resolution of fuzzy regression model

Theory and Methodology

Resolution of fuzzy regression model

Hsiao-Fan Wang *, Ruey-Chyn Tsaur

Department of Industry Engineering, National Tsing Hua University, Hsinchu 30043, Taiwan, ROC

Received 16 November 1998; accepted 11 May 1999

Abstract

Fuzzy linear regression was originally introduced by Tanaka et al. To cope with di�erent types of input±output

information, several approaches to fuzzy linear regression have been proposed. In this paper, a type of problem with

crisp input and fuzzy output described by Tanaka is considered of which a modi®ed fuzzy least square method was

proposed for solution. It shows that with such an approach the predictability in the new model is better than TanakaÕsand its computation e�ciency is better than the conventional fuzzy least square method. Ó 2000 Elsevier Science B.V.

All rights reserved.

Keywords: Fuzzy regression; TanakaÕs model; Fuzzy least square method; Fuzzy constraint

1. Introduction

Fuzzy regression methods have been successfully applied to various problems such as forecasting[7,15,16] and engineering [2,8]. Since in reality, problems with crisp input and the resultant fuzzy output arecommonly existent, most of the research focused on such types of problems with di�erent forms of models.These researches either used TanakaÕs approach to minimize the total spread of the output [3,7,11]; oradopted the fuzzy least square method (FLSM) to minimize the total square error of the output [4,10,13].However, TanakaÕs approach has been criticized to provide too wide ranges in estimation which could notgive much help in applications [15], and the FLSM, though providing narrower range, costs too much ofcomputation time.

Therefore, the advantage of TanakaÕs model is its simplicity in programming and computation; and thatof FLSM is its minimum degree of fuzziness between the collected data and the estimated values. In thisstudy, how to derive a model such that both predictability and computability can be improved is our aim ofstudy.

European Journal of Operational Research 126 (2000) 637±650www.elsevier.com/locate/dsw

* Corresponding author. Fax: +886-35-722685.

E-mail address: [email protected] (H.-F. Wang).

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 3 1 7 - 3

In this paper, Section 2 focuses on the concept of TanakaÕs model with its advantages and disadvantages.In Section 3, FLSM is introduced, especially to describe its advantages and shortcomings by the cohesiveconcept. A comparison between FLSM and TanakaÕs model is presented in Section 4. The proposed modelis addressed in Section 5 with a computation procedure. In Section 6, an application to a decision problemto incorporate a decision makerÕs preference structure is discussed. Finally, evaluation and conclusion aredrawn in Section 7.

2. Tanaka's model

2.1. Concept of Tanaka's model

For a problem with crisp-input and fuzzy-output, Tanaka is the ®rst one to study such kinds of problems[14]. In his model the fuzzy output data are assumed to be a fuzzy number with a symmetric triangularmembership function denoted by ~Y i � �yi; ei�; i � 1; 2; . . . ;M , where yi is the center and ei is the spread ofthe ith collected data as

u~Y i�yi� �

1ÿ yiÿyij jei

; yi ÿ ei6 yi6 yi � ei

0; otherwise

8<: �1�

The basic model assumes a fuzzy linear function as below

~Yi � ~A0 � ~A1Xi1 � � � � � ~AN XiN � ~AXi; �2�where Xi � �Xi0;Xi1; . . . ;XiN �T is a vector of independent variables in the ith data; ~A � � ~A0; ~A1; . . . ; ~AN � is avector of fuzzy parameters presented in the form of symmetric triangular fuzzy numbers denoted by~Aj � �aj; cj�; j � 1; . . . ;N , with its membership function described as (3) below where aj is its central value

and cj is its spread value.

u ~Aj�aj� � 1ÿ ajÿajj j

cj; aj ÿ cj6 aj6 aj � cj;

0; otherwise:

(�3�

Therefore, formula (2) can be rewritten as

~Yi � �a0; c0� � �a1; c1�Xi1 � � � � � �aN ; cN �XiN : �4�By applying the Extension Principle [14], the derived membership function of fuzzy number ~Yi is shown in(5) and each value of the dependent variable can be estimated as a fuzzy number~Yi � Y L

i ; Yh�1

i ; Y Ui

ÿ �; i � 1; 2; . . . ;M , where the lower bound of ~Yi is Y L

i � �aÿ c�TXi; the center value of ~Yi isY h�1

i � aTXi and the upper bound of ~Yi is Y Ui � �a� c�TXi.

u ~Yi�yi� �

1ÿ yiÿaTXij jcTXi

; Xi 6� 0;

1; Xi � 0; yi � 0;

0; otherwise;

8>>><>>>: �5�

where cT � �c0; c1; . . . ; cN �; aT � �a0; a1; . . . ; aN �.The degree of ®tness of the estimated fuzzy linear model ~Yi � ~AXi to the given data ~Y i � �yi; ei� is

measured by index h�i which is the maximum value of h subject to Yhi � Y h

i with

638 H.-F. Wang, R.-C. Tsaur / European Journal of Operational Research 126 (2000) 637±650

Yhi � fyi j u~Y i

�yi�P hg and Y hi � fyi j u ~Yi

�yi�P hg:

The index h�i is plotted in Fig. 1. The degree of the ®tness of the fuzzy linear model to all data ~Y 1; . . . ; ~Y M isde®ned by mini �h�i �.

The vagueness of the fuzzy linear regression is de®ned by

J � c1 � � � � � cn: �6�Therefore, the problem of fuzzy regression is to ®nd fuzzy parameters ~Aj such that the total vagueness J isminimized subject to h�i P h for all i, where h is chosen by the decision maker and the h�i obtained fromFig. 1 as

h�i � 1ÿ yi ÿ aTXij jPNj�1 cj Xij

�� ÿ ei

: �7�

The above analysis leads to the following linear programming model [11]:

Min J � c1 � c2 � � � � � cN

s:t:XN

j�0

ajXij � �1ÿ h�XN

j�0

cj Xij

�� P yi � �1ÿ h�ei 8i � 1; 2; . . . ;M ;

ÿXN

j�0


j�0

cj xij

�� P ÿ yi � �1ÿ h�ei 8i � 1; 2; . . . ;M ;

cj P 0; aj 2 R; Xi0 � 1; 06 h6 1; 8i � 1; 2; . . . ;M ; j � 1; 2; . . . ;N :

�8�

It can be noted that fuzzy regression intervals derived from Tanaka are determined by the collected dataand the value h, thus the interval usually is too wide for prediction.

2.2. Modi®cation of Tanaka's model

In TanakaÕs model of (8), the objective function is resulted from minimizing the total spread of fuzzyparameters ~Aj. Therefore the relation between dependent variable and independent variables is desired to be

Fig. 1. Degree of ®tness of ~Yi to the collected data ~�Y i.

H.-F. Wang, R.-C. Tsaur / European Journal of Operational Research 126 (2000) 637±650 639

as crisp as possible. However, since the collected outputs are also fuzzy, when considering the minimal totalspread, such fuzziness should be taken into account as Min

PNj�1 cj Xij

�� ÿ ei

� �as shown in Fig. 1. Therefore

(8) should be rewritten as follows:

MinXN

j�0

XM

i�1

cj Xij

�� ÿ ei

!

s:t:XN

j�0


j�0

cj Xij

�� P yi � �1ÿ h�ei 8i � 1; 2; . . . ;M ;

ÿXN

j�0


j�0

cj Xij

�� P ÿ yi � �1ÿ h�ei 8i � 1; 2; . . . ;M ;

cj P 0; aj 2 R; Xi0 � 1; 06 h6 1; 8i � 1; 2; . . . ;M ; j � 1; 2; . . . ;N :

�9�

3. Fuzzy least square method

3.1. Concept of fuzzy least square method

In 1988, Diamond [4] proposed the Fuzzy Least Square Method to determine fuzzy parameters. Byadopting the concept of minimum fuzziness between the estimated dependent value ~Yi and the collecteddata ~Y i, a minimum regression interval can be obtained from a single-variate fuzzy linear regression model.That is, if ~Yi � ~A0 � ~A1Xi1 where ~A0 � �a0; c0�; ~A1 � �a1; c1�, a0 and a1 are the center values and c0; c1 are thespread values of ~A0 and ~A1, respectively, the method is to minimize the fuzzy distance between~A0 � ~A1Xi1 and ~Y i as de®ned by r� ~A0 � ~A1Xi1;

~Y i� �P

d� ~A0 � ~A1Xi1;~Y i�2. That is, by expanding

r� ~A0 � ~A1Xi1;~Y i� as follows:

r� ~A0 � ~A1Xi1;~Y i� � �a0 � c0 � a1Xi1 � c1Xi1 ÿ yi ÿ ei�2 � �a0 � a1Xi1 ÿ yi ÿ ei�2

� �a0 ÿ c0 � a1Xi1 ÿ c1Xi1 ÿ yi � ei�2 �10�

the solution can be obtained from the second deviation of (10) where the cohesive condition de®ned in (11)is satis®ed.

De®nition 1 ([4]). A nondegenerate data set Xi1;~Y i � yi; ei� �; i � 1; 2; . . . ;M , where each ~Y i 2 I�R�, is said

to be cohesive if

eXM

i�1

Xi1

ÿ ÿ X�

yi

ÿ ÿ y�P X

XM

i�1

Xi1

ÿ ÿ X�

ei� ÿ e�P 0; �11�

where X is the mean ofPM

i�1 Xi1, y the mean ofPM

i�1 yi and e is the mean ofPM

i�1 ei.

Therefore, the equations arising from the derivatives of r� ~A0 � ~A1Xi1;~Y i� can be set equal to zero to

derive the values of a0, a1 and c0, c1. Since this method only discusses the cohesive condition, of which thetrends between central value and spread value are the same, and the values of a0, a1 and c0, c1 are requiredto be positive, these may not be applied to general cases. Thus the next subsection will derive a more generalmodel of FLSM.


3.2. An extension of FLSM

If the cohesive condition is to be relaxed, all the characters of the parameters a0, a1 and c0, c1 in (10) needto be checked.

Because the � signs of a0, a1 and c0, c1 in each of the four derivative equations of (10) have the followingpossibilities, let us consider each of them as follows:

(a) If a0 > 0; a1 > 0, and c0, c1 are unrestricted, then

or� ~A0 � ~A1Xi1;~Y i�

oa0

� 2XM

i�1

a0f ÿ c0 � �a1 ÿ c1�Xi1 ÿ �yi ÿ ei� � a0 � c0 � �a1 � c1�Xi1 ÿ �yi � ei� � a0 � a1Xi1 ÿ yig

or� ~A0 � ~A1Xi1;~Y i�

oc0

� 2XM

i�1

a0f ÿ c0 � �a1 ÿ c1�Xi1 ÿ �yi ÿ ei�� ÿ 1� � a0 � c0 � �a1 � c1�Xi1 ÿ �yi � ei� � a0 � a1Xi1 ÿ yig

or� ~A0 � ~A1Xi1;~Y i�

oa1

� 2XM

i�1

Xi1 a0f ÿ c0 � �a1 ÿ c1�Xi1 ÿ �yi ÿ ei� � a0 � c0 � �a1 � c1�Xi1 ÿ �yi � ei� � a0 � a1Xi1 ÿ yig

or� ~A0 � ~A1Xi1;~Y i�

oc1

� 2XM

i�1

Xif ÿ a0� ÿ c0 � �a1 ÿ c1�Xi1 ÿ �yi ÿ ei�� a0� � c0 � �a1 � c1�Xi1 ÿ �yi � ei��g:

Solving the above equations by setting them to be zero, we have

a1 � NPM

i�1 Xi1yi ÿPM

i�1 Xi1PM

i�1 yi

NPM

i�1 X 2i1 ÿ

PMi�1 Xi1

ÿ �2; a0 � y ÿ a1X ; c1 � N

PMi�1 Xi1ei ÿ

PMi�1 Xi1

PMi�1 ei

NPM

i�1 X 2i1 ÿ

PMi�1 Xi1

ÿ �2;

and c0 � eÿ c1X , where

e �PM

i�1 ei

M; X �

PMi�1 Xi1

M; y �

PMi�1 yi

M:

(b) If a0 < 0; a1 > 0; c0 < 0 and c1 unrestricted, the result is

a1 � NPM

i�1 Xi1yi ÿPM

i�1 Xi1PM

i�1 yi

NPM

i�1 X 2i1 ÿ

PMi�1 Xi1

ÿ �2; a0 � y ÿ a1X ; c1 � N

PMi�1 Xi1ei ÿ

PMi�1 Xi1

PMi�1 ei

NPM

i�1 X 2i1 ÿ

PMi�1 Xi1

ÿ �2;

and c0 � ÿ�eÿ c1X �.(c) If a0 > 0, a1 < 0, c0 unrestricted, and c1 < 0, the result is

a1 � NPM

i�1 Xi1yi ÿPM

i�1 Xi1PM

i�1 yi

NPM

i�1 X 2i1 ÿ

PMi�1 Xi1

ÿ �2; a0 � y ÿ a1X ; c1 � ÿN

PMi�1 Xi1ei ÿ

PMi�1 Xi1

PMi�1 ei

NPM

i�1 X 2i1 ÿ

PMi�1 Xi1

ÿ �2;

and c0 � eÿ c1X .


(d) If a0 < 0; a1 < 0 and c0 < 0; c1 < 0, the result is that

a1 � NPM

i�1 Xi1yi ÿPM

i�1 Xi1PM

i�1 yi

NPM

i�1 X 2i1 ÿ

PMi�1 Xi1

ÿ �2; a0 � y ÿ a1X ;

c1 � ÿNPM

i�1 Xi1ei ÿPM

i�1 Xi1PM

i�1 ei

NPM

i�1 X 2i1 ÿ

PMi�1 Xi1

ÿ �2; c0 � ÿ�eÿ c1X �:

Besides, (d) also can be divided into the cases of a0 < 0; a1 < 0; c0 < 0 and c1 unrestricted, which is equalto (b), and that of a0 < 0; a1 < 0; c0 unrestricted and c1 < 0 which is equal to (c). Thus the procedure to®nd the solution in the fuzzy least square method can be summarized as follows:1. Calculating the values a0 and a1.2. If a0 > 0; a1 > 0; then the solution is equal to (a).3. If a0 < 0; a1 > 0, then if

XM

i�1

Xi1

XM

i�1

Xi1ei ÿXM

i�1

X 2

i1

XM

i�1

ei P 0;

the solution is as (a); otherwise it is equal to (b).4. If a0 > 0; a1 < 0, then if

XM

i�1

Xi1

XM

i�1

ei ÿ NXM

i�1

Xi1ei P 0;

then the solution is as (a); otherwise it is equal to (c).5. If a0 < 0; a1 < 0, then if

XM

i�1

Xi1

XM

i�1

Xi1ei ÿXM

i�1

X 2

i1

XM

i�1

ei P 0;XM

i�1

Xi1

XM

i�1

ei ÿ NXM

i�1

Xi1ei P 0;

then the solution is as (a); if they are all negative then they are equal to (d); if eitherPMi�1 Xi1

PMi�1 Xi1ei ÿ

PMi�1 X 2

i1

PMi�1 ei < 0 or

PMi�1 Xi1

PMi�1 ei ÿ N

PMi�1 Xi1ei < 0; then they are equal to

(b) and (c), respectively.From the above extension it can be noted that if the ®tted data are multivariate, then using FLSM is not

easy because of the detecting procedure for di�erent signs.

4. Comparison

Based on the above analysis, an example is illustrated to compare the ®tness between the Tanaka modelin (8) and the fuzzy least square method in (10) (see Fig. 2). Table 1 has the data given by [4] and in Tanakamodel, value h is set to be 0 in order to obtain the minimum fuzzy spread.

The corresponding fuzzy linear regressions obtained from TanakaÕs model and the fuzzy least squaremethod are ~Y � 2:112� �0:0356; 0:0924�X and ~Y � 1:538� �0:108; 0:019�X, respectively.

It can be noted that the concept of the fuzzy least square method is similar to the least square method inStatistics when c0; c1 and ei are all zero. Thus its regression interval derived from the minimized distance to


the collected data is narrower than TanakaÕs. However, the computation complexity for TanakaÕs modelwhich is in a form of linear program is O(M2N) [6]; whereas that for the fuzzy least square method in theworse case for M � N , is O(N4M2) [12]. Therefore, from the prediction point of view, FLSM is better;whereas from the computational point of view, TanaksÕs model is more e�cient.

Furthermore, if rewriting Eq. (10) into

�a0 � c0 � a1Xi1 � c1Xi1 ÿ yi ÿ yi ÿ ei�2 � �a0 � a1Xi1 ÿ yi�2

� �a0 ÿ c0 � a1Xi1 ÿ c1Xi1 ÿ yi � ei�2

� ��a0 � a1Xi1 ÿ yi� � �c0 � c1Xi1 ÿ ei��2 � �a0 � a1Xi1 ÿ yi�2

� ��a0 � a1Xi1 ÿ yi� ÿ �c0 � c1Xi1 ÿ ei��2 � 2�c0 � c1Xi1 ÿ ei�2 � 3�a0 � a1Xi1 ÿ yi�2

� 2�cTXi ÿ ei�2 � 3�Y h�1i ÿ yi�2; �12�

one can realize that a fuzzy least square method is in fact to ®nd a fuzzy regression by minimizing thedistance between their center and spread values. Based on this concept, we propose our model in thefollowing section with more e�cient solution procedure.

Fig. 2. Comparison of TanakaÕs model and fuzzy least square.

Table 1

Collected data [4]

~Y i � yi; ei� � Xi1

(4.0, 0.8) 21.0

(3.0, 0.3) 15.0

(3.5, 0.35) 15.0

(2.0, 0.4) 9.0

(3.0, 0.45) 12.0

(3.5, 0.7) 18.0

(3.5, 0.38) 6.0

(2.5, 0.5) 12.0


5. The proposed fuzzy regression model

Eq. (12) has suggested one way to analyze a fuzzy regression model with a narrower interval. In order tobe consistent with the concept of TanakaÕs model, the objective function of the proposed model is ®rstsuggested to minimize the di�erences of spread between collected data and the proposed regression interval;and then, the errors in central values are further minimized.

5.1. Minimization of the spread values

Considering a fuzzy regression by minimizing the square errors of the spread values, the membershipfunction in (3) is revised into (13) below to describe a fuzzy relation between the dependent variable and theindependent variables:

u ~Aj�aj� � 1

(ÿ aj ÿ aj

cj

� �2): �13�

If a fuzzy linear function is assumed to be ~Yi � ~A0 � ~A1Xi1 � � � � � ~AN XiN � ~AXi, then using thefuzzy arithmetics for fuzzy number, the membership function of the estimated output ~Yi can be obtainedby

u ~Yi�yi� � 1

(ÿ yi ÿ aTXi

cTXi ÿ ei

� �2): �14�

Our aim is to minimize the total vagueness of the given data by MinPN

j�0�PM

i�1 cjXij ÿ ei�2 such that themembership degree of each observation is greater than a threshold h as u ~Yi

�yi�P h.This leads to the following nonlinear programming problem:

MinXN

j�0

XM

i�1

cjXij

ÿ ei

!2

s:t:XN

j�0

ajxij � �1ÿ h�1=2XN

j�0

cj xij

�� P yi � �1ÿ h�1=2ei 8i � 1; 2; . . . ;M ;

ÿXN

j�0

ajxij � �1ÿ h�1=2XN

j�0

cj xij

�� P ÿ yi � �1ÿ h�1=2ei 8i � 1; 2; . . . ;M ;

cj P 0; aj 2 R; xi0 � 1; 06 h6 1; 8i � 1; 2; . . . ;M ; j � 1; 2; . . . ;N :

�15�

5.2. Minimization of the central values

Note that we can soften the relations of constraint from model (15) P , 6 into J and [, then a moree�cient solution can be obtained because of a feasible region being extended [11,17,18]. Besides, to obtainthe minimum regression interval that covers the collected data, we set h to be zero. Then, model (15) can bewritten as follows:


MinXN

j�0

XM

i�1

cjXij

ÿ ei

!2

s:t:XN

j�0

ajxij �XN

j�0

cj xij

�� J yi � ei 8i � 1; 2; . . . ;M ;

ÿXN

j�0

ajxij �XN

j�0

cj xij

�� J ÿ yi � ei 8i � 1; 2; . . . ;M ;

cj P 0; aj 2 R; xi0 � 1; 8i � 1; 2; . . . ;M ; j � 1; 2; . . . ;N ;

�16�

where ` J ' is graphically shown in Fig. 3 with membership function

uJ �Xi� �1 if yi � ei6 aTXi � cTXi;

1ÿ �yi�ei�ÿ�aTXi�cTXi�p if yi � ei ÿ p6 aTXi � cTXi6 yi � ei;

0 otherwise:

8<: �17�

Therefore, to satisfy (17) to the maximal degree, we summarize our analysis as follows:

Max k

s:t: �1ÿ k�p ÿXN

j�0

XM

i�1

cjXij

ÿ ei

!2

P ÿ d0;

�1ÿ k�p �XN

j�0

ajxij �XN

j�0

cj xij

�� P yi � ei 8i � 1; 2; . . . ;M ;

�1ÿ k�p ÿXN

j�0

ajxij �XN

j�0

cj xij

�� P ÿ yi � ei 8i � 1; 2; . . . ;M ;

06 k6 1; cj; aj 2 R; xi0 � 1; 8i � 1; 2; . . . ;M ; j � 1; 2; . . . ;N :

�18�

Model (18) says that the optimal value of k is determined by the maximum value of kÕs that satis®es all ofthe constraints. Thus k is treated equally in all constraints. Besides, p is the width of the tolerance interval ofyi � ei, and the parameter d0 represents the desired value of the objective function in (18). In addition, sincevalue cj in model (16) is required to be positive, which cannot explain the situation that the trend of centervalues is increasing, but that of the spread value is decreasing [1,4], therefore, relaxing value cj to be anunrestricted value is necessary in (18). For example [4], consider data pairs

Fig. 3. Membership function of fuzzy relation.


X1 � 1; ~Y 1 � 1;3

4

� �; X2 � 2; ~Y 2 � 15

8;3

8

� �; X3 � 3; ~Y 3 � 13

4;

1

16

� �:

The trend of center value is increasing, but that of the spread is decreasing.After proposing the model, the following theorem shows that the proposed model has better predict-

ability than TanakaÕs.

Theorem 1. For an input-crisp output-fuzzy regression problem, if the distribution of the collected data~Y i �i � 1; 2; . . . ;M� is not all equal, then the fuzzy regression interval derived from the proposed method issmaller than that of Tanaka's model.

Proof.

(i) Since the membership function of J is de®ned as

uJ �Xi� �1 if yi � ei6 aTXi � cTXi;

1ÿ �yi�ei�ÿ�aTXi�cTXi�p if yi � ei ÿ p6 aTXi � cTXi6 yi � ei;

0 otherwise:

8><>:Thus, 9 an i, i � 1; 2; . . . ;M ; 3 yi � ei > aTXi � cTXi if all of the collected data ~Y iare not all equal.(ii) According to TanakaÕs model, all of the collected data are covered by the derived fuzzy regressioninterval, thus yi � ei6 aT�Xi � cT�Xi 8 i � 1; 2; . . . ;M ; where aT�Xi � cT�Xi is the upper bound of the re-gression interval derived from TanakaÕs model.From the discussion in (i) and (ii), it is obvious that aTXi � cTXi < yi � ei6 aT�Xi � cT�Xi, thus the fuzzyregression interval derived by the proposed method is smaller than Tanaka's model.

6. An application to decision problem ± an extension

Based on Utility Theory [5], a preference structure can be described by di�erent utility functions. Infuzzy environment, three utility functions can be expressed in terms of membership functions. Therefore,when a membership function is in general expressed by (19), decision makers as risk-averser �k � 0:5�, risk-neutral �k � 1�, and risk-seeker �k � 2� can be viewed from Fig. 4.

Fig. 4. The membership types of the risk attitude.


uk�aj� � 1ÿ aj ÿ aj

cj

� �k

: �19�

Now when k � 2, it is equivalent to (13) as the proposed method.When k � 0:5 we can derive the corresponding model as follows:

Max k


j�0

XM

i�1

cjXij

ÿ ei

!0:5

P ÿ d0;

�1ÿ k�p �XN

j�0

ajxij �XN

j�0

cj xij

�� P yi � ei 8i � 1; 2; . . . ;M ;

�1ÿ k�p ÿXN

j�0

ajxij �XN

j�0

cj xij

�� P ÿ yi � ei 8i � 1; 2; . . . ;M ;

06 k6 1; cj; aj 2 R; xi0 � 1; 8i � 1; 2; . . . ;M ; j � 1; 2; . . . ;N ;

�20�

and if the decision maker is risk-neutral then the model can be obtained as follows:

Max k


j�0

XM

i�1

cjXij

ÿ ei

!P ÿ d0;

�1ÿ k�p �XN

j�0

ajxij �XN

j�0

cj xij

�� P yi � ei 8i � 1; 2; . . . ;M ;

�1ÿ k�p ÿXN

j�0

ajxij �XN

j�0

cj xij

�� P ÿ yi � ei 8i � 1; 2; . . . ;M ;

06 k6 1; cj; aj 2 R; xi0 � 1; 8i � 1; 2; . . . ;M ; j � 1; 2; . . . ;N :

�21�

Models (18) and (20) can be solved by any nonlinear algorithm. If using the most commonly used algorithmsuch as the Frank±Wolfe algorithm [9], its complexity is O(M2N). Model (21) is in linear programmingform and can be solved by the simplex method with O(M2N) [6]. Thus the e�ciency of the proposed modelis better than the fuzzy least square method.

To show three cases numerically and for comparison, we use the data of Table 1. Each case is illustratedwith the corresponding model and ®gure where Tanaka's model is plotted by `+', the fuzzy least squaremodel is plotted by `)', the proposed model is plotted by `.', and the collected data are plotted by `*'.Besides, value p is set to be 5 and d0 is set to be 0 for the desired value of the total vagueness. The results ofcomparison are presented in Table 2.

Table 2

Comparison results

Cases Fuzzy regression derives from the proposed method Figure

Risk-seeker ~Y � �1:313; 0:533� � �0:1244; 0:0036�X Fig. 5

Risk-neutral ~Y � �1:256; 0:5582� � �0:1307;ÿ0:0027�X Fig. 6

Risk-averser ~Y � �1:4024; 1:002� � �0:1237;ÿ0:0096�X Fig. 7


Fig. 5. The test result as k� 2.

Fig. 6. The test result as k� 1.

Fig. 7. The test result as k� 0.5.


From Table 2 and Figs. 5±7, it can be noted that the fuzzy regression intervals of the fuzzy least squareand the proposed method are similar in that all are narrower than TanakaÕs model. Besides, due to theunrestricted cj, the regression intervals of the proposed model do not diverge as quickly as the increase ofindependent variables. Thus the proposed approach can be appropriately ®t into the collected data with afuzzy regression model of which a more useful regression interval can be used for prediction.

It can be observed from Fig. 7 that the regression interval with risk-averser is wider than those of risk-seeker and risk-neutral. This can be interpreted as that when the decision attitude is risk-averser, he alwaysneeds to consider other situations which cause a larger interval.

7. Evaluation and conclusion

In this study, we proposed a model such that the predictability of TanakaÕs model can be improved andthe computation complexity of the fuzzy least square method can be decreased. This is achieved byadopting the Tanaka modelÕs simplicity and the fuzzy least square methodÕs predictability. Besides, theproposed model can be extended to describe a decision makerÕs preference attitude such that applicabilitycan be increased.

Further studies can be extended to other fuzzy regression problems when center value a is a fuzzynumber or spread value c is fuzzy or both of them are fuzzy to present the vagueness environment indecision making.

Acknowledgements

The authors acknowledge the ®nancial support from National Science Council, Taiwan, ROC, withproject number NSC 86-2213-E007-020.

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resolution of fuzzy regression model

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