Chen mfcyzshueducn
Received 17 July 2013 Revised 12 October 2013 Accepted 14 October 2013
Academic Editor T Warren Liao
Copyright copy 2013 X Liu and Y Chen This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A systematic approach is proposed to optimize ℎ value for fuzzy linear regression (FLR) analysis using minimum fuzziness criteriawith symmetric triangular fuzzy numbers (TFNs) Firstly a new concept of credibility is defined to evaluate the performance ofFLR models with different ℎ values when a set of sample data pairs is given Secondly based on the defined concept of credibility aprogramming model is formulated to optimize the value of ℎ Finally both the numerical study and the real application show thatthe approach proposed in this paper is effective and efficient that is optimal value for ℎ can be determined definitely with respectto a set of given sample data pairs
1 Introduction
Statistic regression analysis and fuzzy regression analysis aretwo types of methods underlying different philosophies toassess the functional relationship between the dependent andindependent variables and determine the best-fit model fordescribing the relationship by exploiting the knowledge fromthe given input-output data pairs In statistical regressionanalysis deviations between the observed values and theestimates are assumed to be random errors disturbed by aprobabilistic distribution Different from statistic regressionanalysis in fuzzy regression analysis the deviations areattributed to the imprecision of the observed values andorthe indefiniteness of model structure Tanaka et al [1] firstlyproposed fuzzy linear regression (FLR) analysis using thefuzzy functions defined by Zadehrsquos extension principle [2]in which the observed values can differ from the estimatedvalues to a certain degree of belief [3] Thus the uncertaintyin this type of regression model becomes fuzziness notrandomness and the disturbance is incorporated into thefuzzy coefficients and the final objective is to adjust the fuzzycoefficients from the available sample data pairs
According to [3] the existing FLR methods can beroughly classified into the following two categories basedon criterion function that is FLR methods using minimum
fuzziness criteria and FLR methods using fuzzy least-squarescriteria By using the first category of FLR methods FLRmodel can be built by minimizing the system vaguenessThe first FLR method in [1] was extended by using othertypes of fuzzy coefficients including general LP-type fuzzycoefficients [4] exponential fuzzy coefficients [5] and tri-angular fuzzy coefficients [6 7] Chen et al depended onsymmetric triangular fuzzy numbers to study determinationMethod for Parameters of Rockrsquos shear strength throughleast absolute linear regression and the analysis of practicalengineering computation and comparison to other methodsshows that the method is reasonable [8] trapezoidal fuzzycoefficients [9] and Kheirfam and Verdegay extend the dualsimplex method to a type of fuzzy linear programmingproblem involving symmetric trapezoidal fuzzy numbersThe results obtained lead to a solution for fuzzy linearprogramming problems and the optimal value function withfuzzy coefficients [10] And interval regression where amodel with interval coefficients is assumed is regarded asthe simplest version of fuzzy regression analysis [11ndash13] Somefuzzy nonlinear regression approaches also were proposed[3 7] a research indicated that prediction performance ofthe nonlinear multiple regression model is higher than thatof the fuzzy inference system model [14] Based on theother direction of FLR methods FLR model will be built
2 Mathematical Problems in Engineering
by minimizing fuzzy distance between the predicted outputsand the observed outputs [15ndash17] Celmins [15] has dealtwith quadratic membership functions based on least squaresfitting with indicators of discord data spread dilator and soforth and Diamond [16] proposed models for least squaresfitting for crisp input fuzzy output and for fuzzy input-outputwhere the distance of fuzzy numbers is defined to measurethe best fit for models Chang [17] formulated fuzzy least-squares regression model by defining the weighted fuzzyarithmetic (WFA) Lv et al proposed a novel least squaressupport vector machine- (LSSVM-) based ensemble learningparadigm to predict NO
119909emission of a coal-fired boiler using
real operation data The result shows that the new soft FCM-LSSVM-PLS ensemble method can predict NO
119909emission
accurately [18]Regarding the first category of FLR methods the value
of ℎ determines the range of the possibility distribution ofthe fuzzy parameters [1 3ndash8 19] so it is important to selecta suitable value for ℎ in FLR analysis Moskowitz and Kim[20] studied the relationship among the ℎ value membershipfunction shape and the spreads of fuzzy parameters inFLR with symmetric fuzzy numbers they also developed ageneral approach to assess the proper ℎ parameter valuesTheir studies showed that the system fuzziness will increasewith the augment of ℎ value And Tanaka and Watada [19]suggested that the selection of the ℎ value should be based onthe sufficiency of the collected dataset When the dataset issufficiently large ℎ = 0 should be used and is increased alongwith the decreasing volume of the collected data Howeverboth did not suggest how to get an optimal ℎ value forthe FLR model when a sample dataset is given In practicalsituations the value of ℎ is usually subjectively preselected bythe decision makers (DMs) [17 18]
In fact if a larger value is given to ℎ the FLR usingtriangular fuzzy coefficients tends to yield large unnecessaryfuzziness and estimated parameters with too large aspirationwhich leads to the fuzzy predictive interval too fuzzier andhas no operational definition or interpretation [8] On theother hand if a smaller ℎ value is used the FLR tends toyield very lower membership degree which leads to the verynarrow fuzzy predictive interval and the reliability of the FLRmodel is doubtable Therefore it is necessary to develop asystematic approach to help DMs determine the optimal ℎvalue for the FLR usingminimum fuzziness criteria To tacklethe problem with the suggestion from Tanaka and Watada[19] a new concept of credibility is introduced tomeasure theperformance of the FLRmodels with different ℎ values in thispaper based on which a systematic approach is formulated tooptimize ℎ values for FLR using minimum fuzziness
The rest of the paper is organized as follows In Section 2Tanakarsquos FLR method is described In Section 3 the conceptof credibility is proposed to measure the performance of FLRmodels with different ℎ values In Section 4 a systematicapproach is formulated to optimize the ℎ value for FLR withsymmetric triangular fuzzy numbers (TFNs) In Section 5 anumerical example is used to show how the optimal value canbe determined using the approach proposed in this paperIn Section 6 a real application is conducted to illustrate
120583A119895(Aj)
A0
1
Sj AC
jAj
Figure 1 Symmetric triangular fuzzy number
the effectiveness of the approach proposed in this paper Theconclusions are drawn in Section 7
2 FLR with Symmetric TFNs
As in a FLR analysis the explained variable is assumedto be a linear combination of the explanatory variablesThis relationship should be obtained from a sample of 119898observations (119910
1 x1) (119910
119894 x119894) (119910
119898 x119898) where 119910
119894is
the 119894th crisp observation and x119894= 1199091198940 1199091198941 119909
119894119895 119909
119894119899
is the 119894th crisp input vector Moreover 1199091198940= 1forall119894 and 119909
119894119895
is the observed value for the 119895th variable in the 119894th case ofthe sample In particular the fuzzy linear function has to beestimated as follows
119910119894= 119891 (x
119894) = 119860
0+ 11986011199091198941sdot sdot sdot 119860119895119909119894119895sdot sdot sdot + 119860
119899119909119894119899=
119899
sum
119895=0
119860119895119909119894119895
(1)
where 119910119894is the fuzzy estimation of 119910
119894 And119860
119895 119895 = 0 1 119899
are fuzzy coefficients in the terms of symmetric TFNs and canbe uniquely defined by 119860
119895= (119886119878
119895 119886119862
119895) 119895 = 0 1 119899 Here
119886119878
119895is the spread value and 119886119862
119895is the centre value of 119860
119895(see
Figure 1)The goal in the fuzzy linear regression is to determine
119891(x119894) by minimizing the system vagueness subject to the
following inclusion condition [1 19]
119910119894isin [119891 (x
119894)]ℎ
119894 = 1 2 119898 (2)
where 119891(x119894)ℎ is the ℎ-level set of the fuzzy output from the
linear fuzzy model 119891(x119894) corresponding to the input vector
x119894 Since ℎ-level set of fuzzy numbers are intervals (2) can be
further given as follows by using Interval arithmetic
119899
sum
119895=0
119886119862
119895119909119894119895minus (1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816
le 119910119894119895le
119899
sum
119895=0
119886119862
119895119909119894119895+ (1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816 119894 = 1 2 119898
(3)
Mathematical Problems in Engineering 3
The system fuzziness in (1) Δ can be given as
Δ =
119898
sum
119894=1
Δ119910119894
(4)
in which Δ119910119894is the fuzziness associated with 119910
119894and can be
given as
Δ119910119894=
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816 (5)
Henceforth according to [1 19]119860119895 119895 = 0 1 119899 in the
form of symmetric TFNs can be determined by solving thefollowing optimal programming model
min Δ =
119898
sum
119894=1
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816 (6a)
subject to
(1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816minus
119899
sum
119895=0
119886119862
119895119909119894119895ge minus119910119894119895 119894 = 1 119898 (6b)
(1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816+
119899
sum
119895=0
119886119862
119895119909119894119895ge 119910119894119895 119894 = 1 119898 (6c)
119886119878
119895ge 0 119895 = 0 1 119899 (6d)
in which the constraints (6b) and (6c) are corresponding toinclusion condition in (2) and the constraint (6d) guaranteesthat the spread values of 119860
119895 119895 = 0 1 119899 are nonnegative
3 Credibility Measure for the FLR Model
As we can see from (6a) to (6d) the value of ℎ determines therange of the possibility distribution of the fuzzy parametersso it is important to select a suitable ℎ value for FLR Todo this the first problem to be solved is how to evaluatethe FLR models with different ℎ values in (1) Thereforea new concept of credibility measurement is introduced inthis section based on which the FLR models associated withdifferent ℎ values can be assessed
Assume that ℎ1and ℎ
2are any two values for ℎ and 0 le
ℎ1lt 1 and 0 le ℎ
2lt 1 And we denote that 1198601
119895and 1198602
119895
119895 = 0 1 119899 are two sets of symmetric TFNs with respecttoℎ1andℎ2 respectively and1199101
119894and1199102119894are the corresponding
estimations of 119910119894from (1) From (1) 1199101
119894and 1199102
119894are calculated
as
1199101
119894=
119899
sum
119895=0
1198601
119895119909119894119895 (7)
1199102
119894=
119899
sum
119895=0
1198602
119895119909119894119895 (8)
Now we are interested in which one out of 1199101119894and 1199102
119894 is
better as the estimation of 119910119894 That is to say which one out of
ℎ1and ℎ2 is better to be used to build a FLR model when the
h
1
0 y
120583yi (yi )
yL1 yC1 yU1 yU2yL2 yC2yi
y1 y2
Figure 2 1205831199101
119894
(119910119894) = 1205831199102
119894
(119910119894)
h
h
1
0y
120583yi (yi )
yL2
1
yC1 yU1 yU2yL1
2
yC2yi
y1 y2
Figure 3 Δ1199101119894= Δ1199102
119894
sample set of data pairs is given To deal with the problemin our opinion two factors that is the estimating fuzzinessΔ119910119894and the membership degree 120583
119910119894
(119910119894) should be taken
into account In general the smaller Δ119910119894is and the higher
120583119910119894
(119910119894) is the better the performance of 119910
119894in representing 119910
119894
will be To illustrate our point of view two special cases areconsidered firstly as follows
(1) If 1205831199101
119894
(119910119894) is equal to 120583
1199102
119894
(119910119894) it is clear that out of 1199101
119894
and1199102119894 the performance of onewith smaller fuzziness
is better than that of the other with larger fuzzinessin representing 119910
119894 As shown in Figure 2 120583
1199101
119894
(119910119894) =
1205831199102
119894
(119910119894) and Δ1199101
119894lt Δ119910
2
119894 therefore 1199101
119894is better than
1199102
119894as the estimation of 119910
119894
(2) IfΔ1199101119894is equal toΔ1199102
119894 it is doubtless that out of1199101
119894and
1199102
119894 the performance of one with higher membership
degree is better than that of the other with lowermembership degree in representing 119910
119894 As shown in
Figure 3 Δ1199101119894= Δ1199102
119894and 120583
1199101
119894
(119910119894) gt 1205831199102
119894
(119910119894) therefore
1199101
119894is better than 1199102
119894as the estimation of 119910
119894
Now let us consider a more general situation that neither1205831199101
119894
(119910119894) is equal to 120583
1199102
119894
(119910119894) nor Δ1199101
119894is equal to Δ1199102
119894 as shown
in Figure 4 To deal with this problem a new concept ofcredibility measure is introduced in this paperWe denote thecredibility of 119910
119894in representing 119910
119894as 119911119894 which is expressed as
119911119894=120583119910119894
(119910119894)
Δ119910119894
(9)
The higher 119911119894is the better the performance of 119910
119894in
representing 119910119894will be Obviously the scenarios of (1) and
(2) are two special cases of (9) respectively
4 Mathematical Problems in Engineering
h
h
1
0y
120583yi (yi )
yL1
1
yC2 yU1 yU2yL2
2
yC1yi
y2 y1
Figure 4 Δ1199101119894= Δ1199102
119894and 120583
1199101
119894
(119910119894) = 1205831199102
119894
(119910119894)
(aS)hlowast2 (aS)h
lowast1 aC
h
h
1
0
1
2
120583A(a)
Figure 5 The case when ℎ2ge ℎ1
As a result based on (9) the total credibility of all sampledata 119911 can be obtained to assess the performance of theobtained FLRmodel in (1) which can be calculated as follows
119911 =
119898
sum
119894=1
119911119894=
119898
sum
119894=1
120583119910119894
(119910119894)
Δ119910119894
(10)
The higher 119911 is the better of the performance of the FLRmodel will be This will help us select the optimal value ofℎ for a FLR model
4 Formulating a Systematic Approach toOptimizing ℎ Value
Based on the concept of credibility measure for FLR modelsintroduced in Section 3 a systematic approach is proposed tooptimize ℎ value for FLR models in this Section
As shown in Figure 5 we denote the optimal solutionof (6a) (6b) (6c) and (6d) with respect to ℎ
1as 119860ℎ
lowast
1
119895=
((119886119878
119895)ℎlowast
1
119886119862
119895) 119895 = 0 1 119899 and Δℎ
lowast
1 and the correspondingfuzziness as Δ119910ℎ
lowast
1
119894 119894 = 1 119898 According to Moskowitz and
Kim [20] the optimal solution of (6a) (6b) (6c) and (6d)and the corresponding fuzziness with regard to ℎ
2can be
obtained as
119860ℎlowast
2
119895= (
1 minus ℎ1
1 minus ℎ2
(119886119878
119895)ℎlowast
1
119886119862
119895) 119895 = 0 1 119899
Δℎlowast
2 =1 minus ℎ1
1 minus ℎ2
Δℎlowast
1
Δ119910ℎlowast
2
119894=1 minus ℎ1
1 minus ℎ2
Δ119910ℎlowast
1
119894 119894 = 1 119898
(11)
Table 1 Testing dataset
119910 119909
14 216 414 618 818 1022 1218 1622 18
This indicates that the central value 119886119862119894119895of each 119860
119894119895will keep
constant when the ℎ value changes from ℎ1to ℎ2 while the
spread value 119886119878119894119895of each 119860
119894119895and the objective function Δ and
the fuzzinessΔ119910119894(119894 = 1 119898) become (1minusℎ
1)(1minusℎ
2) times
simultaneouslyWith respect to ℎ
1 (120583119910119894
(119910119894))ℎ1 119894 = 1 119898 are calculated
as follows
(120583119910119894
(119910119894))ℎ1
= 1 minus
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ1
119894
= 1 minus 119902ℎ1
119894 119894 = 1 119898 (12)
in which 119902ℎ1119894is given as
119902ℎ1
119894=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ1
119894
=
10038161003816100381610038161003816sum119899
119895=0(119886119862
119895) 119909119894119895minus 119910119894
10038161003816100381610038161003816
sum119899
119895=0(119886119878119895)ℎ1 10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816
119894 = 1 119898 (13)
Therefore according to (8) the estimating credibility for allsample data with regard to ℎ
1can be expressed as
119911ℎ1
119894=(120583119910119894
(119910119894))ℎ1
Δ119910ℎ1
119894
=1 minus 119902ℎ1
119894
Δ119910ℎ1
119894
119894 = 1 119898 (14)
Therefore from (9) the total credibility for the FLR modelwith respect to ℎ
1can be obtained as
119911ℎ1 =
119898
sum
119894=1
119911ℎ1
119894=
119898
sum
119894=1
1 minus 119902ℎ1
119894
Δ119910ℎ1
119894
(15)
From (12) with regard to ℎ2 we have
119902ℎ2
119894=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ2
119894
=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
((1 minus ℎ1) (1 minus ℎ
2)) Δ119910ℎ1
119894
=1 minus ℎ2
1 minus ℎ1
119902ℎ1
119894 119894 = 1 119898
(16)
Mathematical Problems in Engineering 5
Table 2 Corresponding results with symmetric triangular fuzzy coefficients (ℎ = 0)
119894 1 2 3 4 5 6 7 8Δ1199100
11989412500 15000 17500 20000 22500 25000 27500 30000
1199020
11989406000 10000 10000 05000 01111 10000 10000 00000
1199010
11989404800 06667 05714 02500 00494 04000 03636 00000
(120583119910119894(119910119894))0
04000 00000 00000 05000 08889 00000 00000 100001199110
11989403200 00000 00000 02500 03951 00000 00000 03333
Therefore according to (11) (120583119910119894
(119910119894))ℎ2 with regard to ℎ
2can
be calculated as
(120583119910119894
(119910119894))ℎ2
= 1 minus 119902ℎ2
119894= 1 minus
1 minus ℎ2
1 minus ℎ1
119902ℎ1
119894
= 1 minus 119902ℎ1
119894+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894
= (120583119910119894
(119910119894))ℎ1
+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894 119894 = 1 119898
(17)
Henceforth according to (8) (10) and (16) the estimatingcredibility for all sample data pairs with regard to ℎ
2can be
expressed as
119911ℎ2
119894=(120583119910119894
(119910119894))ℎ2
Δ119910ℎ2
119894
=(120583119910119894
(119910119894))ℎ1
+ ((ℎ2minus ℎ1) (1 minus ℎ
1)) 119902ℎ1
119894
((1 minus ℎ1) (1 minus ℎ
2)) Δ119910ℎ1
119894
=1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894 119894 = 1 119898
(18)
in which
119901ℎ1
119894=
119902ℎ1
119894
Δ119910ℎ1
119894
=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
(Δ119910ℎ1
119894)2 119894 = 1 119898 (19)
Consequently the estimating credibility for the FLRmodel in(1) with regard to ℎ
2can be obtained as
119911ℎ2 =
119898
sum
119894=1
119911ℎ2
119894=
119898
sum
119894=1
(1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894)
=1 minus ℎ2
1 minus ℎ1
119911ℎ1 +
(1 minus ℎ2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
(20)
in which
119901ℎ1 =
119898
sum
119894=1
119901ℎ1
119894 (21)
For similarity we denote ℎ1= 0 and ℎ
2= ℎ (11) (16) (17)
(18) and (20) can be rewritten as
119860ℎ
119895= (
1
1 minus ℎ(119886119878
119895)0
119886119862
119895) 119895 = 0 1 119899
Δℎ
=1
1 minus ℎΔ0
Δ119910ℎ
119894=
1
1 minus ℎΔ1199100
119894 119894 = 1 119898
(22)
119902ℎ
119894= (1 minus ℎ) 119902
0
119894 (23)
(120583119910119894
(119910119894))ℎ
= (120583119910119894
(119910119894))0
+ ℎ1199020
119894 (24)
119911ℎ
119894= minus1199010
119894ℎ2
+ (1199010
119894minus 1199110
119894) ℎ + 119911
0
119894 (25)
119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ + 1199110
(26)
Based on (26) the optimal ℎ value for the FLR model in(1) can be obtained by maximizing the estimating credibilitythat is the optimal value for ℎ can be obtained by solving thefollowing programming model
max 119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ (27a)
st 0 le ℎ lt 1 (27b)
It is obviously that the programming model in (27a) to (27b)is quadratic and many kinds of optimization algorithmssuch as the gradient descent method can be used to solve theprevious nonlinear programming problem
5 Numerical Example
The numerical example in [21] is used to how the optimal ℎvalue can be obtained using the approach proposed in thispaper The eight testing data pairs are listed in Table 1
When ℎ is specified as 0 the fuzzy coefficients interms of symmetric TFNs can be obtained by solving theprogramming model in (6a) (6b) (6c) and (6d) as 1198600
0=
(10000 120000) and 1198600
1= (01250 06250) And the
corresponding FLR model is given as
1199100
= (10000 120000) + (01250 06250) 119909 (28)
Accordingly the fuzziness Δ1199100119894 parameters 1199020
119894and 1199010
119894 mem-
bership degree (120583119910119894
(119910119894))0 and credibility 1199110
119894 119894 = 1 119898 are
calculated respectively as shown in Table 2
6 Mathematical Problems in Engineering
Table 3 Testing data set
119910 1199091
1199092
13360 2950 799913763 3130 756314786 3760 692519676 3990 627522053 3990 646622325 4030 630923319 4150 615126567 4360 600733516 4570 582241129 4780 584346068 4950 605747796 5010 582347402 5020 580346680 4990 575346616 5000 556846980 5000 552446895 5000 545147624 5090 500849939 5310 500552120 5520 4972
From Table 2 1199110 is calculated as 12984 and 1199010 as 27811According to (26) the total credibility of the FLRmodel withrespect to ℎ can be expressed as follows
119911ℎ
= minus27811ℎ2
+ 14827ℎ + 12984 (29)
According to (27a) and (27b) the optimal value ℎlowast is given as02666 and the optimal coefficients in the form of symmetriccase are obtained as 119860lowast
0= (13635 120000) and 119860
lowast
1=
(01704 06250) Therefore the optimal FLR model is givenas
119910lowast
= (13635 120000) + (01704 06250) 119909 (30)
The total credibility of the model in (30) 119911lowast is calculated as14960 which is higher than that of the model in (28) that is12984
The changes of the fuzzinessΔ119910ℎ119894 themembership degree
(120583119910119894
(119910119894))ℎ the credibility 119911
ℎ
119894 119894 = 1 119898 and the total
credibility 119911ℎ with different ℎ values are depicted in Figures6 7 8 and 9 respectively Notably the vertical axis islogarithmic scale in Figure 6 to enable us to see small changesin fuzziness among eight sample data and the membershipdegrees for sample data 2 3 6 and 7 are overlapped inFigure 7
From Figure 6 we can see that the fuzziness of all sampledata become larger with the augment of ℎ value especiallywhen ℎ value is close to 1 the fuzziness will be extremelylarger From Figure 7 it can be found that the membershipdegree of all sample data becomes higher with the increase ofℎ value and the membership degree will be close to 1 whenℎ value is close to 1 It is clearly shown in Figure 8 that when
0 02 04 06 08 1h value
Fuzz
ines
s
Data1Data2Data3Data4
Data5Data6Data7Data8
103
102
101
100
Figure 6 Fuzziness of the eight sample data
0 02 04 06 08 10
02
04
06
08
1
12
h value
Mem
bers
hip
degr
ee
Data1Data2Data3Data4
Data5Data6Data7Data8
Figure 7 Membership degree of the eight sample data
ℎ value is close to 1 the credibility measures for all sampledata will be close to zero due to the extremely larger of thefuzziness (see Figure 7) Figure 9 shows that when ℎ valueis 02666 the total credibility of the FLR model will achievethe maximum based on which the best FLR model can beobtained
To further demonstrate the effectiveness of the approachproposed in this paper another numerical example with twoindependent variables is given as follows The twenty testingdatasets are listed in Table 3
According to (27a) and (27b) the optimal value ℎlowast isgiven as 03184 When ℎ is specified as 0 03184 and 05respectively the fuzzy coefficients in terms of symmetricTFNs and the corresponding total credibility can be obtained
Mathematical Problems in Engineering 7
Table 4 The corresponding results
h value Fuzzy linear model Total credibility03148 119910
lowast
= (minus6108251 00000) + (191484 00000) 1199091+ (14018 12301) 119909
201371
00 1199100
= (minus6108233 00000) + (191484 00000) 1199091+ (14017 08429) 119909
201082
05 11991005
= (minus6108240 00000) + (191484 00000) 1199091+ (14017 16857) 119909
201271
Data1Data2Data3Data4
Data5Data6Data7Data8
0 02 04 06 08 10
005
01
015
0202
h value
Cred
ibili
ty m
easu
re
Figure 8 Credibility of the eight sample data
0 02 04 06 08 10
05
1
15
h value
Tota
l cre
dibi
lity
Figure 9 Total credibility of the FLR models
by solving the programming model in (6a) (6b) (6c) and(6d) The results are summarized in Table 4
From Table 4 we can see that when ℎ value is set to03148 the total credibility of the FLR model will achieve themaximum that is 01371 which indicates that the approachproposed in this paper is effective and reasonable
Table 5 The experiment results
Run no 119909 (gcm3) 119910 (horsepower)1 0760 512 0780 623 0820 754 0815 705 0790 716 0785 687 0770 598 0765 569 0788 9010 0769 59
Table 6 The FLR models with the corresponding total credibility
h value Fuzzy linear model Totalcredibility
02057 119910 = (minus256000 08765) + (406962 00000)119909 727850 119910 = (minus256000 06962) + (406962 00000)119909 6790305 119910 = (minus256000 13924) + (406962 00000)119909 62793
6 Real Application
To investigate the effectiveness of the approach proposed inthis paper modeling welding process for electronic manu-facturing using fuzzy linear regression was studied An elec-tronic company is a famous OEM company of printed circuitboard (PCB) in PR China The engineers in this companywanted to improve the welding quality by investigating therelationship between the pull strength (119910) of welding line andthe proportion of colophony (119909) in welding fluid And anengineering experiment was conducted and the experimentresults are shown in Table 5
With the use of the 10 experimental datasets pairs theoptimal ℎ value is calculated as 02057 If ℎ value is givenas 02057 (the optimal value) 0 and 05 respectively theFLR models with the corresponding total credibility aresummarized in Table 6
It is not surprise that the total credibility of the FLRmodelwith the optimal ℎ value achieves the highest among the threeFLRmodels (see Table 6) In fact when the optimal ℎ value isused the total credibility of the FLRmodel will improve 719comparing with ℎ = 0 and 1591 comparing with ℎ = 05
To further investigate the modeling performance of theapproach proposed in this paper we divided the sample datapairs to two groups one group is for modeling that is 80of all the sample data pairs were used to build the FLRmodelthe other group is for testing that is the left 20 of all the
8 Mathematical Problems in Engineering
Table 7 Testing results
Testing data ℎlowast
ℎ = ℎlowast
ℎ = 0 ℎ = 05
Total Total Totalcredibility credibility credibility
9 10 02206 15721 15434 130851 4 02820 19803 15332 114725 6 02018 02841 00000 028742 9 02602 18206 20378 142554 7 02761 10588 12084 106141 2 03066 21009 17199 161282 5 02963 14407 15948 130633 7 02743 18868 12155 145811 5 02647 12111 12000 120006 10 02139 15273 14654 12908
Average 14833 13518 12098
sample data were used to test the performance of the FLRmodel Therefore each time two datasets were randomlyselected from ten datasets as testing datasets while the resteight datasets were used to develop FLRmodels when ℎ valueis specified as the optimal value 0 and 05 respectively Theircorresponding total credibility was calculated The previousprocedures were repeated for ten times Table 7 summarizesthe testing results
From Table 7 it can be seen that the predictive perfor-mance of the FLR model with the optimal ℎ value is the bestamong those of the other models In fact in general whenthe optimal ℎ value is used the total credibility of the FLRmodel will improve 973 comparing with ℎ = 0 and 2261comparing with ℎ = 05Therefore the approach proposed inthis paper is effective
7 Conclusions
In this paper a systematic approach is proposed to selectoptimal value of ℎ for FLR analysis with symmetric TFNsFirstly a new concept of credibility is introduced by theconsideration of system fuzziness and membership degreewhich can be used to assess the performance of FLR analysiswith different ℎ valueswhen a set of sample data pairs is givenSecondly a procedure to obtain optimal ℎ value is formulatedfor FLR with symmetric TFN by maximizing the totalcredibility of FLR models By using the approach proposedin this paper the optimal value of ℎ can be determineddefinitely with respect to a set of sample data pairs Boththe numerical example and real application demonstrate thatthe approach proposed in this paper is effective and efficientThe further work in this direction involves extending theapproach proposed in this paper for FLR analysis with fuzzyoutputs that is developing a procedure to select optimalℎ value for FLR with fuzzy observations and applying theproposed approach to practical problems
Acknowledgments
Thework described in this paper is supported by aGrant fromthe National Nature Science Foundation of Chinese (Projectno NSFC 71272177G020902) and the funds of InnovationProgram of Shanghai Municipal Education Commission(Project no 12ZS101)
References
[1] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982
[2] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978
[3] D Zhang L-F Deng K-Y Cai and A So ldquoFuzzy nonlinearregression with fuzzified radial basis function networkrdquo IEEETransactions on Fuzzy Systems vol 13 no 6 pp 742ndash760 2005
[4] A Bardossy ldquoNote on fuzzy regressionrdquo Fuzzy Sets and Systemsvol 37 no 1 pp 65ndash75 1990
[5] H Tanaka H Ishibuchi and S Yoshikawa ldquoExponential possi-bility regression analysisrdquo Fuzzy Sets and Systems vol 69 no 3pp 305ndash318 1995
[6] K K Yen S Ghoshray and G Roig ldquoA linear regression modelusing triangular fuzzy number coefficientsrdquo Fuzzy Sets andSystems vol 106 no 2 pp 167ndash177 1999
[7] Y Chen J Tang R Y K Fung and Z Ren ldquoFuzzy regression-based mathematical programming model for quality functiondeploymentrdquo International Journal of Production Research vol42 no 5 pp 1009ndash1027 2004
[8] Z B Chen B Lin and Q Fang ldquoStudy on determinationmethod for parameters of Rockrsquos shear strength through leastabsolute linear regression based on symmetric triangular fuzzynumbersrdquo Applied Mechanics and Materials vol 170 pp 804ndash807 2012
[9] R Y K Fung Y Chen and J Tang ldquoEstimating the functionalrelationships for quality function deployment under uncertain-tiesrdquo Fuzzy Sets and Systems vol 157 no 1 pp 98ndash120 2006
[10] B Kheirfam and J-L Verdegay ldquoThe dual simplex method andsensitivity analysis for fuzzy linear programming with sym-metric trapezoidal numbersrdquo Fuzzy Optimization and DecisionMaking vol 12 no 2 pp 171ndash189 2013
[11] PDiamond andHTanaka ldquoFuzzy regression analysisrdquo inFuzzySets in Decision Analysis Operations Research and Statisticsvol 1 of Handbooks of Fuzzy Sets Series pp 349ndash387 KluwerAcademic Boston Mass USA 1998
[12] H Tanaka andH Lee ldquoInterval regression analysis by quadraticprogramming approachrdquo IEEE Transactions on Fuzzy Systemsvol 6 no 4 pp 473ndash481 1998
[13] C Hwang andD H Hong ldquoAnalyzing the nonlinear time seriesof turbulent flows with kernel interval regression machinerdquoInternational Journal of Uncertainty Fuzziness and Knowledge-Based Systems vol 13 no 4 pp 365ndash380 2005
[14] S Yagiz and C Gokceoglu ldquoApplication of fuzzy inferencesystem and nonlinear regression models for predicting rockbrittlenessrdquo Expert Systems with Applications vol 37 no 3 pp2265ndash2272 2010
[15] A Celmins ldquoLeast squares model fitting to fuzzy vector datardquoFuzzy Sets and Systems vol 22 no 3 pp 245ndash269 1987
[16] P Diamond ldquoFuzzy least squaresrdquo Information Sciences vol 46no 3 pp 141ndash157 1988
Mathematical Problems in Engineering 9
[17] Y-H O Chang ldquoHybrid fuzzy least-squares regression analysisand its reliability measuresrdquo Fuzzy Sets and Systems vol 119 no2 pp 225ndash246 2001
[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for emission prediction of acoal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013
[19] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988
[20] H Moskowitz and K Kim ldquoOn assessing the 119867 value in fuzzylinear regressionrdquo Fuzzy Sets and Systems vol 58 no 3 pp 303ndash327 1993
[21] Y-H O Chang and B M Ayyub ldquoFuzzy regression methodsmdasha comparative assessmentrdquo Fuzzy Sets and Systems vol 119 no2 pp 187ndash203 2001
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Complex AnalysisJournal of
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International Journal of
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Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
by minimizing fuzzy distance between the predicted outputsand the observed outputs [15ndash17] Celmins [15] has dealtwith quadratic membership functions based on least squaresfitting with indicators of discord data spread dilator and soforth and Diamond [16] proposed models for least squaresfitting for crisp input fuzzy output and for fuzzy input-outputwhere the distance of fuzzy numbers is defined to measurethe best fit for models Chang [17] formulated fuzzy least-squares regression model by defining the weighted fuzzyarithmetic (WFA) Lv et al proposed a novel least squaressupport vector machine- (LSSVM-) based ensemble learningparadigm to predict NO
119909emission of a coal-fired boiler using
real operation data The result shows that the new soft FCM-LSSVM-PLS ensemble method can predict NO
119909emission
accurately [18]Regarding the first category of FLR methods the value
of ℎ determines the range of the possibility distribution ofthe fuzzy parameters [1 3ndash8 19] so it is important to selecta suitable value for ℎ in FLR analysis Moskowitz and Kim[20] studied the relationship among the ℎ value membershipfunction shape and the spreads of fuzzy parameters inFLR with symmetric fuzzy numbers they also developed ageneral approach to assess the proper ℎ parameter valuesTheir studies showed that the system fuzziness will increasewith the augment of ℎ value And Tanaka and Watada [19]suggested that the selection of the ℎ value should be based onthe sufficiency of the collected dataset When the dataset issufficiently large ℎ = 0 should be used and is increased alongwith the decreasing volume of the collected data Howeverboth did not suggest how to get an optimal ℎ value forthe FLR model when a sample dataset is given In practicalsituations the value of ℎ is usually subjectively preselected bythe decision makers (DMs) [17 18]
In fact if a larger value is given to ℎ the FLR usingtriangular fuzzy coefficients tends to yield large unnecessaryfuzziness and estimated parameters with too large aspirationwhich leads to the fuzzy predictive interval too fuzzier andhas no operational definition or interpretation [8] On theother hand if a smaller ℎ value is used the FLR tends toyield very lower membership degree which leads to the verynarrow fuzzy predictive interval and the reliability of the FLRmodel is doubtable Therefore it is necessary to develop asystematic approach to help DMs determine the optimal ℎvalue for the FLR usingminimum fuzziness criteria To tacklethe problem with the suggestion from Tanaka and Watada[19] a new concept of credibility is introduced tomeasure theperformance of the FLRmodels with different ℎ values in thispaper based on which a systematic approach is formulated tooptimize ℎ values for FLR using minimum fuzziness
The rest of the paper is organized as follows In Section 2Tanakarsquos FLR method is described In Section 3 the conceptof credibility is proposed to measure the performance of FLRmodels with different ℎ values In Section 4 a systematicapproach is formulated to optimize the ℎ value for FLR withsymmetric triangular fuzzy numbers (TFNs) In Section 5 anumerical example is used to show how the optimal value canbe determined using the approach proposed in this paperIn Section 6 a real application is conducted to illustrate
120583A119895(Aj)
A0
1
Sj AC
jAj
Figure 1 Symmetric triangular fuzzy number
the effectiveness of the approach proposed in this paper Theconclusions are drawn in Section 7
2 FLR with Symmetric TFNs
As in a FLR analysis the explained variable is assumedto be a linear combination of the explanatory variablesThis relationship should be obtained from a sample of 119898observations (119910
1 x1) (119910
119894 x119894) (119910
119898 x119898) where 119910
119894is
the 119894th crisp observation and x119894= 1199091198940 1199091198941 119909
119894119895 119909
119894119899
is the 119894th crisp input vector Moreover 1199091198940= 1forall119894 and 119909
119894119895
is the observed value for the 119895th variable in the 119894th case ofthe sample In particular the fuzzy linear function has to beestimated as follows
119910119894= 119891 (x
119894) = 119860
0+ 11986011199091198941sdot sdot sdot 119860119895119909119894119895sdot sdot sdot + 119860
119899119909119894119899=
119899
sum
119895=0
119860119895119909119894119895
(1)
where 119910119894is the fuzzy estimation of 119910
119894 And119860
119895 119895 = 0 1 119899
are fuzzy coefficients in the terms of symmetric TFNs and canbe uniquely defined by 119860
119895= (119886119878
119895 119886119862
119895) 119895 = 0 1 119899 Here
119886119878
119895is the spread value and 119886119862
119895is the centre value of 119860
119895(see
Figure 1)The goal in the fuzzy linear regression is to determine
119891(x119894) by minimizing the system vagueness subject to the
following inclusion condition [1 19]
119910119894isin [119891 (x
119894)]ℎ
119894 = 1 2 119898 (2)
where 119891(x119894)ℎ is the ℎ-level set of the fuzzy output from the
linear fuzzy model 119891(x119894) corresponding to the input vector
x119894 Since ℎ-level set of fuzzy numbers are intervals (2) can be
further given as follows by using Interval arithmetic
119899
sum
119895=0
119886119862
119895119909119894119895minus (1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816
le 119910119894119895le
119899
sum
119895=0
119886119862
119895119909119894119895+ (1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816 119894 = 1 2 119898
(3)
Mathematical Problems in Engineering 3
The system fuzziness in (1) Δ can be given as
Δ =
119898
sum
119894=1
Δ119910119894
(4)
in which Δ119910119894is the fuzziness associated with 119910
119894and can be
given as
Δ119910119894=
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816 (5)
Henceforth according to [1 19]119860119895 119895 = 0 1 119899 in the
form of symmetric TFNs can be determined by solving thefollowing optimal programming model
min Δ =
119898
sum
119894=1
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816 (6a)
subject to
(1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816minus
119899
sum
119895=0
119886119862
119895119909119894119895ge minus119910119894119895 119894 = 1 119898 (6b)
(1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816+
119899
sum
119895=0
119886119862
119895119909119894119895ge 119910119894119895 119894 = 1 119898 (6c)
119886119878
119895ge 0 119895 = 0 1 119899 (6d)
in which the constraints (6b) and (6c) are corresponding toinclusion condition in (2) and the constraint (6d) guaranteesthat the spread values of 119860
119895 119895 = 0 1 119899 are nonnegative
3 Credibility Measure for the FLR Model
As we can see from (6a) to (6d) the value of ℎ determines therange of the possibility distribution of the fuzzy parametersso it is important to select a suitable ℎ value for FLR Todo this the first problem to be solved is how to evaluatethe FLR models with different ℎ values in (1) Thereforea new concept of credibility measurement is introduced inthis section based on which the FLR models associated withdifferent ℎ values can be assessed
Assume that ℎ1and ℎ
2are any two values for ℎ and 0 le
ℎ1lt 1 and 0 le ℎ
2lt 1 And we denote that 1198601
119895and 1198602
119895
119895 = 0 1 119899 are two sets of symmetric TFNs with respecttoℎ1andℎ2 respectively and1199101
119894and1199102119894are the corresponding
estimations of 119910119894from (1) From (1) 1199101
119894and 1199102
119894are calculated
as
1199101
119894=
119899
sum
119895=0
1198601
119895119909119894119895 (7)
1199102
119894=
119899
sum
119895=0
1198602
119895119909119894119895 (8)
Now we are interested in which one out of 1199101119894and 1199102
119894 is
better as the estimation of 119910119894 That is to say which one out of
ℎ1and ℎ2 is better to be used to build a FLR model when the
h
1
0 y
120583yi (yi )
yL1 yC1 yU1 yU2yL2 yC2yi
y1 y2
Figure 2 1205831199101
119894
(119910119894) = 1205831199102
119894
(119910119894)
h
h
1
0y
120583yi (yi )
yL2
1
yC1 yU1 yU2yL1
2
yC2yi
y1 y2
Figure 3 Δ1199101119894= Δ1199102
119894
sample set of data pairs is given To deal with the problemin our opinion two factors that is the estimating fuzzinessΔ119910119894and the membership degree 120583
119910119894
(119910119894) should be taken
into account In general the smaller Δ119910119894is and the higher
120583119910119894
(119910119894) is the better the performance of 119910
119894in representing 119910
119894
will be To illustrate our point of view two special cases areconsidered firstly as follows
(1) If 1205831199101
119894
(119910119894) is equal to 120583
1199102
119894
(119910119894) it is clear that out of 1199101
119894
and1199102119894 the performance of onewith smaller fuzziness
is better than that of the other with larger fuzzinessin representing 119910
119894 As shown in Figure 2 120583
1199101
119894
(119910119894) =
1205831199102
119894
(119910119894) and Δ1199101
119894lt Δ119910
2
119894 therefore 1199101
119894is better than
1199102
119894as the estimation of 119910
119894
(2) IfΔ1199101119894is equal toΔ1199102
119894 it is doubtless that out of1199101
119894and
1199102
119894 the performance of one with higher membership
degree is better than that of the other with lowermembership degree in representing 119910
119894 As shown in
Figure 3 Δ1199101119894= Δ1199102
119894and 120583
1199101
119894
(119910119894) gt 1205831199102
119894
(119910119894) therefore
1199101
119894is better than 1199102
119894as the estimation of 119910
119894
Now let us consider a more general situation that neither1205831199101
119894
(119910119894) is equal to 120583
1199102
119894
(119910119894) nor Δ1199101
119894is equal to Δ1199102
119894 as shown
in Figure 4 To deal with this problem a new concept ofcredibility measure is introduced in this paperWe denote thecredibility of 119910
119894in representing 119910
119894as 119911119894 which is expressed as
119911119894=120583119910119894
(119910119894)
Δ119910119894
(9)
The higher 119911119894is the better the performance of 119910
119894in
representing 119910119894will be Obviously the scenarios of (1) and
(2) are two special cases of (9) respectively
4 Mathematical Problems in Engineering
h
h
1
0y
120583yi (yi )
yL1
1
yC2 yU1 yU2yL2
2
yC1yi
y2 y1
Figure 4 Δ1199101119894= Δ1199102
119894and 120583
1199101
119894
(119910119894) = 1205831199102
119894
(119910119894)
(aS)hlowast2 (aS)h
lowast1 aC
h
h
1
0
1
2
120583A(a)
Figure 5 The case when ℎ2ge ℎ1
As a result based on (9) the total credibility of all sampledata 119911 can be obtained to assess the performance of theobtained FLRmodel in (1) which can be calculated as follows
119911 =
119898
sum
119894=1
119911119894=
119898
sum
119894=1
120583119910119894
(119910119894)
Δ119910119894
(10)
The higher 119911 is the better of the performance of the FLRmodel will be This will help us select the optimal value ofℎ for a FLR model
4 Formulating a Systematic Approach toOptimizing ℎ Value
Based on the concept of credibility measure for FLR modelsintroduced in Section 3 a systematic approach is proposed tooptimize ℎ value for FLR models in this Section
As shown in Figure 5 we denote the optimal solutionof (6a) (6b) (6c) and (6d) with respect to ℎ
1as 119860ℎ
lowast
1
119895=
((119886119878
119895)ℎlowast
1
119886119862
119895) 119895 = 0 1 119899 and Δℎ
lowast
1 and the correspondingfuzziness as Δ119910ℎ
lowast
1
119894 119894 = 1 119898 According to Moskowitz and
Kim [20] the optimal solution of (6a) (6b) (6c) and (6d)and the corresponding fuzziness with regard to ℎ
2can be
obtained as
119860ℎlowast
2
119895= (
1 minus ℎ1
1 minus ℎ2
(119886119878
119895)ℎlowast
1
119886119862
119895) 119895 = 0 1 119899
Δℎlowast
2 =1 minus ℎ1
1 minus ℎ2
Δℎlowast
1
Δ119910ℎlowast
2
119894=1 minus ℎ1
1 minus ℎ2
Δ119910ℎlowast
1
119894 119894 = 1 119898
(11)
Table 1 Testing dataset
119910 119909
14 216 414 618 818 1022 1218 1622 18
This indicates that the central value 119886119862119894119895of each 119860
119894119895will keep
constant when the ℎ value changes from ℎ1to ℎ2 while the
spread value 119886119878119894119895of each 119860
119894119895and the objective function Δ and
the fuzzinessΔ119910119894(119894 = 1 119898) become (1minusℎ
1)(1minusℎ
2) times
simultaneouslyWith respect to ℎ
1 (120583119910119894
(119910119894))ℎ1 119894 = 1 119898 are calculated
as follows
(120583119910119894
(119910119894))ℎ1
= 1 minus
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ1
119894
= 1 minus 119902ℎ1
119894 119894 = 1 119898 (12)
in which 119902ℎ1119894is given as
119902ℎ1
119894=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ1
119894
=
10038161003816100381610038161003816sum119899
119895=0(119886119862
119895) 119909119894119895minus 119910119894
10038161003816100381610038161003816
sum119899
119895=0(119886119878119895)ℎ1 10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816
119894 = 1 119898 (13)
Therefore according to (8) the estimating credibility for allsample data with regard to ℎ
1can be expressed as
119911ℎ1
119894=(120583119910119894
(119910119894))ℎ1
Δ119910ℎ1
119894
=1 minus 119902ℎ1
119894
Δ119910ℎ1
119894
119894 = 1 119898 (14)
Therefore from (9) the total credibility for the FLR modelwith respect to ℎ
1can be obtained as
119911ℎ1 =
119898
sum
119894=1
119911ℎ1
119894=
119898
sum
119894=1
1 minus 119902ℎ1
119894
Δ119910ℎ1
119894
(15)
From (12) with regard to ℎ2 we have
119902ℎ2
119894=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ2
119894
=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
((1 minus ℎ1) (1 minus ℎ
2)) Δ119910ℎ1
119894
=1 minus ℎ2
1 minus ℎ1
119902ℎ1
119894 119894 = 1 119898
(16)
Mathematical Problems in Engineering 5
Table 2 Corresponding results with symmetric triangular fuzzy coefficients (ℎ = 0)
119894 1 2 3 4 5 6 7 8Δ1199100
11989412500 15000 17500 20000 22500 25000 27500 30000
1199020
11989406000 10000 10000 05000 01111 10000 10000 00000
1199010
11989404800 06667 05714 02500 00494 04000 03636 00000
(120583119910119894(119910119894))0
04000 00000 00000 05000 08889 00000 00000 100001199110
11989403200 00000 00000 02500 03951 00000 00000 03333
Therefore according to (11) (120583119910119894
(119910119894))ℎ2 with regard to ℎ
2can
be calculated as
(120583119910119894
(119910119894))ℎ2
= 1 minus 119902ℎ2
119894= 1 minus
1 minus ℎ2
1 minus ℎ1
119902ℎ1
119894
= 1 minus 119902ℎ1
119894+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894
= (120583119910119894
(119910119894))ℎ1
+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894 119894 = 1 119898
(17)
Henceforth according to (8) (10) and (16) the estimatingcredibility for all sample data pairs with regard to ℎ
2can be
expressed as
119911ℎ2
119894=(120583119910119894
(119910119894))ℎ2
Δ119910ℎ2
119894
=(120583119910119894
(119910119894))ℎ1
+ ((ℎ2minus ℎ1) (1 minus ℎ
1)) 119902ℎ1
119894
((1 minus ℎ1) (1 minus ℎ
2)) Δ119910ℎ1
119894
=1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894 119894 = 1 119898
(18)
in which
119901ℎ1
119894=
119902ℎ1
119894
Δ119910ℎ1
119894
=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
(Δ119910ℎ1
119894)2 119894 = 1 119898 (19)
Consequently the estimating credibility for the FLRmodel in(1) with regard to ℎ
2can be obtained as
119911ℎ2 =
119898
sum
119894=1
119911ℎ2
119894=
119898
sum
119894=1
(1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894)
=1 minus ℎ2
1 minus ℎ1
119911ℎ1 +
(1 minus ℎ2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
(20)
in which
119901ℎ1 =
119898
sum
119894=1
119901ℎ1
119894 (21)
For similarity we denote ℎ1= 0 and ℎ
2= ℎ (11) (16) (17)
(18) and (20) can be rewritten as
119860ℎ
119895= (
1
1 minus ℎ(119886119878
119895)0
119886119862
119895) 119895 = 0 1 119899
Δℎ
=1
1 minus ℎΔ0
Δ119910ℎ
119894=
1
1 minus ℎΔ1199100
119894 119894 = 1 119898
(22)
119902ℎ
119894= (1 minus ℎ) 119902
0
119894 (23)
(120583119910119894
(119910119894))ℎ
= (120583119910119894
(119910119894))0
+ ℎ1199020
119894 (24)
119911ℎ
119894= minus1199010
119894ℎ2
+ (1199010
119894minus 1199110
119894) ℎ + 119911
0
119894 (25)
119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ + 1199110
(26)
Based on (26) the optimal ℎ value for the FLR model in(1) can be obtained by maximizing the estimating credibilitythat is the optimal value for ℎ can be obtained by solving thefollowing programming model
max 119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ (27a)
st 0 le ℎ lt 1 (27b)
It is obviously that the programming model in (27a) to (27b)is quadratic and many kinds of optimization algorithmssuch as the gradient descent method can be used to solve theprevious nonlinear programming problem
5 Numerical Example
The numerical example in [21] is used to how the optimal ℎvalue can be obtained using the approach proposed in thispaper The eight testing data pairs are listed in Table 1
When ℎ is specified as 0 the fuzzy coefficients interms of symmetric TFNs can be obtained by solving theprogramming model in (6a) (6b) (6c) and (6d) as 1198600
0=
(10000 120000) and 1198600
1= (01250 06250) And the
corresponding FLR model is given as
1199100
= (10000 120000) + (01250 06250) 119909 (28)
Accordingly the fuzziness Δ1199100119894 parameters 1199020
119894and 1199010
119894 mem-
bership degree (120583119910119894
(119910119894))0 and credibility 1199110
119894 119894 = 1 119898 are
calculated respectively as shown in Table 2
6 Mathematical Problems in Engineering
Table 3 Testing data set
119910 1199091
1199092
13360 2950 799913763 3130 756314786 3760 692519676 3990 627522053 3990 646622325 4030 630923319 4150 615126567 4360 600733516 4570 582241129 4780 584346068 4950 605747796 5010 582347402 5020 580346680 4990 575346616 5000 556846980 5000 552446895 5000 545147624 5090 500849939 5310 500552120 5520 4972
From Table 2 1199110 is calculated as 12984 and 1199010 as 27811According to (26) the total credibility of the FLRmodel withrespect to ℎ can be expressed as follows
119911ℎ
= minus27811ℎ2
+ 14827ℎ + 12984 (29)
According to (27a) and (27b) the optimal value ℎlowast is given as02666 and the optimal coefficients in the form of symmetriccase are obtained as 119860lowast
0= (13635 120000) and 119860
lowast
1=
(01704 06250) Therefore the optimal FLR model is givenas
119910lowast
= (13635 120000) + (01704 06250) 119909 (30)
The total credibility of the model in (30) 119911lowast is calculated as14960 which is higher than that of the model in (28) that is12984
The changes of the fuzzinessΔ119910ℎ119894 themembership degree
(120583119910119894
(119910119894))ℎ the credibility 119911
ℎ
119894 119894 = 1 119898 and the total
credibility 119911ℎ with different ℎ values are depicted in Figures6 7 8 and 9 respectively Notably the vertical axis islogarithmic scale in Figure 6 to enable us to see small changesin fuzziness among eight sample data and the membershipdegrees for sample data 2 3 6 and 7 are overlapped inFigure 7
From Figure 6 we can see that the fuzziness of all sampledata become larger with the augment of ℎ value especiallywhen ℎ value is close to 1 the fuzziness will be extremelylarger From Figure 7 it can be found that the membershipdegree of all sample data becomes higher with the increase ofℎ value and the membership degree will be close to 1 whenℎ value is close to 1 It is clearly shown in Figure 8 that when
0 02 04 06 08 1h value
Fuzz
ines
s
Data1Data2Data3Data4
Data5Data6Data7Data8
103
102
101
100
Figure 6 Fuzziness of the eight sample data
0 02 04 06 08 10
02
04
06
08
1
12
h value
Mem
bers
hip
degr
ee
Data1Data2Data3Data4
Data5Data6Data7Data8
Figure 7 Membership degree of the eight sample data
ℎ value is close to 1 the credibility measures for all sampledata will be close to zero due to the extremely larger of thefuzziness (see Figure 7) Figure 9 shows that when ℎ valueis 02666 the total credibility of the FLR model will achievethe maximum based on which the best FLR model can beobtained
To further demonstrate the effectiveness of the approachproposed in this paper another numerical example with twoindependent variables is given as follows The twenty testingdatasets are listed in Table 3
According to (27a) and (27b) the optimal value ℎlowast isgiven as 03184 When ℎ is specified as 0 03184 and 05respectively the fuzzy coefficients in terms of symmetricTFNs and the corresponding total credibility can be obtained
Mathematical Problems in Engineering 7
Table 4 The corresponding results
h value Fuzzy linear model Total credibility03148 119910
lowast
= (minus6108251 00000) + (191484 00000) 1199091+ (14018 12301) 119909
201371
00 1199100
= (minus6108233 00000) + (191484 00000) 1199091+ (14017 08429) 119909
201082
05 11991005
= (minus6108240 00000) + (191484 00000) 1199091+ (14017 16857) 119909
201271
Data1Data2Data3Data4
Data5Data6Data7Data8
0 02 04 06 08 10
005
01
015
0202
h value
Cred
ibili
ty m
easu
re
Figure 8 Credibility of the eight sample data
0 02 04 06 08 10
05
1
15
h value
Tota
l cre
dibi
lity
Figure 9 Total credibility of the FLR models
by solving the programming model in (6a) (6b) (6c) and(6d) The results are summarized in Table 4
From Table 4 we can see that when ℎ value is set to03148 the total credibility of the FLR model will achieve themaximum that is 01371 which indicates that the approachproposed in this paper is effective and reasonable
Table 5 The experiment results
Run no 119909 (gcm3) 119910 (horsepower)1 0760 512 0780 623 0820 754 0815 705 0790 716 0785 687 0770 598 0765 569 0788 9010 0769 59
Table 6 The FLR models with the corresponding total credibility
h value Fuzzy linear model Totalcredibility
02057 119910 = (minus256000 08765) + (406962 00000)119909 727850 119910 = (minus256000 06962) + (406962 00000)119909 6790305 119910 = (minus256000 13924) + (406962 00000)119909 62793
6 Real Application
To investigate the effectiveness of the approach proposed inthis paper modeling welding process for electronic manu-facturing using fuzzy linear regression was studied An elec-tronic company is a famous OEM company of printed circuitboard (PCB) in PR China The engineers in this companywanted to improve the welding quality by investigating therelationship between the pull strength (119910) of welding line andthe proportion of colophony (119909) in welding fluid And anengineering experiment was conducted and the experimentresults are shown in Table 5
With the use of the 10 experimental datasets pairs theoptimal ℎ value is calculated as 02057 If ℎ value is givenas 02057 (the optimal value) 0 and 05 respectively theFLR models with the corresponding total credibility aresummarized in Table 6
It is not surprise that the total credibility of the FLRmodelwith the optimal ℎ value achieves the highest among the threeFLRmodels (see Table 6) In fact when the optimal ℎ value isused the total credibility of the FLRmodel will improve 719comparing with ℎ = 0 and 1591 comparing with ℎ = 05
To further investigate the modeling performance of theapproach proposed in this paper we divided the sample datapairs to two groups one group is for modeling that is 80of all the sample data pairs were used to build the FLRmodelthe other group is for testing that is the left 20 of all the
8 Mathematical Problems in Engineering
Table 7 Testing results
Testing data ℎlowast
ℎ = ℎlowast
ℎ = 0 ℎ = 05
Total Total Totalcredibility credibility credibility
9 10 02206 15721 15434 130851 4 02820 19803 15332 114725 6 02018 02841 00000 028742 9 02602 18206 20378 142554 7 02761 10588 12084 106141 2 03066 21009 17199 161282 5 02963 14407 15948 130633 7 02743 18868 12155 145811 5 02647 12111 12000 120006 10 02139 15273 14654 12908
Average 14833 13518 12098
sample data were used to test the performance of the FLRmodel Therefore each time two datasets were randomlyselected from ten datasets as testing datasets while the resteight datasets were used to develop FLRmodels when ℎ valueis specified as the optimal value 0 and 05 respectively Theircorresponding total credibility was calculated The previousprocedures were repeated for ten times Table 7 summarizesthe testing results
From Table 7 it can be seen that the predictive perfor-mance of the FLR model with the optimal ℎ value is the bestamong those of the other models In fact in general whenthe optimal ℎ value is used the total credibility of the FLRmodel will improve 973 comparing with ℎ = 0 and 2261comparing with ℎ = 05Therefore the approach proposed inthis paper is effective
7 Conclusions
In this paper a systematic approach is proposed to selectoptimal value of ℎ for FLR analysis with symmetric TFNsFirstly a new concept of credibility is introduced by theconsideration of system fuzziness and membership degreewhich can be used to assess the performance of FLR analysiswith different ℎ valueswhen a set of sample data pairs is givenSecondly a procedure to obtain optimal ℎ value is formulatedfor FLR with symmetric TFN by maximizing the totalcredibility of FLR models By using the approach proposedin this paper the optimal value of ℎ can be determineddefinitely with respect to a set of sample data pairs Boththe numerical example and real application demonstrate thatthe approach proposed in this paper is effective and efficientThe further work in this direction involves extending theapproach proposed in this paper for FLR analysis with fuzzyoutputs that is developing a procedure to select optimalℎ value for FLR with fuzzy observations and applying theproposed approach to practical problems
Acknowledgments
Thework described in this paper is supported by aGrant fromthe National Nature Science Foundation of Chinese (Projectno NSFC 71272177G020902) and the funds of InnovationProgram of Shanghai Municipal Education Commission(Project no 12ZS101)
References
[1] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982
[2] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978
[3] D Zhang L-F Deng K-Y Cai and A So ldquoFuzzy nonlinearregression with fuzzified radial basis function networkrdquo IEEETransactions on Fuzzy Systems vol 13 no 6 pp 742ndash760 2005
[4] A Bardossy ldquoNote on fuzzy regressionrdquo Fuzzy Sets and Systemsvol 37 no 1 pp 65ndash75 1990
[5] H Tanaka H Ishibuchi and S Yoshikawa ldquoExponential possi-bility regression analysisrdquo Fuzzy Sets and Systems vol 69 no 3pp 305ndash318 1995
[6] K K Yen S Ghoshray and G Roig ldquoA linear regression modelusing triangular fuzzy number coefficientsrdquo Fuzzy Sets andSystems vol 106 no 2 pp 167ndash177 1999
[7] Y Chen J Tang R Y K Fung and Z Ren ldquoFuzzy regression-based mathematical programming model for quality functiondeploymentrdquo International Journal of Production Research vol42 no 5 pp 1009ndash1027 2004
[8] Z B Chen B Lin and Q Fang ldquoStudy on determinationmethod for parameters of Rockrsquos shear strength through leastabsolute linear regression based on symmetric triangular fuzzynumbersrdquo Applied Mechanics and Materials vol 170 pp 804ndash807 2012
[9] R Y K Fung Y Chen and J Tang ldquoEstimating the functionalrelationships for quality function deployment under uncertain-tiesrdquo Fuzzy Sets and Systems vol 157 no 1 pp 98ndash120 2006
[10] B Kheirfam and J-L Verdegay ldquoThe dual simplex method andsensitivity analysis for fuzzy linear programming with sym-metric trapezoidal numbersrdquo Fuzzy Optimization and DecisionMaking vol 12 no 2 pp 171ndash189 2013
[11] PDiamond andHTanaka ldquoFuzzy regression analysisrdquo inFuzzySets in Decision Analysis Operations Research and Statisticsvol 1 of Handbooks of Fuzzy Sets Series pp 349ndash387 KluwerAcademic Boston Mass USA 1998
[12] H Tanaka andH Lee ldquoInterval regression analysis by quadraticprogramming approachrdquo IEEE Transactions on Fuzzy Systemsvol 6 no 4 pp 473ndash481 1998
[13] C Hwang andD H Hong ldquoAnalyzing the nonlinear time seriesof turbulent flows with kernel interval regression machinerdquoInternational Journal of Uncertainty Fuzziness and Knowledge-Based Systems vol 13 no 4 pp 365ndash380 2005
[14] S Yagiz and C Gokceoglu ldquoApplication of fuzzy inferencesystem and nonlinear regression models for predicting rockbrittlenessrdquo Expert Systems with Applications vol 37 no 3 pp2265ndash2272 2010
[15] A Celmins ldquoLeast squares model fitting to fuzzy vector datardquoFuzzy Sets and Systems vol 22 no 3 pp 245ndash269 1987
[16] P Diamond ldquoFuzzy least squaresrdquo Information Sciences vol 46no 3 pp 141ndash157 1988
Mathematical Problems in Engineering 9
[17] Y-H O Chang ldquoHybrid fuzzy least-squares regression analysisand its reliability measuresrdquo Fuzzy Sets and Systems vol 119 no2 pp 225ndash246 2001
[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for emission prediction of acoal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013
[19] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988
[20] H Moskowitz and K Kim ldquoOn assessing the 119867 value in fuzzylinear regressionrdquo Fuzzy Sets and Systems vol 58 no 3 pp 303ndash327 1993
[21] Y-H O Chang and B M Ayyub ldquoFuzzy regression methodsmdasha comparative assessmentrdquo Fuzzy Sets and Systems vol 119 no2 pp 187ndash203 2001
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
The system fuzziness in (1) Δ can be given as
Δ =
119898
sum
119894=1
Δ119910119894
(4)
in which Δ119910119894is the fuzziness associated with 119910
119894and can be
given as
Δ119910119894=
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816 (5)
Henceforth according to [1 19]119860119895 119895 = 0 1 119899 in the
form of symmetric TFNs can be determined by solving thefollowing optimal programming model
min Δ =
119898
sum
119894=1
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816 (6a)
subject to
(1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816minus
119899
sum
119895=0
119886119862
119895119909119894119895ge minus119910119894119895 119894 = 1 119898 (6b)
(1 minus ℎ)
119899
sum
119895=0
119886119878
119895
10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816+
119899
sum
119895=0
119886119862
119895119909119894119895ge 119910119894119895 119894 = 1 119898 (6c)
119886119878
119895ge 0 119895 = 0 1 119899 (6d)
in which the constraints (6b) and (6c) are corresponding toinclusion condition in (2) and the constraint (6d) guaranteesthat the spread values of 119860
119895 119895 = 0 1 119899 are nonnegative
3 Credibility Measure for the FLR Model
As we can see from (6a) to (6d) the value of ℎ determines therange of the possibility distribution of the fuzzy parametersso it is important to select a suitable ℎ value for FLR Todo this the first problem to be solved is how to evaluatethe FLR models with different ℎ values in (1) Thereforea new concept of credibility measurement is introduced inthis section based on which the FLR models associated withdifferent ℎ values can be assessed
Assume that ℎ1and ℎ
2are any two values for ℎ and 0 le
ℎ1lt 1 and 0 le ℎ
2lt 1 And we denote that 1198601
119895and 1198602
119895
119895 = 0 1 119899 are two sets of symmetric TFNs with respecttoℎ1andℎ2 respectively and1199101
119894and1199102119894are the corresponding
estimations of 119910119894from (1) From (1) 1199101
119894and 1199102
119894are calculated
as
1199101
119894=
119899
sum
119895=0
1198601
119895119909119894119895 (7)
1199102
119894=
119899
sum
119895=0
1198602
119895119909119894119895 (8)
Now we are interested in which one out of 1199101119894and 1199102
119894 is
better as the estimation of 119910119894 That is to say which one out of
ℎ1and ℎ2 is better to be used to build a FLR model when the
h
1
0 y
120583yi (yi )
yL1 yC1 yU1 yU2yL2 yC2yi
y1 y2
Figure 2 1205831199101
119894
(119910119894) = 1205831199102
119894
(119910119894)
h
h
1
0y
120583yi (yi )
yL2
1
yC1 yU1 yU2yL1
2
yC2yi
y1 y2
Figure 3 Δ1199101119894= Δ1199102
119894
sample set of data pairs is given To deal with the problemin our opinion two factors that is the estimating fuzzinessΔ119910119894and the membership degree 120583
119910119894
(119910119894) should be taken
into account In general the smaller Δ119910119894is and the higher
120583119910119894
(119910119894) is the better the performance of 119910
119894in representing 119910
119894
will be To illustrate our point of view two special cases areconsidered firstly as follows
(1) If 1205831199101
119894
(119910119894) is equal to 120583
1199102
119894
(119910119894) it is clear that out of 1199101
119894
and1199102119894 the performance of onewith smaller fuzziness
is better than that of the other with larger fuzzinessin representing 119910
119894 As shown in Figure 2 120583
1199101
119894
(119910119894) =
1205831199102
119894
(119910119894) and Δ1199101
119894lt Δ119910
2
119894 therefore 1199101
119894is better than
1199102
119894as the estimation of 119910
119894
(2) IfΔ1199101119894is equal toΔ1199102
119894 it is doubtless that out of1199101
119894and
1199102
119894 the performance of one with higher membership
degree is better than that of the other with lowermembership degree in representing 119910
119894 As shown in
Figure 3 Δ1199101119894= Δ1199102
119894and 120583
1199101
119894
(119910119894) gt 1205831199102
119894
(119910119894) therefore
1199101
119894is better than 1199102
119894as the estimation of 119910
119894
Now let us consider a more general situation that neither1205831199101
119894
(119910119894) is equal to 120583
1199102
119894
(119910119894) nor Δ1199101
119894is equal to Δ1199102
119894 as shown
in Figure 4 To deal with this problem a new concept ofcredibility measure is introduced in this paperWe denote thecredibility of 119910
119894in representing 119910
119894as 119911119894 which is expressed as
119911119894=120583119910119894
(119910119894)
Δ119910119894
(9)
The higher 119911119894is the better the performance of 119910
119894in
representing 119910119894will be Obviously the scenarios of (1) and
(2) are two special cases of (9) respectively
4 Mathematical Problems in Engineering
h
h
1
0y
120583yi (yi )
yL1
1
yC2 yU1 yU2yL2
2
yC1yi
y2 y1
Figure 4 Δ1199101119894= Δ1199102
119894and 120583
1199101
119894
(119910119894) = 1205831199102
119894
(119910119894)
(aS)hlowast2 (aS)h
lowast1 aC
h
h
1
0
1
2
120583A(a)
Figure 5 The case when ℎ2ge ℎ1
As a result based on (9) the total credibility of all sampledata 119911 can be obtained to assess the performance of theobtained FLRmodel in (1) which can be calculated as follows
119911 =
119898
sum
119894=1
119911119894=
119898
sum
119894=1
120583119910119894
(119910119894)
Δ119910119894
(10)
The higher 119911 is the better of the performance of the FLRmodel will be This will help us select the optimal value ofℎ for a FLR model
4 Formulating a Systematic Approach toOptimizing ℎ Value
Based on the concept of credibility measure for FLR modelsintroduced in Section 3 a systematic approach is proposed tooptimize ℎ value for FLR models in this Section
As shown in Figure 5 we denote the optimal solutionof (6a) (6b) (6c) and (6d) with respect to ℎ
1as 119860ℎ
lowast
1
119895=
((119886119878
119895)ℎlowast
1
119886119862
119895) 119895 = 0 1 119899 and Δℎ
lowast
1 and the correspondingfuzziness as Δ119910ℎ
lowast
1
119894 119894 = 1 119898 According to Moskowitz and
Kim [20] the optimal solution of (6a) (6b) (6c) and (6d)and the corresponding fuzziness with regard to ℎ
2can be
obtained as
119860ℎlowast
2
119895= (
1 minus ℎ1
1 minus ℎ2
(119886119878
119895)ℎlowast
1
119886119862
119895) 119895 = 0 1 119899
Δℎlowast
2 =1 minus ℎ1
1 minus ℎ2
Δℎlowast
1
Δ119910ℎlowast
2
119894=1 minus ℎ1
1 minus ℎ2
Δ119910ℎlowast
1
119894 119894 = 1 119898
(11)
Table 1 Testing dataset
119910 119909
14 216 414 618 818 1022 1218 1622 18
This indicates that the central value 119886119862119894119895of each 119860
119894119895will keep
constant when the ℎ value changes from ℎ1to ℎ2 while the
spread value 119886119878119894119895of each 119860
119894119895and the objective function Δ and
the fuzzinessΔ119910119894(119894 = 1 119898) become (1minusℎ
1)(1minusℎ
2) times
simultaneouslyWith respect to ℎ
1 (120583119910119894
(119910119894))ℎ1 119894 = 1 119898 are calculated
as follows
(120583119910119894
(119910119894))ℎ1
= 1 minus
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ1
119894
= 1 minus 119902ℎ1
119894 119894 = 1 119898 (12)
in which 119902ℎ1119894is given as
119902ℎ1
119894=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ1
119894
=
10038161003816100381610038161003816sum119899
119895=0(119886119862
119895) 119909119894119895minus 119910119894
10038161003816100381610038161003816
sum119899
119895=0(119886119878119895)ℎ1 10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816
119894 = 1 119898 (13)
Therefore according to (8) the estimating credibility for allsample data with regard to ℎ
1can be expressed as
119911ℎ1
119894=(120583119910119894
(119910119894))ℎ1
Δ119910ℎ1
119894
=1 minus 119902ℎ1
119894
Δ119910ℎ1
119894
119894 = 1 119898 (14)
Therefore from (9) the total credibility for the FLR modelwith respect to ℎ
1can be obtained as
119911ℎ1 =
119898
sum
119894=1
119911ℎ1
119894=
119898
sum
119894=1
1 minus 119902ℎ1
119894
Δ119910ℎ1
119894
(15)
From (12) with regard to ℎ2 we have
119902ℎ2
119894=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ2
119894
=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
((1 minus ℎ1) (1 minus ℎ
2)) Δ119910ℎ1
119894
=1 minus ℎ2
1 minus ℎ1
119902ℎ1
119894 119894 = 1 119898
(16)
Mathematical Problems in Engineering 5
Table 2 Corresponding results with symmetric triangular fuzzy coefficients (ℎ = 0)
119894 1 2 3 4 5 6 7 8Δ1199100
11989412500 15000 17500 20000 22500 25000 27500 30000
1199020
11989406000 10000 10000 05000 01111 10000 10000 00000
1199010
11989404800 06667 05714 02500 00494 04000 03636 00000
(120583119910119894(119910119894))0
04000 00000 00000 05000 08889 00000 00000 100001199110
11989403200 00000 00000 02500 03951 00000 00000 03333
Therefore according to (11) (120583119910119894
(119910119894))ℎ2 with regard to ℎ
2can
be calculated as
(120583119910119894
(119910119894))ℎ2
= 1 minus 119902ℎ2
119894= 1 minus
1 minus ℎ2
1 minus ℎ1
119902ℎ1
119894
= 1 minus 119902ℎ1
119894+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894
= (120583119910119894
(119910119894))ℎ1
+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894 119894 = 1 119898
(17)
Henceforth according to (8) (10) and (16) the estimatingcredibility for all sample data pairs with regard to ℎ
2can be
expressed as
119911ℎ2
119894=(120583119910119894
(119910119894))ℎ2
Δ119910ℎ2
119894
=(120583119910119894
(119910119894))ℎ1
+ ((ℎ2minus ℎ1) (1 minus ℎ
1)) 119902ℎ1
119894
((1 minus ℎ1) (1 minus ℎ
2)) Δ119910ℎ1
119894
=1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894 119894 = 1 119898
(18)
in which
119901ℎ1
119894=
119902ℎ1
119894
Δ119910ℎ1
119894
=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
(Δ119910ℎ1
119894)2 119894 = 1 119898 (19)
Consequently the estimating credibility for the FLRmodel in(1) with regard to ℎ
2can be obtained as
119911ℎ2 =
119898
sum
119894=1
119911ℎ2
119894=
119898
sum
119894=1
(1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894)
=1 minus ℎ2
1 minus ℎ1
119911ℎ1 +
(1 minus ℎ2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
(20)
in which
119901ℎ1 =
119898
sum
119894=1
119901ℎ1
119894 (21)
For similarity we denote ℎ1= 0 and ℎ
2= ℎ (11) (16) (17)
(18) and (20) can be rewritten as
119860ℎ
119895= (
1
1 minus ℎ(119886119878
119895)0
119886119862
119895) 119895 = 0 1 119899
Δℎ
=1
1 minus ℎΔ0
Δ119910ℎ
119894=
1
1 minus ℎΔ1199100
119894 119894 = 1 119898
(22)
119902ℎ
119894= (1 minus ℎ) 119902
0
119894 (23)
(120583119910119894
(119910119894))ℎ
= (120583119910119894
(119910119894))0
+ ℎ1199020
119894 (24)
119911ℎ
119894= minus1199010
119894ℎ2
+ (1199010
119894minus 1199110
119894) ℎ + 119911
0
119894 (25)
119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ + 1199110
(26)
Based on (26) the optimal ℎ value for the FLR model in(1) can be obtained by maximizing the estimating credibilitythat is the optimal value for ℎ can be obtained by solving thefollowing programming model
max 119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ (27a)
st 0 le ℎ lt 1 (27b)
It is obviously that the programming model in (27a) to (27b)is quadratic and many kinds of optimization algorithmssuch as the gradient descent method can be used to solve theprevious nonlinear programming problem
5 Numerical Example
The numerical example in [21] is used to how the optimal ℎvalue can be obtained using the approach proposed in thispaper The eight testing data pairs are listed in Table 1
When ℎ is specified as 0 the fuzzy coefficients interms of symmetric TFNs can be obtained by solving theprogramming model in (6a) (6b) (6c) and (6d) as 1198600
0=
(10000 120000) and 1198600
1= (01250 06250) And the
corresponding FLR model is given as
1199100
= (10000 120000) + (01250 06250) 119909 (28)
Accordingly the fuzziness Δ1199100119894 parameters 1199020
119894and 1199010
119894 mem-
bership degree (120583119910119894
(119910119894))0 and credibility 1199110
119894 119894 = 1 119898 are
calculated respectively as shown in Table 2
6 Mathematical Problems in Engineering
Table 3 Testing data set
119910 1199091
1199092
13360 2950 799913763 3130 756314786 3760 692519676 3990 627522053 3990 646622325 4030 630923319 4150 615126567 4360 600733516 4570 582241129 4780 584346068 4950 605747796 5010 582347402 5020 580346680 4990 575346616 5000 556846980 5000 552446895 5000 545147624 5090 500849939 5310 500552120 5520 4972
From Table 2 1199110 is calculated as 12984 and 1199010 as 27811According to (26) the total credibility of the FLRmodel withrespect to ℎ can be expressed as follows
119911ℎ
= minus27811ℎ2
+ 14827ℎ + 12984 (29)
According to (27a) and (27b) the optimal value ℎlowast is given as02666 and the optimal coefficients in the form of symmetriccase are obtained as 119860lowast
0= (13635 120000) and 119860
lowast
1=
(01704 06250) Therefore the optimal FLR model is givenas
119910lowast
= (13635 120000) + (01704 06250) 119909 (30)
The total credibility of the model in (30) 119911lowast is calculated as14960 which is higher than that of the model in (28) that is12984
The changes of the fuzzinessΔ119910ℎ119894 themembership degree
(120583119910119894
(119910119894))ℎ the credibility 119911
ℎ
119894 119894 = 1 119898 and the total
credibility 119911ℎ with different ℎ values are depicted in Figures6 7 8 and 9 respectively Notably the vertical axis islogarithmic scale in Figure 6 to enable us to see small changesin fuzziness among eight sample data and the membershipdegrees for sample data 2 3 6 and 7 are overlapped inFigure 7
From Figure 6 we can see that the fuzziness of all sampledata become larger with the augment of ℎ value especiallywhen ℎ value is close to 1 the fuzziness will be extremelylarger From Figure 7 it can be found that the membershipdegree of all sample data becomes higher with the increase ofℎ value and the membership degree will be close to 1 whenℎ value is close to 1 It is clearly shown in Figure 8 that when
0 02 04 06 08 1h value
Fuzz
ines
s
Data1Data2Data3Data4
Data5Data6Data7Data8
103
102
101
100
Figure 6 Fuzziness of the eight sample data
0 02 04 06 08 10
02
04
06
08
1
12
h value
Mem
bers
hip
degr
ee
Data1Data2Data3Data4
Data5Data6Data7Data8
Figure 7 Membership degree of the eight sample data
ℎ value is close to 1 the credibility measures for all sampledata will be close to zero due to the extremely larger of thefuzziness (see Figure 7) Figure 9 shows that when ℎ valueis 02666 the total credibility of the FLR model will achievethe maximum based on which the best FLR model can beobtained
To further demonstrate the effectiveness of the approachproposed in this paper another numerical example with twoindependent variables is given as follows The twenty testingdatasets are listed in Table 3
According to (27a) and (27b) the optimal value ℎlowast isgiven as 03184 When ℎ is specified as 0 03184 and 05respectively the fuzzy coefficients in terms of symmetricTFNs and the corresponding total credibility can be obtained
Mathematical Problems in Engineering 7
Table 4 The corresponding results
h value Fuzzy linear model Total credibility03148 119910
lowast
= (minus6108251 00000) + (191484 00000) 1199091+ (14018 12301) 119909
201371
00 1199100
= (minus6108233 00000) + (191484 00000) 1199091+ (14017 08429) 119909
201082
05 11991005
= (minus6108240 00000) + (191484 00000) 1199091+ (14017 16857) 119909
201271
Data1Data2Data3Data4
Data5Data6Data7Data8
0 02 04 06 08 10
005
01
015
0202
h value
Cred
ibili
ty m
easu
re
Figure 8 Credibility of the eight sample data
0 02 04 06 08 10
05
1
15
h value
Tota
l cre
dibi
lity
Figure 9 Total credibility of the FLR models
by solving the programming model in (6a) (6b) (6c) and(6d) The results are summarized in Table 4
From Table 4 we can see that when ℎ value is set to03148 the total credibility of the FLR model will achieve themaximum that is 01371 which indicates that the approachproposed in this paper is effective and reasonable
Table 5 The experiment results
Run no 119909 (gcm3) 119910 (horsepower)1 0760 512 0780 623 0820 754 0815 705 0790 716 0785 687 0770 598 0765 569 0788 9010 0769 59
Table 6 The FLR models with the corresponding total credibility
h value Fuzzy linear model Totalcredibility
02057 119910 = (minus256000 08765) + (406962 00000)119909 727850 119910 = (minus256000 06962) + (406962 00000)119909 6790305 119910 = (minus256000 13924) + (406962 00000)119909 62793
6 Real Application
To investigate the effectiveness of the approach proposed inthis paper modeling welding process for electronic manu-facturing using fuzzy linear regression was studied An elec-tronic company is a famous OEM company of printed circuitboard (PCB) in PR China The engineers in this companywanted to improve the welding quality by investigating therelationship between the pull strength (119910) of welding line andthe proportion of colophony (119909) in welding fluid And anengineering experiment was conducted and the experimentresults are shown in Table 5
With the use of the 10 experimental datasets pairs theoptimal ℎ value is calculated as 02057 If ℎ value is givenas 02057 (the optimal value) 0 and 05 respectively theFLR models with the corresponding total credibility aresummarized in Table 6
It is not surprise that the total credibility of the FLRmodelwith the optimal ℎ value achieves the highest among the threeFLRmodels (see Table 6) In fact when the optimal ℎ value isused the total credibility of the FLRmodel will improve 719comparing with ℎ = 0 and 1591 comparing with ℎ = 05
To further investigate the modeling performance of theapproach proposed in this paper we divided the sample datapairs to two groups one group is for modeling that is 80of all the sample data pairs were used to build the FLRmodelthe other group is for testing that is the left 20 of all the
8 Mathematical Problems in Engineering
Table 7 Testing results
Testing data ℎlowast
ℎ = ℎlowast
ℎ = 0 ℎ = 05
Total Total Totalcredibility credibility credibility
9 10 02206 15721 15434 130851 4 02820 19803 15332 114725 6 02018 02841 00000 028742 9 02602 18206 20378 142554 7 02761 10588 12084 106141 2 03066 21009 17199 161282 5 02963 14407 15948 130633 7 02743 18868 12155 145811 5 02647 12111 12000 120006 10 02139 15273 14654 12908
Average 14833 13518 12098
sample data were used to test the performance of the FLRmodel Therefore each time two datasets were randomlyselected from ten datasets as testing datasets while the resteight datasets were used to develop FLRmodels when ℎ valueis specified as the optimal value 0 and 05 respectively Theircorresponding total credibility was calculated The previousprocedures were repeated for ten times Table 7 summarizesthe testing results
From Table 7 it can be seen that the predictive perfor-mance of the FLR model with the optimal ℎ value is the bestamong those of the other models In fact in general whenthe optimal ℎ value is used the total credibility of the FLRmodel will improve 973 comparing with ℎ = 0 and 2261comparing with ℎ = 05Therefore the approach proposed inthis paper is effective
7 Conclusions
In this paper a systematic approach is proposed to selectoptimal value of ℎ for FLR analysis with symmetric TFNsFirstly a new concept of credibility is introduced by theconsideration of system fuzziness and membership degreewhich can be used to assess the performance of FLR analysiswith different ℎ valueswhen a set of sample data pairs is givenSecondly a procedure to obtain optimal ℎ value is formulatedfor FLR with symmetric TFN by maximizing the totalcredibility of FLR models By using the approach proposedin this paper the optimal value of ℎ can be determineddefinitely with respect to a set of sample data pairs Boththe numerical example and real application demonstrate thatthe approach proposed in this paper is effective and efficientThe further work in this direction involves extending theapproach proposed in this paper for FLR analysis with fuzzyoutputs that is developing a procedure to select optimalℎ value for FLR with fuzzy observations and applying theproposed approach to practical problems
Acknowledgments
Thework described in this paper is supported by aGrant fromthe National Nature Science Foundation of Chinese (Projectno NSFC 71272177G020902) and the funds of InnovationProgram of Shanghai Municipal Education Commission(Project no 12ZS101)
References
[1] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982
[2] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978
[3] D Zhang L-F Deng K-Y Cai and A So ldquoFuzzy nonlinearregression with fuzzified radial basis function networkrdquo IEEETransactions on Fuzzy Systems vol 13 no 6 pp 742ndash760 2005
[4] A Bardossy ldquoNote on fuzzy regressionrdquo Fuzzy Sets and Systemsvol 37 no 1 pp 65ndash75 1990
[5] H Tanaka H Ishibuchi and S Yoshikawa ldquoExponential possi-bility regression analysisrdquo Fuzzy Sets and Systems vol 69 no 3pp 305ndash318 1995
[6] K K Yen S Ghoshray and G Roig ldquoA linear regression modelusing triangular fuzzy number coefficientsrdquo Fuzzy Sets andSystems vol 106 no 2 pp 167ndash177 1999
[7] Y Chen J Tang R Y K Fung and Z Ren ldquoFuzzy regression-based mathematical programming model for quality functiondeploymentrdquo International Journal of Production Research vol42 no 5 pp 1009ndash1027 2004
[8] Z B Chen B Lin and Q Fang ldquoStudy on determinationmethod for parameters of Rockrsquos shear strength through leastabsolute linear regression based on symmetric triangular fuzzynumbersrdquo Applied Mechanics and Materials vol 170 pp 804ndash807 2012
[9] R Y K Fung Y Chen and J Tang ldquoEstimating the functionalrelationships for quality function deployment under uncertain-tiesrdquo Fuzzy Sets and Systems vol 157 no 1 pp 98ndash120 2006
[10] B Kheirfam and J-L Verdegay ldquoThe dual simplex method andsensitivity analysis for fuzzy linear programming with sym-metric trapezoidal numbersrdquo Fuzzy Optimization and DecisionMaking vol 12 no 2 pp 171ndash189 2013
[11] PDiamond andHTanaka ldquoFuzzy regression analysisrdquo inFuzzySets in Decision Analysis Operations Research and Statisticsvol 1 of Handbooks of Fuzzy Sets Series pp 349ndash387 KluwerAcademic Boston Mass USA 1998
[12] H Tanaka andH Lee ldquoInterval regression analysis by quadraticprogramming approachrdquo IEEE Transactions on Fuzzy Systemsvol 6 no 4 pp 473ndash481 1998
[13] C Hwang andD H Hong ldquoAnalyzing the nonlinear time seriesof turbulent flows with kernel interval regression machinerdquoInternational Journal of Uncertainty Fuzziness and Knowledge-Based Systems vol 13 no 4 pp 365ndash380 2005
[14] S Yagiz and C Gokceoglu ldquoApplication of fuzzy inferencesystem and nonlinear regression models for predicting rockbrittlenessrdquo Expert Systems with Applications vol 37 no 3 pp2265ndash2272 2010
[15] A Celmins ldquoLeast squares model fitting to fuzzy vector datardquoFuzzy Sets and Systems vol 22 no 3 pp 245ndash269 1987
[16] P Diamond ldquoFuzzy least squaresrdquo Information Sciences vol 46no 3 pp 141ndash157 1988
Mathematical Problems in Engineering 9
[17] Y-H O Chang ldquoHybrid fuzzy least-squares regression analysisand its reliability measuresrdquo Fuzzy Sets and Systems vol 119 no2 pp 225ndash246 2001
[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for emission prediction of acoal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013
[19] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988
[20] H Moskowitz and K Kim ldquoOn assessing the 119867 value in fuzzylinear regressionrdquo Fuzzy Sets and Systems vol 58 no 3 pp 303ndash327 1993
[21] Y-H O Chang and B M Ayyub ldquoFuzzy regression methodsmdasha comparative assessmentrdquo Fuzzy Sets and Systems vol 119 no2 pp 187ndash203 2001
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
h
h
1
0y
120583yi (yi )
yL1
1
yC2 yU1 yU2yL2
2
yC1yi
y2 y1
Figure 4 Δ1199101119894= Δ1199102
119894and 120583
1199101
119894
(119910119894) = 1205831199102
119894
(119910119894)
(aS)hlowast2 (aS)h
lowast1 aC
h
h
1
0
1
2
120583A(a)
Figure 5 The case when ℎ2ge ℎ1
As a result based on (9) the total credibility of all sampledata 119911 can be obtained to assess the performance of theobtained FLRmodel in (1) which can be calculated as follows
119911 =
119898
sum
119894=1
119911119894=
119898
sum
119894=1
120583119910119894
(119910119894)
Δ119910119894
(10)
The higher 119911 is the better of the performance of the FLRmodel will be This will help us select the optimal value ofℎ for a FLR model
4 Formulating a Systematic Approach toOptimizing ℎ Value
Based on the concept of credibility measure for FLR modelsintroduced in Section 3 a systematic approach is proposed tooptimize ℎ value for FLR models in this Section
As shown in Figure 5 we denote the optimal solutionof (6a) (6b) (6c) and (6d) with respect to ℎ
1as 119860ℎ
lowast
1
119895=
((119886119878
119895)ℎlowast
1
119886119862
119895) 119895 = 0 1 119899 and Δℎ
lowast
1 and the correspondingfuzziness as Δ119910ℎ
lowast
1
119894 119894 = 1 119898 According to Moskowitz and
Kim [20] the optimal solution of (6a) (6b) (6c) and (6d)and the corresponding fuzziness with regard to ℎ
2can be
obtained as
119860ℎlowast
2
119895= (
1 minus ℎ1
1 minus ℎ2
(119886119878
119895)ℎlowast
1
119886119862
119895) 119895 = 0 1 119899
Δℎlowast
2 =1 minus ℎ1
1 minus ℎ2
Δℎlowast
1
Δ119910ℎlowast
2
119894=1 minus ℎ1
1 minus ℎ2
Δ119910ℎlowast
1
119894 119894 = 1 119898
(11)
Table 1 Testing dataset
119910 119909
14 216 414 618 818 1022 1218 1622 18
This indicates that the central value 119886119862119894119895of each 119860
119894119895will keep
constant when the ℎ value changes from ℎ1to ℎ2 while the
spread value 119886119878119894119895of each 119860
119894119895and the objective function Δ and
the fuzzinessΔ119910119894(119894 = 1 119898) become (1minusℎ
1)(1minusℎ
2) times
simultaneouslyWith respect to ℎ
1 (120583119910119894
(119910119894))ℎ1 119894 = 1 119898 are calculated
as follows
(120583119910119894
(119910119894))ℎ1
= 1 minus
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ1
119894
= 1 minus 119902ℎ1
119894 119894 = 1 119898 (12)
in which 119902ℎ1119894is given as
119902ℎ1
119894=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ1
119894
=
10038161003816100381610038161003816sum119899
119895=0(119886119862
119895) 119909119894119895minus 119910119894
10038161003816100381610038161003816
sum119899
119895=0(119886119878119895)ℎ1 10038161003816100381610038161003816119909119894119895
10038161003816100381610038161003816
119894 = 1 119898 (13)
Therefore according to (8) the estimating credibility for allsample data with regard to ℎ
1can be expressed as
119911ℎ1
119894=(120583119910119894
(119910119894))ℎ1
Δ119910ℎ1
119894
=1 minus 119902ℎ1
119894
Δ119910ℎ1
119894
119894 = 1 119898 (14)
Therefore from (9) the total credibility for the FLR modelwith respect to ℎ
1can be obtained as
119911ℎ1 =
119898
sum
119894=1
119911ℎ1
119894=
119898
sum
119894=1
1 minus 119902ℎ1
119894
Δ119910ℎ1
119894
(15)
From (12) with regard to ℎ2 we have
119902ℎ2
119894=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
Δ119910ℎ2
119894
=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
((1 minus ℎ1) (1 minus ℎ
2)) Δ119910ℎ1
119894
=1 minus ℎ2
1 minus ℎ1
119902ℎ1
119894 119894 = 1 119898
(16)
Mathematical Problems in Engineering 5
Table 2 Corresponding results with symmetric triangular fuzzy coefficients (ℎ = 0)
119894 1 2 3 4 5 6 7 8Δ1199100
11989412500 15000 17500 20000 22500 25000 27500 30000
1199020
11989406000 10000 10000 05000 01111 10000 10000 00000
1199010
11989404800 06667 05714 02500 00494 04000 03636 00000
(120583119910119894(119910119894))0
04000 00000 00000 05000 08889 00000 00000 100001199110
11989403200 00000 00000 02500 03951 00000 00000 03333
Therefore according to (11) (120583119910119894
(119910119894))ℎ2 with regard to ℎ
2can
be calculated as
(120583119910119894
(119910119894))ℎ2
= 1 minus 119902ℎ2
119894= 1 minus
1 minus ℎ2
1 minus ℎ1
119902ℎ1
119894
= 1 minus 119902ℎ1
119894+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894
= (120583119910119894
(119910119894))ℎ1
+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894 119894 = 1 119898
(17)
Henceforth according to (8) (10) and (16) the estimatingcredibility for all sample data pairs with regard to ℎ
2can be
expressed as
119911ℎ2
119894=(120583119910119894
(119910119894))ℎ2
Δ119910ℎ2
119894
=(120583119910119894
(119910119894))ℎ1
+ ((ℎ2minus ℎ1) (1 minus ℎ
1)) 119902ℎ1
119894
((1 minus ℎ1) (1 minus ℎ
2)) Δ119910ℎ1
119894
=1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894 119894 = 1 119898
(18)
in which
119901ℎ1
119894=
119902ℎ1
119894
Δ119910ℎ1
119894
=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
(Δ119910ℎ1
119894)2 119894 = 1 119898 (19)
Consequently the estimating credibility for the FLRmodel in(1) with regard to ℎ
2can be obtained as
119911ℎ2 =
119898
sum
119894=1
119911ℎ2
119894=
119898
sum
119894=1
(1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894)
=1 minus ℎ2
1 minus ℎ1
119911ℎ1 +
(1 minus ℎ2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
(20)
in which
119901ℎ1 =
119898
sum
119894=1
119901ℎ1
119894 (21)
For similarity we denote ℎ1= 0 and ℎ
2= ℎ (11) (16) (17)
(18) and (20) can be rewritten as
119860ℎ
119895= (
1
1 minus ℎ(119886119878
119895)0
119886119862
119895) 119895 = 0 1 119899
Δℎ
=1
1 minus ℎΔ0
Δ119910ℎ
119894=
1
1 minus ℎΔ1199100
119894 119894 = 1 119898
(22)
119902ℎ
119894= (1 minus ℎ) 119902
0
119894 (23)
(120583119910119894
(119910119894))ℎ
= (120583119910119894
(119910119894))0
+ ℎ1199020
119894 (24)
119911ℎ
119894= minus1199010
119894ℎ2
+ (1199010
119894minus 1199110
119894) ℎ + 119911
0
119894 (25)
119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ + 1199110
(26)
Based on (26) the optimal ℎ value for the FLR model in(1) can be obtained by maximizing the estimating credibilitythat is the optimal value for ℎ can be obtained by solving thefollowing programming model
max 119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ (27a)
st 0 le ℎ lt 1 (27b)
It is obviously that the programming model in (27a) to (27b)is quadratic and many kinds of optimization algorithmssuch as the gradient descent method can be used to solve theprevious nonlinear programming problem
5 Numerical Example
The numerical example in [21] is used to how the optimal ℎvalue can be obtained using the approach proposed in thispaper The eight testing data pairs are listed in Table 1
When ℎ is specified as 0 the fuzzy coefficients interms of symmetric TFNs can be obtained by solving theprogramming model in (6a) (6b) (6c) and (6d) as 1198600
0=
(10000 120000) and 1198600
1= (01250 06250) And the
corresponding FLR model is given as
1199100
= (10000 120000) + (01250 06250) 119909 (28)
Accordingly the fuzziness Δ1199100119894 parameters 1199020
119894and 1199010
119894 mem-
bership degree (120583119910119894
(119910119894))0 and credibility 1199110
119894 119894 = 1 119898 are
calculated respectively as shown in Table 2
6 Mathematical Problems in Engineering
Table 3 Testing data set
119910 1199091
1199092
13360 2950 799913763 3130 756314786 3760 692519676 3990 627522053 3990 646622325 4030 630923319 4150 615126567 4360 600733516 4570 582241129 4780 584346068 4950 605747796 5010 582347402 5020 580346680 4990 575346616 5000 556846980 5000 552446895 5000 545147624 5090 500849939 5310 500552120 5520 4972
From Table 2 1199110 is calculated as 12984 and 1199010 as 27811According to (26) the total credibility of the FLRmodel withrespect to ℎ can be expressed as follows
119911ℎ
= minus27811ℎ2
+ 14827ℎ + 12984 (29)
According to (27a) and (27b) the optimal value ℎlowast is given as02666 and the optimal coefficients in the form of symmetriccase are obtained as 119860lowast
0= (13635 120000) and 119860
lowast
1=
(01704 06250) Therefore the optimal FLR model is givenas
119910lowast
= (13635 120000) + (01704 06250) 119909 (30)
The total credibility of the model in (30) 119911lowast is calculated as14960 which is higher than that of the model in (28) that is12984
The changes of the fuzzinessΔ119910ℎ119894 themembership degree
(120583119910119894
(119910119894))ℎ the credibility 119911
ℎ
119894 119894 = 1 119898 and the total
credibility 119911ℎ with different ℎ values are depicted in Figures6 7 8 and 9 respectively Notably the vertical axis islogarithmic scale in Figure 6 to enable us to see small changesin fuzziness among eight sample data and the membershipdegrees for sample data 2 3 6 and 7 are overlapped inFigure 7
From Figure 6 we can see that the fuzziness of all sampledata become larger with the augment of ℎ value especiallywhen ℎ value is close to 1 the fuzziness will be extremelylarger From Figure 7 it can be found that the membershipdegree of all sample data becomes higher with the increase ofℎ value and the membership degree will be close to 1 whenℎ value is close to 1 It is clearly shown in Figure 8 that when
0 02 04 06 08 1h value
Fuzz
ines
s
Data1Data2Data3Data4
Data5Data6Data7Data8
103
102
101
100
Figure 6 Fuzziness of the eight sample data
0 02 04 06 08 10
02
04
06
08
1
12
h value
Mem
bers
hip
degr
ee
Data1Data2Data3Data4
Data5Data6Data7Data8
Figure 7 Membership degree of the eight sample data
ℎ value is close to 1 the credibility measures for all sampledata will be close to zero due to the extremely larger of thefuzziness (see Figure 7) Figure 9 shows that when ℎ valueis 02666 the total credibility of the FLR model will achievethe maximum based on which the best FLR model can beobtained
To further demonstrate the effectiveness of the approachproposed in this paper another numerical example with twoindependent variables is given as follows The twenty testingdatasets are listed in Table 3
According to (27a) and (27b) the optimal value ℎlowast isgiven as 03184 When ℎ is specified as 0 03184 and 05respectively the fuzzy coefficients in terms of symmetricTFNs and the corresponding total credibility can be obtained
Mathematical Problems in Engineering 7
Table 4 The corresponding results
h value Fuzzy linear model Total credibility03148 119910
lowast
= (minus6108251 00000) + (191484 00000) 1199091+ (14018 12301) 119909
201371
00 1199100
= (minus6108233 00000) + (191484 00000) 1199091+ (14017 08429) 119909
201082
05 11991005
= (minus6108240 00000) + (191484 00000) 1199091+ (14017 16857) 119909
201271
Data1Data2Data3Data4
Data5Data6Data7Data8
0 02 04 06 08 10
005
01
015
0202
h value
Cred
ibili
ty m
easu
re
Figure 8 Credibility of the eight sample data
0 02 04 06 08 10
05
1
15
h value
Tota
l cre
dibi
lity
Figure 9 Total credibility of the FLR models
by solving the programming model in (6a) (6b) (6c) and(6d) The results are summarized in Table 4
From Table 4 we can see that when ℎ value is set to03148 the total credibility of the FLR model will achieve themaximum that is 01371 which indicates that the approachproposed in this paper is effective and reasonable
Table 5 The experiment results
Run no 119909 (gcm3) 119910 (horsepower)1 0760 512 0780 623 0820 754 0815 705 0790 716 0785 687 0770 598 0765 569 0788 9010 0769 59
Table 6 The FLR models with the corresponding total credibility
h value Fuzzy linear model Totalcredibility
02057 119910 = (minus256000 08765) + (406962 00000)119909 727850 119910 = (minus256000 06962) + (406962 00000)119909 6790305 119910 = (minus256000 13924) + (406962 00000)119909 62793
6 Real Application
To investigate the effectiveness of the approach proposed inthis paper modeling welding process for electronic manu-facturing using fuzzy linear regression was studied An elec-tronic company is a famous OEM company of printed circuitboard (PCB) in PR China The engineers in this companywanted to improve the welding quality by investigating therelationship between the pull strength (119910) of welding line andthe proportion of colophony (119909) in welding fluid And anengineering experiment was conducted and the experimentresults are shown in Table 5
With the use of the 10 experimental datasets pairs theoptimal ℎ value is calculated as 02057 If ℎ value is givenas 02057 (the optimal value) 0 and 05 respectively theFLR models with the corresponding total credibility aresummarized in Table 6
It is not surprise that the total credibility of the FLRmodelwith the optimal ℎ value achieves the highest among the threeFLRmodels (see Table 6) In fact when the optimal ℎ value isused the total credibility of the FLRmodel will improve 719comparing with ℎ = 0 and 1591 comparing with ℎ = 05
To further investigate the modeling performance of theapproach proposed in this paper we divided the sample datapairs to two groups one group is for modeling that is 80of all the sample data pairs were used to build the FLRmodelthe other group is for testing that is the left 20 of all the
8 Mathematical Problems in Engineering
Table 7 Testing results
Testing data ℎlowast
ℎ = ℎlowast
ℎ = 0 ℎ = 05
Total Total Totalcredibility credibility credibility
9 10 02206 15721 15434 130851 4 02820 19803 15332 114725 6 02018 02841 00000 028742 9 02602 18206 20378 142554 7 02761 10588 12084 106141 2 03066 21009 17199 161282 5 02963 14407 15948 130633 7 02743 18868 12155 145811 5 02647 12111 12000 120006 10 02139 15273 14654 12908
Average 14833 13518 12098
sample data were used to test the performance of the FLRmodel Therefore each time two datasets were randomlyselected from ten datasets as testing datasets while the resteight datasets were used to develop FLRmodels when ℎ valueis specified as the optimal value 0 and 05 respectively Theircorresponding total credibility was calculated The previousprocedures were repeated for ten times Table 7 summarizesthe testing results
From Table 7 it can be seen that the predictive perfor-mance of the FLR model with the optimal ℎ value is the bestamong those of the other models In fact in general whenthe optimal ℎ value is used the total credibility of the FLRmodel will improve 973 comparing with ℎ = 0 and 2261comparing with ℎ = 05Therefore the approach proposed inthis paper is effective
7 Conclusions
In this paper a systematic approach is proposed to selectoptimal value of ℎ for FLR analysis with symmetric TFNsFirstly a new concept of credibility is introduced by theconsideration of system fuzziness and membership degreewhich can be used to assess the performance of FLR analysiswith different ℎ valueswhen a set of sample data pairs is givenSecondly a procedure to obtain optimal ℎ value is formulatedfor FLR with symmetric TFN by maximizing the totalcredibility of FLR models By using the approach proposedin this paper the optimal value of ℎ can be determineddefinitely with respect to a set of sample data pairs Boththe numerical example and real application demonstrate thatthe approach proposed in this paper is effective and efficientThe further work in this direction involves extending theapproach proposed in this paper for FLR analysis with fuzzyoutputs that is developing a procedure to select optimalℎ value for FLR with fuzzy observations and applying theproposed approach to practical problems
Acknowledgments
Thework described in this paper is supported by aGrant fromthe National Nature Science Foundation of Chinese (Projectno NSFC 71272177G020902) and the funds of InnovationProgram of Shanghai Municipal Education Commission(Project no 12ZS101)
References
[1] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982
[2] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978
[3] D Zhang L-F Deng K-Y Cai and A So ldquoFuzzy nonlinearregression with fuzzified radial basis function networkrdquo IEEETransactions on Fuzzy Systems vol 13 no 6 pp 742ndash760 2005
[4] A Bardossy ldquoNote on fuzzy regressionrdquo Fuzzy Sets and Systemsvol 37 no 1 pp 65ndash75 1990
[5] H Tanaka H Ishibuchi and S Yoshikawa ldquoExponential possi-bility regression analysisrdquo Fuzzy Sets and Systems vol 69 no 3pp 305ndash318 1995
[6] K K Yen S Ghoshray and G Roig ldquoA linear regression modelusing triangular fuzzy number coefficientsrdquo Fuzzy Sets andSystems vol 106 no 2 pp 167ndash177 1999
[7] Y Chen J Tang R Y K Fung and Z Ren ldquoFuzzy regression-based mathematical programming model for quality functiondeploymentrdquo International Journal of Production Research vol42 no 5 pp 1009ndash1027 2004
[8] Z B Chen B Lin and Q Fang ldquoStudy on determinationmethod for parameters of Rockrsquos shear strength through leastabsolute linear regression based on symmetric triangular fuzzynumbersrdquo Applied Mechanics and Materials vol 170 pp 804ndash807 2012
[9] R Y K Fung Y Chen and J Tang ldquoEstimating the functionalrelationships for quality function deployment under uncertain-tiesrdquo Fuzzy Sets and Systems vol 157 no 1 pp 98ndash120 2006
[10] B Kheirfam and J-L Verdegay ldquoThe dual simplex method andsensitivity analysis for fuzzy linear programming with sym-metric trapezoidal numbersrdquo Fuzzy Optimization and DecisionMaking vol 12 no 2 pp 171ndash189 2013
[11] PDiamond andHTanaka ldquoFuzzy regression analysisrdquo inFuzzySets in Decision Analysis Operations Research and Statisticsvol 1 of Handbooks of Fuzzy Sets Series pp 349ndash387 KluwerAcademic Boston Mass USA 1998
[12] H Tanaka andH Lee ldquoInterval regression analysis by quadraticprogramming approachrdquo IEEE Transactions on Fuzzy Systemsvol 6 no 4 pp 473ndash481 1998
[13] C Hwang andD H Hong ldquoAnalyzing the nonlinear time seriesof turbulent flows with kernel interval regression machinerdquoInternational Journal of Uncertainty Fuzziness and Knowledge-Based Systems vol 13 no 4 pp 365ndash380 2005
[14] S Yagiz and C Gokceoglu ldquoApplication of fuzzy inferencesystem and nonlinear regression models for predicting rockbrittlenessrdquo Expert Systems with Applications vol 37 no 3 pp2265ndash2272 2010
[15] A Celmins ldquoLeast squares model fitting to fuzzy vector datardquoFuzzy Sets and Systems vol 22 no 3 pp 245ndash269 1987
[16] P Diamond ldquoFuzzy least squaresrdquo Information Sciences vol 46no 3 pp 141ndash157 1988
Mathematical Problems in Engineering 9
[17] Y-H O Chang ldquoHybrid fuzzy least-squares regression analysisand its reliability measuresrdquo Fuzzy Sets and Systems vol 119 no2 pp 225ndash246 2001
[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for emission prediction of acoal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013
[19] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988
[20] H Moskowitz and K Kim ldquoOn assessing the 119867 value in fuzzylinear regressionrdquo Fuzzy Sets and Systems vol 58 no 3 pp 303ndash327 1993
[21] Y-H O Chang and B M Ayyub ldquoFuzzy regression methodsmdasha comparative assessmentrdquo Fuzzy Sets and Systems vol 119 no2 pp 187ndash203 2001
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Mathematical Problems in Engineering
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Table 2 Corresponding results with symmetric triangular fuzzy coefficients (ℎ = 0)
119894 1 2 3 4 5 6 7 8Δ1199100
11989412500 15000 17500 20000 22500 25000 27500 30000
1199020
11989406000 10000 10000 05000 01111 10000 10000 00000
1199010
11989404800 06667 05714 02500 00494 04000 03636 00000
(120583119910119894(119910119894))0
04000 00000 00000 05000 08889 00000 00000 100001199110
11989403200 00000 00000 02500 03951 00000 00000 03333
Therefore according to (11) (120583119910119894
(119910119894))ℎ2 with regard to ℎ
2can
be calculated as
(120583119910119894
(119910119894))ℎ2
= 1 minus 119902ℎ2
119894= 1 minus
1 minus ℎ2
1 minus ℎ1
119902ℎ1
119894
= 1 minus 119902ℎ1
119894+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894
= (120583119910119894
(119910119894))ℎ1
+ℎ2minus ℎ1
1 minus ℎ1
119902ℎ1
119894 119894 = 1 119898
(17)
Henceforth according to (8) (10) and (16) the estimatingcredibility for all sample data pairs with regard to ℎ
2can be
expressed as
119911ℎ2
119894=(120583119910119894
(119910119894))ℎ2
Δ119910ℎ2
119894
=(120583119910119894
(119910119894))ℎ1
+ ((ℎ2minus ℎ1) (1 minus ℎ
1)) 119902ℎ1
119894
((1 minus ℎ1) (1 minus ℎ
2)) Δ119910ℎ1
119894
=1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894 119894 = 1 119898
(18)
in which
119901ℎ1
119894=
119902ℎ1
119894
Δ119910ℎ1
119894
=
10038161003816100381610038161003816119910119862
119894minus 119910119894
10038161003816100381610038161003816
(Δ119910ℎ1
119894)2 119894 = 1 119898 (19)
Consequently the estimating credibility for the FLRmodel in(1) with regard to ℎ
2can be obtained as
119911ℎ2 =
119898
sum
119894=1
119911ℎ2
119894=
119898
sum
119894=1
(1 minus ℎ2
1 minus ℎ1
119911ℎ1
119894+(1 minus ℎ
2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
119894)
=1 minus ℎ2
1 minus ℎ1
119911ℎ1 +
(1 minus ℎ2) (ℎ2minus ℎ1)
(1 minus ℎ1)2
119901ℎ1
(20)
in which
119901ℎ1 =
119898
sum
119894=1
119901ℎ1
119894 (21)
For similarity we denote ℎ1= 0 and ℎ
2= ℎ (11) (16) (17)
(18) and (20) can be rewritten as
119860ℎ
119895= (
1
1 minus ℎ(119886119878
119895)0
119886119862
119895) 119895 = 0 1 119899
Δℎ
=1
1 minus ℎΔ0
Δ119910ℎ
119894=
1
1 minus ℎΔ1199100
119894 119894 = 1 119898
(22)
119902ℎ
119894= (1 minus ℎ) 119902
0
119894 (23)
(120583119910119894
(119910119894))ℎ
= (120583119910119894
(119910119894))0
+ ℎ1199020
119894 (24)
119911ℎ
119894= minus1199010
119894ℎ2
+ (1199010
119894minus 1199110
119894) ℎ + 119911
0
119894 (25)
119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ + 1199110
(26)
Based on (26) the optimal ℎ value for the FLR model in(1) can be obtained by maximizing the estimating credibilitythat is the optimal value for ℎ can be obtained by solving thefollowing programming model
max 119911ℎ
= minus1199010
ℎ2
+ (1199010
minus 1199110
) ℎ (27a)
st 0 le ℎ lt 1 (27b)
It is obviously that the programming model in (27a) to (27b)is quadratic and many kinds of optimization algorithmssuch as the gradient descent method can be used to solve theprevious nonlinear programming problem
5 Numerical Example
The numerical example in [21] is used to how the optimal ℎvalue can be obtained using the approach proposed in thispaper The eight testing data pairs are listed in Table 1
When ℎ is specified as 0 the fuzzy coefficients interms of symmetric TFNs can be obtained by solving theprogramming model in (6a) (6b) (6c) and (6d) as 1198600
0=
(10000 120000) and 1198600
1= (01250 06250) And the
corresponding FLR model is given as
1199100
= (10000 120000) + (01250 06250) 119909 (28)
Accordingly the fuzziness Δ1199100119894 parameters 1199020
119894and 1199010
119894 mem-
bership degree (120583119910119894
(119910119894))0 and credibility 1199110
119894 119894 = 1 119898 are
calculated respectively as shown in Table 2
6 Mathematical Problems in Engineering
Table 3 Testing data set
119910 1199091
1199092
13360 2950 799913763 3130 756314786 3760 692519676 3990 627522053 3990 646622325 4030 630923319 4150 615126567 4360 600733516 4570 582241129 4780 584346068 4950 605747796 5010 582347402 5020 580346680 4990 575346616 5000 556846980 5000 552446895 5000 545147624 5090 500849939 5310 500552120 5520 4972
From Table 2 1199110 is calculated as 12984 and 1199010 as 27811According to (26) the total credibility of the FLRmodel withrespect to ℎ can be expressed as follows
119911ℎ
= minus27811ℎ2
+ 14827ℎ + 12984 (29)
According to (27a) and (27b) the optimal value ℎlowast is given as02666 and the optimal coefficients in the form of symmetriccase are obtained as 119860lowast
0= (13635 120000) and 119860
lowast
1=
(01704 06250) Therefore the optimal FLR model is givenas
119910lowast
= (13635 120000) + (01704 06250) 119909 (30)
The total credibility of the model in (30) 119911lowast is calculated as14960 which is higher than that of the model in (28) that is12984
The changes of the fuzzinessΔ119910ℎ119894 themembership degree
(120583119910119894
(119910119894))ℎ the credibility 119911
ℎ
119894 119894 = 1 119898 and the total
credibility 119911ℎ with different ℎ values are depicted in Figures6 7 8 and 9 respectively Notably the vertical axis islogarithmic scale in Figure 6 to enable us to see small changesin fuzziness among eight sample data and the membershipdegrees for sample data 2 3 6 and 7 are overlapped inFigure 7
From Figure 6 we can see that the fuzziness of all sampledata become larger with the augment of ℎ value especiallywhen ℎ value is close to 1 the fuzziness will be extremelylarger From Figure 7 it can be found that the membershipdegree of all sample data becomes higher with the increase ofℎ value and the membership degree will be close to 1 whenℎ value is close to 1 It is clearly shown in Figure 8 that when
0 02 04 06 08 1h value
Fuzz
ines
s
Data1Data2Data3Data4
Data5Data6Data7Data8
103
102
101
100
Figure 6 Fuzziness of the eight sample data
0 02 04 06 08 10
02
04
06
08
1
12
h value
Mem
bers
hip
degr
ee
Data1Data2Data3Data4
Data5Data6Data7Data8
Figure 7 Membership degree of the eight sample data
ℎ value is close to 1 the credibility measures for all sampledata will be close to zero due to the extremely larger of thefuzziness (see Figure 7) Figure 9 shows that when ℎ valueis 02666 the total credibility of the FLR model will achievethe maximum based on which the best FLR model can beobtained
To further demonstrate the effectiveness of the approachproposed in this paper another numerical example with twoindependent variables is given as follows The twenty testingdatasets are listed in Table 3
According to (27a) and (27b) the optimal value ℎlowast isgiven as 03184 When ℎ is specified as 0 03184 and 05respectively the fuzzy coefficients in terms of symmetricTFNs and the corresponding total credibility can be obtained
Mathematical Problems in Engineering 7
Table 4 The corresponding results
h value Fuzzy linear model Total credibility03148 119910
lowast
= (minus6108251 00000) + (191484 00000) 1199091+ (14018 12301) 119909
201371
00 1199100
= (minus6108233 00000) + (191484 00000) 1199091+ (14017 08429) 119909
201082
05 11991005
= (minus6108240 00000) + (191484 00000) 1199091+ (14017 16857) 119909
201271
Data1Data2Data3Data4
Data5Data6Data7Data8
0 02 04 06 08 10
005
01
015
0202
h value
Cred
ibili
ty m
easu
re
Figure 8 Credibility of the eight sample data
0 02 04 06 08 10
05
1
15
h value
Tota
l cre
dibi
lity
Figure 9 Total credibility of the FLR models
by solving the programming model in (6a) (6b) (6c) and(6d) The results are summarized in Table 4
From Table 4 we can see that when ℎ value is set to03148 the total credibility of the FLR model will achieve themaximum that is 01371 which indicates that the approachproposed in this paper is effective and reasonable
Table 5 The experiment results
Run no 119909 (gcm3) 119910 (horsepower)1 0760 512 0780 623 0820 754 0815 705 0790 716 0785 687 0770 598 0765 569 0788 9010 0769 59
Table 6 The FLR models with the corresponding total credibility
h value Fuzzy linear model Totalcredibility
02057 119910 = (minus256000 08765) + (406962 00000)119909 727850 119910 = (minus256000 06962) + (406962 00000)119909 6790305 119910 = (minus256000 13924) + (406962 00000)119909 62793
6 Real Application
To investigate the effectiveness of the approach proposed inthis paper modeling welding process for electronic manu-facturing using fuzzy linear regression was studied An elec-tronic company is a famous OEM company of printed circuitboard (PCB) in PR China The engineers in this companywanted to improve the welding quality by investigating therelationship between the pull strength (119910) of welding line andthe proportion of colophony (119909) in welding fluid And anengineering experiment was conducted and the experimentresults are shown in Table 5
With the use of the 10 experimental datasets pairs theoptimal ℎ value is calculated as 02057 If ℎ value is givenas 02057 (the optimal value) 0 and 05 respectively theFLR models with the corresponding total credibility aresummarized in Table 6
It is not surprise that the total credibility of the FLRmodelwith the optimal ℎ value achieves the highest among the threeFLRmodels (see Table 6) In fact when the optimal ℎ value isused the total credibility of the FLRmodel will improve 719comparing with ℎ = 0 and 1591 comparing with ℎ = 05
To further investigate the modeling performance of theapproach proposed in this paper we divided the sample datapairs to two groups one group is for modeling that is 80of all the sample data pairs were used to build the FLRmodelthe other group is for testing that is the left 20 of all the
8 Mathematical Problems in Engineering
Table 7 Testing results
Testing data ℎlowast
ℎ = ℎlowast
ℎ = 0 ℎ = 05
Total Total Totalcredibility credibility credibility
9 10 02206 15721 15434 130851 4 02820 19803 15332 114725 6 02018 02841 00000 028742 9 02602 18206 20378 142554 7 02761 10588 12084 106141 2 03066 21009 17199 161282 5 02963 14407 15948 130633 7 02743 18868 12155 145811 5 02647 12111 12000 120006 10 02139 15273 14654 12908
Average 14833 13518 12098
sample data were used to test the performance of the FLRmodel Therefore each time two datasets were randomlyselected from ten datasets as testing datasets while the resteight datasets were used to develop FLRmodels when ℎ valueis specified as the optimal value 0 and 05 respectively Theircorresponding total credibility was calculated The previousprocedures were repeated for ten times Table 7 summarizesthe testing results
From Table 7 it can be seen that the predictive perfor-mance of the FLR model with the optimal ℎ value is the bestamong those of the other models In fact in general whenthe optimal ℎ value is used the total credibility of the FLRmodel will improve 973 comparing with ℎ = 0 and 2261comparing with ℎ = 05Therefore the approach proposed inthis paper is effective
7 Conclusions
In this paper a systematic approach is proposed to selectoptimal value of ℎ for FLR analysis with symmetric TFNsFirstly a new concept of credibility is introduced by theconsideration of system fuzziness and membership degreewhich can be used to assess the performance of FLR analysiswith different ℎ valueswhen a set of sample data pairs is givenSecondly a procedure to obtain optimal ℎ value is formulatedfor FLR with symmetric TFN by maximizing the totalcredibility of FLR models By using the approach proposedin this paper the optimal value of ℎ can be determineddefinitely with respect to a set of sample data pairs Boththe numerical example and real application demonstrate thatthe approach proposed in this paper is effective and efficientThe further work in this direction involves extending theapproach proposed in this paper for FLR analysis with fuzzyoutputs that is developing a procedure to select optimalℎ value for FLR with fuzzy observations and applying theproposed approach to practical problems
Acknowledgments
Thework described in this paper is supported by aGrant fromthe National Nature Science Foundation of Chinese (Projectno NSFC 71272177G020902) and the funds of InnovationProgram of Shanghai Municipal Education Commission(Project no 12ZS101)
References
[1] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982
[2] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978
[3] D Zhang L-F Deng K-Y Cai and A So ldquoFuzzy nonlinearregression with fuzzified radial basis function networkrdquo IEEETransactions on Fuzzy Systems vol 13 no 6 pp 742ndash760 2005
[4] A Bardossy ldquoNote on fuzzy regressionrdquo Fuzzy Sets and Systemsvol 37 no 1 pp 65ndash75 1990
[5] H Tanaka H Ishibuchi and S Yoshikawa ldquoExponential possi-bility regression analysisrdquo Fuzzy Sets and Systems vol 69 no 3pp 305ndash318 1995
[6] K K Yen S Ghoshray and G Roig ldquoA linear regression modelusing triangular fuzzy number coefficientsrdquo Fuzzy Sets andSystems vol 106 no 2 pp 167ndash177 1999
[7] Y Chen J Tang R Y K Fung and Z Ren ldquoFuzzy regression-based mathematical programming model for quality functiondeploymentrdquo International Journal of Production Research vol42 no 5 pp 1009ndash1027 2004
[8] Z B Chen B Lin and Q Fang ldquoStudy on determinationmethod for parameters of Rockrsquos shear strength through leastabsolute linear regression based on symmetric triangular fuzzynumbersrdquo Applied Mechanics and Materials vol 170 pp 804ndash807 2012
[9] R Y K Fung Y Chen and J Tang ldquoEstimating the functionalrelationships for quality function deployment under uncertain-tiesrdquo Fuzzy Sets and Systems vol 157 no 1 pp 98ndash120 2006
[10] B Kheirfam and J-L Verdegay ldquoThe dual simplex method andsensitivity analysis for fuzzy linear programming with sym-metric trapezoidal numbersrdquo Fuzzy Optimization and DecisionMaking vol 12 no 2 pp 171ndash189 2013
[11] PDiamond andHTanaka ldquoFuzzy regression analysisrdquo inFuzzySets in Decision Analysis Operations Research and Statisticsvol 1 of Handbooks of Fuzzy Sets Series pp 349ndash387 KluwerAcademic Boston Mass USA 1998
[12] H Tanaka andH Lee ldquoInterval regression analysis by quadraticprogramming approachrdquo IEEE Transactions on Fuzzy Systemsvol 6 no 4 pp 473ndash481 1998
[13] C Hwang andD H Hong ldquoAnalyzing the nonlinear time seriesof turbulent flows with kernel interval regression machinerdquoInternational Journal of Uncertainty Fuzziness and Knowledge-Based Systems vol 13 no 4 pp 365ndash380 2005
[14] S Yagiz and C Gokceoglu ldquoApplication of fuzzy inferencesystem and nonlinear regression models for predicting rockbrittlenessrdquo Expert Systems with Applications vol 37 no 3 pp2265ndash2272 2010
[15] A Celmins ldquoLeast squares model fitting to fuzzy vector datardquoFuzzy Sets and Systems vol 22 no 3 pp 245ndash269 1987
[16] P Diamond ldquoFuzzy least squaresrdquo Information Sciences vol 46no 3 pp 141ndash157 1988
Mathematical Problems in Engineering 9
[17] Y-H O Chang ldquoHybrid fuzzy least-squares regression analysisand its reliability measuresrdquo Fuzzy Sets and Systems vol 119 no2 pp 225ndash246 2001
[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for emission prediction of acoal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013
[19] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988
[20] H Moskowitz and K Kim ldquoOn assessing the 119867 value in fuzzylinear regressionrdquo Fuzzy Sets and Systems vol 58 no 3 pp 303ndash327 1993
[21] Y-H O Chang and B M Ayyub ldquoFuzzy regression methodsmdasha comparative assessmentrdquo Fuzzy Sets and Systems vol 119 no2 pp 187ndash203 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Table 3 Testing data set
119910 1199091
1199092
13360 2950 799913763 3130 756314786 3760 692519676 3990 627522053 3990 646622325 4030 630923319 4150 615126567 4360 600733516 4570 582241129 4780 584346068 4950 605747796 5010 582347402 5020 580346680 4990 575346616 5000 556846980 5000 552446895 5000 545147624 5090 500849939 5310 500552120 5520 4972
From Table 2 1199110 is calculated as 12984 and 1199010 as 27811According to (26) the total credibility of the FLRmodel withrespect to ℎ can be expressed as follows
119911ℎ
= minus27811ℎ2
+ 14827ℎ + 12984 (29)
According to (27a) and (27b) the optimal value ℎlowast is given as02666 and the optimal coefficients in the form of symmetriccase are obtained as 119860lowast
0= (13635 120000) and 119860
lowast
1=
(01704 06250) Therefore the optimal FLR model is givenas
119910lowast
= (13635 120000) + (01704 06250) 119909 (30)
The total credibility of the model in (30) 119911lowast is calculated as14960 which is higher than that of the model in (28) that is12984
The changes of the fuzzinessΔ119910ℎ119894 themembership degree
(120583119910119894
(119910119894))ℎ the credibility 119911
ℎ
119894 119894 = 1 119898 and the total
credibility 119911ℎ with different ℎ values are depicted in Figures6 7 8 and 9 respectively Notably the vertical axis islogarithmic scale in Figure 6 to enable us to see small changesin fuzziness among eight sample data and the membershipdegrees for sample data 2 3 6 and 7 are overlapped inFigure 7
From Figure 6 we can see that the fuzziness of all sampledata become larger with the augment of ℎ value especiallywhen ℎ value is close to 1 the fuzziness will be extremelylarger From Figure 7 it can be found that the membershipdegree of all sample data becomes higher with the increase ofℎ value and the membership degree will be close to 1 whenℎ value is close to 1 It is clearly shown in Figure 8 that when
0 02 04 06 08 1h value
Fuzz
ines
s
Data1Data2Data3Data4
Data5Data6Data7Data8
103
102
101
100
Figure 6 Fuzziness of the eight sample data
0 02 04 06 08 10
02
04
06
08
1
12
h value
Mem
bers
hip
degr
ee
Data1Data2Data3Data4
Data5Data6Data7Data8
Figure 7 Membership degree of the eight sample data
ℎ value is close to 1 the credibility measures for all sampledata will be close to zero due to the extremely larger of thefuzziness (see Figure 7) Figure 9 shows that when ℎ valueis 02666 the total credibility of the FLR model will achievethe maximum based on which the best FLR model can beobtained
To further demonstrate the effectiveness of the approachproposed in this paper another numerical example with twoindependent variables is given as follows The twenty testingdatasets are listed in Table 3
According to (27a) and (27b) the optimal value ℎlowast isgiven as 03184 When ℎ is specified as 0 03184 and 05respectively the fuzzy coefficients in terms of symmetricTFNs and the corresponding total credibility can be obtained
Mathematical Problems in Engineering 7
Table 4 The corresponding results
h value Fuzzy linear model Total credibility03148 119910
lowast
= (minus6108251 00000) + (191484 00000) 1199091+ (14018 12301) 119909
201371
00 1199100
= (minus6108233 00000) + (191484 00000) 1199091+ (14017 08429) 119909
201082
05 11991005
= (minus6108240 00000) + (191484 00000) 1199091+ (14017 16857) 119909
201271
Data1Data2Data3Data4
Data5Data6Data7Data8
0 02 04 06 08 10
005
01
015
0202
h value
Cred
ibili
ty m
easu
re
Figure 8 Credibility of the eight sample data
0 02 04 06 08 10
05
1
15
h value
Tota
l cre
dibi
lity
Figure 9 Total credibility of the FLR models
by solving the programming model in (6a) (6b) (6c) and(6d) The results are summarized in Table 4
From Table 4 we can see that when ℎ value is set to03148 the total credibility of the FLR model will achieve themaximum that is 01371 which indicates that the approachproposed in this paper is effective and reasonable
Table 5 The experiment results
Run no 119909 (gcm3) 119910 (horsepower)1 0760 512 0780 623 0820 754 0815 705 0790 716 0785 687 0770 598 0765 569 0788 9010 0769 59
Table 6 The FLR models with the corresponding total credibility
h value Fuzzy linear model Totalcredibility
02057 119910 = (minus256000 08765) + (406962 00000)119909 727850 119910 = (minus256000 06962) + (406962 00000)119909 6790305 119910 = (minus256000 13924) + (406962 00000)119909 62793
6 Real Application
To investigate the effectiveness of the approach proposed inthis paper modeling welding process for electronic manu-facturing using fuzzy linear regression was studied An elec-tronic company is a famous OEM company of printed circuitboard (PCB) in PR China The engineers in this companywanted to improve the welding quality by investigating therelationship between the pull strength (119910) of welding line andthe proportion of colophony (119909) in welding fluid And anengineering experiment was conducted and the experimentresults are shown in Table 5
With the use of the 10 experimental datasets pairs theoptimal ℎ value is calculated as 02057 If ℎ value is givenas 02057 (the optimal value) 0 and 05 respectively theFLR models with the corresponding total credibility aresummarized in Table 6
It is not surprise that the total credibility of the FLRmodelwith the optimal ℎ value achieves the highest among the threeFLRmodels (see Table 6) In fact when the optimal ℎ value isused the total credibility of the FLRmodel will improve 719comparing with ℎ = 0 and 1591 comparing with ℎ = 05
To further investigate the modeling performance of theapproach proposed in this paper we divided the sample datapairs to two groups one group is for modeling that is 80of all the sample data pairs were used to build the FLRmodelthe other group is for testing that is the left 20 of all the
8 Mathematical Problems in Engineering
Table 7 Testing results
Testing data ℎlowast
ℎ = ℎlowast
ℎ = 0 ℎ = 05
Total Total Totalcredibility credibility credibility
9 10 02206 15721 15434 130851 4 02820 19803 15332 114725 6 02018 02841 00000 028742 9 02602 18206 20378 142554 7 02761 10588 12084 106141 2 03066 21009 17199 161282 5 02963 14407 15948 130633 7 02743 18868 12155 145811 5 02647 12111 12000 120006 10 02139 15273 14654 12908
Average 14833 13518 12098
sample data were used to test the performance of the FLRmodel Therefore each time two datasets were randomlyselected from ten datasets as testing datasets while the resteight datasets were used to develop FLRmodels when ℎ valueis specified as the optimal value 0 and 05 respectively Theircorresponding total credibility was calculated The previousprocedures were repeated for ten times Table 7 summarizesthe testing results
From Table 7 it can be seen that the predictive perfor-mance of the FLR model with the optimal ℎ value is the bestamong those of the other models In fact in general whenthe optimal ℎ value is used the total credibility of the FLRmodel will improve 973 comparing with ℎ = 0 and 2261comparing with ℎ = 05Therefore the approach proposed inthis paper is effective
7 Conclusions
In this paper a systematic approach is proposed to selectoptimal value of ℎ for FLR analysis with symmetric TFNsFirstly a new concept of credibility is introduced by theconsideration of system fuzziness and membership degreewhich can be used to assess the performance of FLR analysiswith different ℎ valueswhen a set of sample data pairs is givenSecondly a procedure to obtain optimal ℎ value is formulatedfor FLR with symmetric TFN by maximizing the totalcredibility of FLR models By using the approach proposedin this paper the optimal value of ℎ can be determineddefinitely with respect to a set of sample data pairs Boththe numerical example and real application demonstrate thatthe approach proposed in this paper is effective and efficientThe further work in this direction involves extending theapproach proposed in this paper for FLR analysis with fuzzyoutputs that is developing a procedure to select optimalℎ value for FLR with fuzzy observations and applying theproposed approach to practical problems
Acknowledgments
Thework described in this paper is supported by aGrant fromthe National Nature Science Foundation of Chinese (Projectno NSFC 71272177G020902) and the funds of InnovationProgram of Shanghai Municipal Education Commission(Project no 12ZS101)
References
[1] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982
[2] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978
[3] D Zhang L-F Deng K-Y Cai and A So ldquoFuzzy nonlinearregression with fuzzified radial basis function networkrdquo IEEETransactions on Fuzzy Systems vol 13 no 6 pp 742ndash760 2005
[4] A Bardossy ldquoNote on fuzzy regressionrdquo Fuzzy Sets and Systemsvol 37 no 1 pp 65ndash75 1990
[5] H Tanaka H Ishibuchi and S Yoshikawa ldquoExponential possi-bility regression analysisrdquo Fuzzy Sets and Systems vol 69 no 3pp 305ndash318 1995
[6] K K Yen S Ghoshray and G Roig ldquoA linear regression modelusing triangular fuzzy number coefficientsrdquo Fuzzy Sets andSystems vol 106 no 2 pp 167ndash177 1999
[7] Y Chen J Tang R Y K Fung and Z Ren ldquoFuzzy regression-based mathematical programming model for quality functiondeploymentrdquo International Journal of Production Research vol42 no 5 pp 1009ndash1027 2004
[8] Z B Chen B Lin and Q Fang ldquoStudy on determinationmethod for parameters of Rockrsquos shear strength through leastabsolute linear regression based on symmetric triangular fuzzynumbersrdquo Applied Mechanics and Materials vol 170 pp 804ndash807 2012
[9] R Y K Fung Y Chen and J Tang ldquoEstimating the functionalrelationships for quality function deployment under uncertain-tiesrdquo Fuzzy Sets and Systems vol 157 no 1 pp 98ndash120 2006
[10] B Kheirfam and J-L Verdegay ldquoThe dual simplex method andsensitivity analysis for fuzzy linear programming with sym-metric trapezoidal numbersrdquo Fuzzy Optimization and DecisionMaking vol 12 no 2 pp 171ndash189 2013
[11] PDiamond andHTanaka ldquoFuzzy regression analysisrdquo inFuzzySets in Decision Analysis Operations Research and Statisticsvol 1 of Handbooks of Fuzzy Sets Series pp 349ndash387 KluwerAcademic Boston Mass USA 1998
[12] H Tanaka andH Lee ldquoInterval regression analysis by quadraticprogramming approachrdquo IEEE Transactions on Fuzzy Systemsvol 6 no 4 pp 473ndash481 1998
[13] C Hwang andD H Hong ldquoAnalyzing the nonlinear time seriesof turbulent flows with kernel interval regression machinerdquoInternational Journal of Uncertainty Fuzziness and Knowledge-Based Systems vol 13 no 4 pp 365ndash380 2005
[14] S Yagiz and C Gokceoglu ldquoApplication of fuzzy inferencesystem and nonlinear regression models for predicting rockbrittlenessrdquo Expert Systems with Applications vol 37 no 3 pp2265ndash2272 2010
[15] A Celmins ldquoLeast squares model fitting to fuzzy vector datardquoFuzzy Sets and Systems vol 22 no 3 pp 245ndash269 1987
[16] P Diamond ldquoFuzzy least squaresrdquo Information Sciences vol 46no 3 pp 141ndash157 1988
Mathematical Problems in Engineering 9
[17] Y-H O Chang ldquoHybrid fuzzy least-squares regression analysisand its reliability measuresrdquo Fuzzy Sets and Systems vol 119 no2 pp 225ndash246 2001
[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for emission prediction of acoal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013
[19] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988
[20] H Moskowitz and K Kim ldquoOn assessing the 119867 value in fuzzylinear regressionrdquo Fuzzy Sets and Systems vol 58 no 3 pp 303ndash327 1993
[21] Y-H O Chang and B M Ayyub ldquoFuzzy regression methodsmdasha comparative assessmentrdquo Fuzzy Sets and Systems vol 119 no2 pp 187ndash203 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
Table 4 The corresponding results
h value Fuzzy linear model Total credibility03148 119910
lowast
= (minus6108251 00000) + (191484 00000) 1199091+ (14018 12301) 119909
201371
00 1199100
= (minus6108233 00000) + (191484 00000) 1199091+ (14017 08429) 119909
201082
05 11991005
= (minus6108240 00000) + (191484 00000) 1199091+ (14017 16857) 119909
201271
Data1Data2Data3Data4
Data5Data6Data7Data8
0 02 04 06 08 10
005
01
015
0202
h value
Cred
ibili
ty m
easu
re
Figure 8 Credibility of the eight sample data
0 02 04 06 08 10
05
1
15
h value
Tota
l cre
dibi
lity
Figure 9 Total credibility of the FLR models
by solving the programming model in (6a) (6b) (6c) and(6d) The results are summarized in Table 4
From Table 4 we can see that when ℎ value is set to03148 the total credibility of the FLR model will achieve themaximum that is 01371 which indicates that the approachproposed in this paper is effective and reasonable
Table 5 The experiment results
Run no 119909 (gcm3) 119910 (horsepower)1 0760 512 0780 623 0820 754 0815 705 0790 716 0785 687 0770 598 0765 569 0788 9010 0769 59
Table 6 The FLR models with the corresponding total credibility
h value Fuzzy linear model Totalcredibility
02057 119910 = (minus256000 08765) + (406962 00000)119909 727850 119910 = (minus256000 06962) + (406962 00000)119909 6790305 119910 = (minus256000 13924) + (406962 00000)119909 62793
6 Real Application
To investigate the effectiveness of the approach proposed inthis paper modeling welding process for electronic manu-facturing using fuzzy linear regression was studied An elec-tronic company is a famous OEM company of printed circuitboard (PCB) in PR China The engineers in this companywanted to improve the welding quality by investigating therelationship between the pull strength (119910) of welding line andthe proportion of colophony (119909) in welding fluid And anengineering experiment was conducted and the experimentresults are shown in Table 5
With the use of the 10 experimental datasets pairs theoptimal ℎ value is calculated as 02057 If ℎ value is givenas 02057 (the optimal value) 0 and 05 respectively theFLR models with the corresponding total credibility aresummarized in Table 6
It is not surprise that the total credibility of the FLRmodelwith the optimal ℎ value achieves the highest among the threeFLRmodels (see Table 6) In fact when the optimal ℎ value isused the total credibility of the FLRmodel will improve 719comparing with ℎ = 0 and 1591 comparing with ℎ = 05
To further investigate the modeling performance of theapproach proposed in this paper we divided the sample datapairs to two groups one group is for modeling that is 80of all the sample data pairs were used to build the FLRmodelthe other group is for testing that is the left 20 of all the
8 Mathematical Problems in Engineering
Table 7 Testing results
Testing data ℎlowast
ℎ = ℎlowast
ℎ = 0 ℎ = 05
Total Total Totalcredibility credibility credibility
9 10 02206 15721 15434 130851 4 02820 19803 15332 114725 6 02018 02841 00000 028742 9 02602 18206 20378 142554 7 02761 10588 12084 106141 2 03066 21009 17199 161282 5 02963 14407 15948 130633 7 02743 18868 12155 145811 5 02647 12111 12000 120006 10 02139 15273 14654 12908
Average 14833 13518 12098
sample data were used to test the performance of the FLRmodel Therefore each time two datasets were randomlyselected from ten datasets as testing datasets while the resteight datasets were used to develop FLRmodels when ℎ valueis specified as the optimal value 0 and 05 respectively Theircorresponding total credibility was calculated The previousprocedures were repeated for ten times Table 7 summarizesthe testing results
From Table 7 it can be seen that the predictive perfor-mance of the FLR model with the optimal ℎ value is the bestamong those of the other models In fact in general whenthe optimal ℎ value is used the total credibility of the FLRmodel will improve 973 comparing with ℎ = 0 and 2261comparing with ℎ = 05Therefore the approach proposed inthis paper is effective
7 Conclusions
In this paper a systematic approach is proposed to selectoptimal value of ℎ for FLR analysis with symmetric TFNsFirstly a new concept of credibility is introduced by theconsideration of system fuzziness and membership degreewhich can be used to assess the performance of FLR analysiswith different ℎ valueswhen a set of sample data pairs is givenSecondly a procedure to obtain optimal ℎ value is formulatedfor FLR with symmetric TFN by maximizing the totalcredibility of FLR models By using the approach proposedin this paper the optimal value of ℎ can be determineddefinitely with respect to a set of sample data pairs Boththe numerical example and real application demonstrate thatthe approach proposed in this paper is effective and efficientThe further work in this direction involves extending theapproach proposed in this paper for FLR analysis with fuzzyoutputs that is developing a procedure to select optimalℎ value for FLR with fuzzy observations and applying theproposed approach to practical problems
Acknowledgments
Thework described in this paper is supported by aGrant fromthe National Nature Science Foundation of Chinese (Projectno NSFC 71272177G020902) and the funds of InnovationProgram of Shanghai Municipal Education Commission(Project no 12ZS101)
References
[1] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982
[2] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978
[3] D Zhang L-F Deng K-Y Cai and A So ldquoFuzzy nonlinearregression with fuzzified radial basis function networkrdquo IEEETransactions on Fuzzy Systems vol 13 no 6 pp 742ndash760 2005
[4] A Bardossy ldquoNote on fuzzy regressionrdquo Fuzzy Sets and Systemsvol 37 no 1 pp 65ndash75 1990
[5] H Tanaka H Ishibuchi and S Yoshikawa ldquoExponential possi-bility regression analysisrdquo Fuzzy Sets and Systems vol 69 no 3pp 305ndash318 1995
[6] K K Yen S Ghoshray and G Roig ldquoA linear regression modelusing triangular fuzzy number coefficientsrdquo Fuzzy Sets andSystems vol 106 no 2 pp 167ndash177 1999
[7] Y Chen J Tang R Y K Fung and Z Ren ldquoFuzzy regression-based mathematical programming model for quality functiondeploymentrdquo International Journal of Production Research vol42 no 5 pp 1009ndash1027 2004
[8] Z B Chen B Lin and Q Fang ldquoStudy on determinationmethod for parameters of Rockrsquos shear strength through leastabsolute linear regression based on symmetric triangular fuzzynumbersrdquo Applied Mechanics and Materials vol 170 pp 804ndash807 2012
[9] R Y K Fung Y Chen and J Tang ldquoEstimating the functionalrelationships for quality function deployment under uncertain-tiesrdquo Fuzzy Sets and Systems vol 157 no 1 pp 98ndash120 2006
[10] B Kheirfam and J-L Verdegay ldquoThe dual simplex method andsensitivity analysis for fuzzy linear programming with sym-metric trapezoidal numbersrdquo Fuzzy Optimization and DecisionMaking vol 12 no 2 pp 171ndash189 2013
[11] PDiamond andHTanaka ldquoFuzzy regression analysisrdquo inFuzzySets in Decision Analysis Operations Research and Statisticsvol 1 of Handbooks of Fuzzy Sets Series pp 349ndash387 KluwerAcademic Boston Mass USA 1998
[12] H Tanaka andH Lee ldquoInterval regression analysis by quadraticprogramming approachrdquo IEEE Transactions on Fuzzy Systemsvol 6 no 4 pp 473ndash481 1998
[13] C Hwang andD H Hong ldquoAnalyzing the nonlinear time seriesof turbulent flows with kernel interval regression machinerdquoInternational Journal of Uncertainty Fuzziness and Knowledge-Based Systems vol 13 no 4 pp 365ndash380 2005
[14] S Yagiz and C Gokceoglu ldquoApplication of fuzzy inferencesystem and nonlinear regression models for predicting rockbrittlenessrdquo Expert Systems with Applications vol 37 no 3 pp2265ndash2272 2010
[15] A Celmins ldquoLeast squares model fitting to fuzzy vector datardquoFuzzy Sets and Systems vol 22 no 3 pp 245ndash269 1987
[16] P Diamond ldquoFuzzy least squaresrdquo Information Sciences vol 46no 3 pp 141ndash157 1988
Mathematical Problems in Engineering 9
[17] Y-H O Chang ldquoHybrid fuzzy least-squares regression analysisand its reliability measuresrdquo Fuzzy Sets and Systems vol 119 no2 pp 225ndash246 2001
[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for emission prediction of acoal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013
[19] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988
[20] H Moskowitz and K Kim ldquoOn assessing the 119867 value in fuzzylinear regressionrdquo Fuzzy Sets and Systems vol 58 no 3 pp 303ndash327 1993
[21] Y-H O Chang and B M Ayyub ldquoFuzzy regression methodsmdasha comparative assessmentrdquo Fuzzy Sets and Systems vol 119 no2 pp 187ndash203 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table 7 Testing results
Testing data ℎlowast
ℎ = ℎlowast
ℎ = 0 ℎ = 05
Total Total Totalcredibility credibility credibility
9 10 02206 15721 15434 130851 4 02820 19803 15332 114725 6 02018 02841 00000 028742 9 02602 18206 20378 142554 7 02761 10588 12084 106141 2 03066 21009 17199 161282 5 02963 14407 15948 130633 7 02743 18868 12155 145811 5 02647 12111 12000 120006 10 02139 15273 14654 12908
Average 14833 13518 12098
sample data were used to test the performance of the FLRmodel Therefore each time two datasets were randomlyselected from ten datasets as testing datasets while the resteight datasets were used to develop FLRmodels when ℎ valueis specified as the optimal value 0 and 05 respectively Theircorresponding total credibility was calculated The previousprocedures were repeated for ten times Table 7 summarizesthe testing results
From Table 7 it can be seen that the predictive perfor-mance of the FLR model with the optimal ℎ value is the bestamong those of the other models In fact in general whenthe optimal ℎ value is used the total credibility of the FLRmodel will improve 973 comparing with ℎ = 0 and 2261comparing with ℎ = 05Therefore the approach proposed inthis paper is effective
7 Conclusions
In this paper a systematic approach is proposed to selectoptimal value of ℎ for FLR analysis with symmetric TFNsFirstly a new concept of credibility is introduced by theconsideration of system fuzziness and membership degreewhich can be used to assess the performance of FLR analysiswith different ℎ valueswhen a set of sample data pairs is givenSecondly a procedure to obtain optimal ℎ value is formulatedfor FLR with symmetric TFN by maximizing the totalcredibility of FLR models By using the approach proposedin this paper the optimal value of ℎ can be determineddefinitely with respect to a set of sample data pairs Boththe numerical example and real application demonstrate thatthe approach proposed in this paper is effective and efficientThe further work in this direction involves extending theapproach proposed in this paper for FLR analysis with fuzzyoutputs that is developing a procedure to select optimalℎ value for FLR with fuzzy observations and applying theproposed approach to practical problems
Acknowledgments
Thework described in this paper is supported by aGrant fromthe National Nature Science Foundation of Chinese (Projectno NSFC 71272177G020902) and the funds of InnovationProgram of Shanghai Municipal Education Commission(Project no 12ZS101)
References
[1] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions on Systems Man andCybernetics vol 12 no 6 pp 903ndash907 1982
[2] L A Zadeh ldquoFuzzy sets as a basis for a theory of possibilityrdquoFuzzy Sets and Systems vol 1 no 1 pp 3ndash28 1978
[3] D Zhang L-F Deng K-Y Cai and A So ldquoFuzzy nonlinearregression with fuzzified radial basis function networkrdquo IEEETransactions on Fuzzy Systems vol 13 no 6 pp 742ndash760 2005
[4] A Bardossy ldquoNote on fuzzy regressionrdquo Fuzzy Sets and Systemsvol 37 no 1 pp 65ndash75 1990
[5] H Tanaka H Ishibuchi and S Yoshikawa ldquoExponential possi-bility regression analysisrdquo Fuzzy Sets and Systems vol 69 no 3pp 305ndash318 1995
[6] K K Yen S Ghoshray and G Roig ldquoA linear regression modelusing triangular fuzzy number coefficientsrdquo Fuzzy Sets andSystems vol 106 no 2 pp 167ndash177 1999
[7] Y Chen J Tang R Y K Fung and Z Ren ldquoFuzzy regression-based mathematical programming model for quality functiondeploymentrdquo International Journal of Production Research vol42 no 5 pp 1009ndash1027 2004
[8] Z B Chen B Lin and Q Fang ldquoStudy on determinationmethod for parameters of Rockrsquos shear strength through leastabsolute linear regression based on symmetric triangular fuzzynumbersrdquo Applied Mechanics and Materials vol 170 pp 804ndash807 2012
[9] R Y K Fung Y Chen and J Tang ldquoEstimating the functionalrelationships for quality function deployment under uncertain-tiesrdquo Fuzzy Sets and Systems vol 157 no 1 pp 98ndash120 2006
[10] B Kheirfam and J-L Verdegay ldquoThe dual simplex method andsensitivity analysis for fuzzy linear programming with sym-metric trapezoidal numbersrdquo Fuzzy Optimization and DecisionMaking vol 12 no 2 pp 171ndash189 2013
[11] PDiamond andHTanaka ldquoFuzzy regression analysisrdquo inFuzzySets in Decision Analysis Operations Research and Statisticsvol 1 of Handbooks of Fuzzy Sets Series pp 349ndash387 KluwerAcademic Boston Mass USA 1998
[12] H Tanaka andH Lee ldquoInterval regression analysis by quadraticprogramming approachrdquo IEEE Transactions on Fuzzy Systemsvol 6 no 4 pp 473ndash481 1998
[13] C Hwang andD H Hong ldquoAnalyzing the nonlinear time seriesof turbulent flows with kernel interval regression machinerdquoInternational Journal of Uncertainty Fuzziness and Knowledge-Based Systems vol 13 no 4 pp 365ndash380 2005
[14] S Yagiz and C Gokceoglu ldquoApplication of fuzzy inferencesystem and nonlinear regression models for predicting rockbrittlenessrdquo Expert Systems with Applications vol 37 no 3 pp2265ndash2272 2010
[15] A Celmins ldquoLeast squares model fitting to fuzzy vector datardquoFuzzy Sets and Systems vol 22 no 3 pp 245ndash269 1987
[16] P Diamond ldquoFuzzy least squaresrdquo Information Sciences vol 46no 3 pp 141ndash157 1988
Mathematical Problems in Engineering 9
[17] Y-H O Chang ldquoHybrid fuzzy least-squares regression analysisand its reliability measuresrdquo Fuzzy Sets and Systems vol 119 no2 pp 225ndash246 2001
[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for emission prediction of acoal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013
[19] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988
[20] H Moskowitz and K Kim ldquoOn assessing the 119867 value in fuzzylinear regressionrdquo Fuzzy Sets and Systems vol 58 no 3 pp 303ndash327 1993
[21] Y-H O Chang and B M Ayyub ldquoFuzzy regression methodsmdasha comparative assessmentrdquo Fuzzy Sets and Systems vol 119 no2 pp 187ndash203 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
[17] Y-H O Chang ldquoHybrid fuzzy least-squares regression analysisand its reliability measuresrdquo Fuzzy Sets and Systems vol 119 no2 pp 225ndash246 2001
[18] Y Lv J Liu T Yang andD Zeng ldquoA novel least squares supportvector machine ensemble model for emission prediction of acoal-fired boilerrdquo Energy vol 55 pp 319ndash329 2013
[19] H Tanaka and J Watada ldquoPossibilistic linear systems and theirapplication to the linear regression modelrdquo Fuzzy Sets andSystems vol 27 no 3 pp 275ndash289 1988
[20] H Moskowitz and K Kim ldquoOn assessing the 119867 value in fuzzylinear regressionrdquo Fuzzy Sets and Systems vol 58 no 3 pp 303ndash327 1993
[21] Y-H O Chang and B M Ayyub ldquoFuzzy regression methodsmdasha comparative assessmentrdquo Fuzzy Sets and Systems vol 119 no2 pp 187ndash203 2001
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
