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A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED OPERATIONS
 

A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED OPERATIONS

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A least absolute approach to multiple fuzzy regression using Tw-norm based arithmetic operations is ...

A least absolute approach to multiple fuzzy regression using Tw-norm based arithmetic operations is
discussed by using the generalized Hausdorff metric and it is investigated for the crisp input- fuzzy output
data. A comparative study based on two data sets are presented using the proposed method using shape
preserving operations with other existing method.

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    A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED OPERATIONS A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED OPERATIONS Document Transcript

    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013DOI : 10.5121/ijfls.2013.3206 73A LEAST ABSOLUTE APPROACH TOMULTIPLE FUZZY REGRESSION USING Tw-NORM BASED OPERATIONSB. Pushpa1and R. Vasuki21Manonmaniam Sundaranar Universty, Tirunelveli, India1Panimalar Institute of Technology, Poonamallee, Chennai, India.pushpajuly14@gmail.com2SIVET College, Gowrivakkam, Chennai, India.vasukidevi06@gmail.comABSTRACTA least absolute approach to multiple fuzzy regression using Tw-norm based arithmetic operations isdiscussed by using the generalized Hausdorff metric and it is investigated for the crisp input- fuzzy outputdata. A comparative study based on two data sets are presented using the proposed method using shapepreserving operations with other existing method.KEYWORDSFuzzy Regression, Hausdorff- metric, Fuzzy Linear Programming, Fuzzy parameters, Crisp input data,Fuzzy output data, shape preserving operations.1. INTRODUCTIONRegression analysis has a wide spread applications in various fields such as business,engineering, agriculture, health sciences, biology and economics to explore the statisticalrelationship between input (independent or explanatory) and output (dependent or response)variables. Fuzzy regression models were proposed to model the relationship between thevariables, when the data available are imprecise (fuzzy) quantities and/or the relationship betweenthe variables are fuzzy. Regression analysis based on the method of least -absolute deviation hasbeen used as a robust method. When outlier exists in the response variable, the least absolutedeviation is more robust than the least square deviations estimators. Some recent works on thistopic are as follows: Chang and Lee [1] studied the fuzzy least absolute deviation regressionbased on the ranking method for fuzzy numbers. Kim et al. [2] proposed a two stage method toconstruct the fuzzy linear regression models, using a least absolutes deviations method. Torabiand Behboodian [3] investigated the usage of ordinary least absolute deviation method to estimatethe fuzzy coefficients in a linear regression model with fuzzy input – fuzzy output observations.Considering a certain fuzzy regression model, Chen and Hsueh [4] developed a mathematicalprogramming method to determine the crisp coefficients as well as an adjusted term for a fuzzyregression model, based on L1 norm (absolute norm) criteria. Choi and Buckley [5] suggested twomethods to obtain the least absolute deviation estimators for common fuzzy linear regressionmodels using TM based arithmetic operations. Taheri and Kelkinnama [6,7] introduced someleast absolute regression models, based on crisp input- fuzzy output and fuzzy input-fuzzy outputdata respectively.In a regression model, multiplication of fuzzy numbers are done by arithmetic operations such asα-levels of multiplication of fuzzy numbers and the approximate formula for multiplication offuzzy numbers. Apart from these two, we know that using the weakest T – norm (Tw), the shape
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 201374of fuzzy numbers in multiplication will be preserved. In this regard, Hong et al. [8] presented amethod to evaluate fuzzy linear regression models based on a possibilistic approach, using Tw -norm based arithmetic operations.The objective of this study is to develop a least absolute multiple fuzzy regression model tohandle the functional dependence of crisp inputs-fuzzy output variables using the generalizedHausdorff- metric between fuzzy numbers as well as linear programming approach.In this paper, section 2 focuses on some important preliminary definitions and basics on fuzzyarithmetic operations based on the weakest T-norm. In section 3, the new approach based onHausdorff metric is presented using the shape preserving operations on fuzzy numbers and it isanalyzed with crisp input and fuzzy output and discussed the goodness of fit of the proposedmodel. In section 4, by using numerical examples we provide some comparative studies to showthe performance of the proposed method.2. PRELIMINARIESA fuzzy number is a convex subset of the real line with a normalized membership function. Atriangular fuzzy number ( , , )a a α β=% is defined by1 ,( ) 1 ,0 ,a tif a t aa ta t if a t aotherwiseααββ −− − ≤ ≤ −= − ≤ ≤ +%where a∈ is the center, 0α > is the left spread and 0β > is the right spread of a% .If α β= , then the triangular fuzzy number is called a symmetric triangular fuzzynumber and it is denoted by ( , )a α . A fuzzy number ( , , )LRa a α β=% of type L-R is afunction from real number into the interval ( 0 , 1) satisfying,( ) ,0 ,t aR a t aa ta t L a t aotherwiseββαα  −≤ ≤ +    −  = − ≤ ≤   %where L and R are non increasing and continuous functions from (0,1) to (0,1) satisfyingL(0)=R(0)=1 and L(1)=R(1)=0.A binary operation T on the unit interval is said to be atriangular norm [9] (t-norm) if and only if f T is associative, commutative, non-decreasing andT(x,1)=x for each [0,1]x∈ . Moreover, every t-norm satisfies the inequality,( , ) ( , ) ( , ) min( , )w MT a b T x y T a b a b≤ ≤ = where, 1( , ) , 10 ,wa if bT a b b if aotherwise== =The critical importance of min( , ), , max(0, 1) ( , )wa b a b a b and T a b+ − is emphasized froma mathematical point of view in Ling [9]. The usual arithmetic operations on real numbers can be
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 201375extended to the arithmetic operations on fuzzy numbers by means of extension principle by Zadeh[10], which is based on a triangular norm T. Let A% and B% be fuzzy numbers of the real line .The fuzzy number arithmetic operations are summarized as follows:Fuzzy number addition⊕ :( ) x,y,x y zA B (z) sup T A(x),B(y) .+ = ⊕ =  % %% %Fuzzy number multiplication ⊗ :( ) x,y,x.y zA B (z) sup T A(x),B(y) .= ⊗ =  % %% %The addition (subtraction) rule for L-R fuzzy number is well known in case of TM based addition,in which the resulting sum is again an L-R fuzzy number, i.e., the shape is preserved. Let( , , ) , ( , , )A A LR B B LRA a B b= α β = α β% % . Then using TM in the above definition,( , , )M A B A B LRA B a b α α β β⊕ = + + +% %It is also known that the wT based addition and multiplication preserves the shape of L-R fuzzynumbers[11,12,13,14]. We know that TM based multiplication does not preserve the shape of L-R fuzzy numbers. In this section, we consider wT based multiplication of L-R fuzzy numbers.Let T= wT be the weakest t-norm and let ( , , ) , ( , , )LR LRA A B BA a B b= α β = α β% % be two L-Rfuzzy numbers, then the addition and multiplication of ( , , ) , ( , , )LRA A LR B BA a B b= α β = α β% %is defined as[15],( ,max( , ),max( , ))W A B A B LRA B a b⊕ = +% % α α β β( )( )( ,max( , ),max( , )) , 0( ,max( , ),max( , )) , , 0( ,max( , ),max( , )) , 0, 00, , , 0, 00, , , 0, 0(0,0,0) , 0, 0A B A B LRA B A B RLA B A B LLWA A LRA A RLLRab b a b a for a bab b a b a for a bab b a b a for a bA Bb b for a bb b for a bfor a bα α β ββ β α βα β β αα ββ α><< >⊗ = = >− − = <= =% %In particular, if ( , ), ( , )A BA a B b= α = α% % are symmetric fuzzy numbers, then the multiplication ofA and B% % is written as, ( ,max( , ))w A B LLA B ab b aα α⊗ =% %
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 201376A distance between fuzzy numbers: Several metrics are defined on the family of all fuzzynumbers[16]. The generalized Hausdorff metric fulfil many good properties, also easy tocompute in terms of generalized mid and spread functions. The concept of generalized mid andspread may be useful in working with (fuzzy) convex compact sets [17].),( αα BAdH is the Hausdorff metric between crisp sets αα BandA given by,−−=∈∈∈∈babaBAdBbAaAaBbHαααααα infsup,infsupmax),(If [ ] [ ]212211 ,, bbIandaaI == are two intervals, then( ) { } 2121221121 ,max, IsprIsprImidImidbabaIIdH −+−=−−=Where2,2121211aasprIaaImid−=+= [16].The generalized Hausdorff metric between TT bBaA ),(~,),(~βα == is then,βαβα −+−=−+−= ∞ baBADbaBAD )~,~(,5.0)~,~(13. FUZZY LINEAR REGRESSION USING THE PROPOSEDAPPROACHIn this section, we are discussing fuzzy linear regression about the proposed approach based onHausdorff metric using Tw norm based operations with crisp input- fuzzy output data, in whichthe coefficients of the models are also considered as fuzzy numbers.Consider the set of observed data { }( , ), 1,...,i iX Y i n=% % where ( , )ii iX x= γ% and ( , )i i iY y e=%are symmetric fuzzy numbers. Our aim is to fit a fuzzy regression model with fuzzycoefficients to the aforementioned data set as follows:0 1 1i w w i w w p w ip w iˆY A ( A X ) ..... ( A X ) A X= ⊕ ⊗ ⊕ ⊕ ⊗ = ⊗% % % %% % % %, 1,....=i n (1)where ( ), , 1,...j j jA a j p= α =% are symmetric fuzzy numbers and the arithmeticoperations are based on the weakest Tw norm.Consider the least absolute optimization problem as follows:Minimizes 1( , )w jD Y A X⊗%% % i.e.,Min ( )1 0 1 00.5 max ,n k n ki ij j i j ij ij ji j i jy x a e a x= = = =− + − γ α∑ ∑ ∑ ∑ (2)subject to
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 201377( )( )( )1 110 11 110 11( ) max , ( )( ) max , ( )0 1,max , 0, 1,2,...n nj ij j ij ij j i ij pjn nj ij j ij ij j i ij pjj ij ij jj pa x L h a x y L h ea x L h a x y L h eha x i m− −≤ ≤=− −≤ ≤=≤ ≤+ γ α ≥ −− γ α ≤ +< <γ α ≥ ∀ =∑ ∑∑ ∑(4)Solving this optimization problem using LINGO, we can estimate the fuzzy coefficientsof the model. Several methods have been proposed to detect the presence of outliers,relying on graphical representation and/or on analytical procedures. The box plot or box-and-whisker plot describes the key features of data through the five-number summaries:the smallest observation, lower quartile (Q1), median (Q2), upper quartile (Q3), and thelargest observation (sample maximum). A box plot indicates the abnormal observations.Usually, outlier cut offs are set at 1.5 times the inter-quartile range [18, 19]. In a fuzzyframework, we can draw box plots side by side to detect outliers in the distributions ofthe centers, of the spreads and of the input variables. To overcome limitations in previousapproaches, Hung and Yang [20] consider the effect of each observation on the value ofthe objective function in Tanaka’s approach. Let JM be the optimal value of the objectivefunction and JM(i)is the corresponding value obtained deleting the ithobservation. Theratio( )iM MiMJ JrJ−= is a synthetic evaluation of the impact of the ithobservation on thevalue of the objective function. Observations with large ir value are more likely to beanomalous. Combining these ratios with box plots, or with other suitable graphicalrepresentations, provides an effective way to detect a single outlier, with respect to theinput variables and/or to the centers or the spreads of the fuzzy response variable. Thisapproach could be generalized to the inspection of multiple outliers, but the processbecomes more computing demanding as the sample size and/or the number of outliersincreases.3.1. Evaluation of Regression modelsTo investigate the performance of the fuzzy regression models, we use similarity measurebased on the Graded mean integration representation of distance proposed by Hsieh andChen[21]1( , )1 ( , )S A Bd A B=+% %% %, ( , ) ( ) ( )d A B P A P B= −% %% % , ( )P A% and ( )P B% are thegraded mean integration representation distance.Also to evaluate the goodness of fit between the observed and estimated values from [22], if( , )A a σ=% and ( , )B b τ=% be two normal fuzzy numbers, then2( , ) e x pa bA Bσ τ − = −  +  % %is the goodness of fit of observed and estimated fuzzynumbers A% and B% .
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 2013784. EXAMPLESIn this section, we discuss our proposed method with multiple inputs in the following examples.For the example 1, we study the sensitivity of the proposed approach with respect to outlier datapoints with Hung and Yang [20] outlier treatment.EXAMPLE-1Consider the dataset in Table1 in which the observations are crisp multiple inputs andfuzzy outputs without modifying the outlier in the spread using Hung and Yang omissionapproach [20].S.No.Explanatory Variables response variable ( )iM MiMJ JrJ−=x1 x2 x3 y eps1 2 0.5 15.25 5.83 3.56 0.0956262 0 5 14.13 0.85 0.52 0.1631093 1.13 1.5 14.13 13.93* 8.5* 0.4454394 2 1.25 13.63 4 2.44 -0.309795 2.19 3.75 14.75 1.65 1.01 -0.225096 0.25 3.5 13.75 1.58 0.96 0.0779277 0.75 5.25 15.25 8.18 4.99 0.1984058 4.25 2 13.5 1.85 1.13 -0.25154* indicates outlierTable 1. Dataset with crisp input and fuzzy output with outlierFigure 1. Fuzzy regression model with outlier for the dataset in Table 1 using the proposed approach.The fuzzy regression model obtained by the proposed approach,1 2 3(0,3.8212) (0,0) (0,0) (0.3013,0)w w w w w wY X X X= ⊕ ⊗ ⊕ ⊗ ⊕ ⊗% withoptimum value h=0.215 and the graph is given in Fig.1. In table 1, third data is an
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 201379abnormal data, after identifying the abnormal data using Hung and Yang [20] method, deletingthe data which yields a better result. The fuzzy regression model obtained by proposed approachis 1 2 3(2.482,0) (0,0) (0,0) (0.2042,0)w w w w w wY X X X= ⊕ ⊗ ⊕ ⊗ ⊕ ⊗%with optimum level h= 0.322 (shown in Figure 2.)Figure 2. Using Hung and Yang [20] method of omission approach for the dataset with outlier in table1using the proposed approachComparing the performance of the proposed with the some other existing methods, using theChoi and Buckley’s [5] method, the optimal model is1 2 32.8273 0.3878 1.0125 0.6185 (0,1.0696,2.0042)Y x x x= − ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ ⊕%Chen Hseuh [23] proposed a least square approach to fuzzy regression models with crispcoefficients. 1 2 316.7956 1.0989 1.1798 1.8559 (0,2.8888)Y x x x= − ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ ⊕%Hassanpour et al. [24] proposed least absolute regression method that minimizes the differencebetween centers of the observed and estimated fuzzy responses and also between the spreads ofthem, using goal programming approach. They took into account fuzzy coefficients for crispinputs in their model. Employing their model for the given example yields the following model,1 2 3( 2.8273,0.0000) (0.3877,0.0000) (1.0125,0.000) (0.6185,0.1790)Y x x x= − ⊕ ⊗ ⊕ ⊗ ⊕ ⊗%Using the graded mean integration representation, the similarity measure for the proposed andabove existing models is given in table 2.
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 201380Methods ProposedChoi andBuckley[5]Chen and Hseuh[23]Hassanpour [24]Mean Similaritymeasure betweenthe observed andthe existingmethods0.326389 0.307915 0.122202 0.209148Table 2. Similarity measure between various models with multiple inputsThe table 2 illustrates that the mean similarity measure for the proposed model is 0.3264 whichhas effective performance with other existing methods using Hung and Yang method of omissionapproach with outlier.EXAMPLE-2We now apply this model to analyse the effect of the composition of Portland cement on heatevolved during hardening. The data shown in Table 3 except is are taken from [25]. Thevalues are assumed by R.Xu et al. [26].Obs.No. ( , )i i iy y s=% 1ix 2ix 3ix ( , )i i iY Y σ=% Goodnessof fit1 (78.5,6.9) 7 26 6 (78.33,0.709) 0.99952 (74.3,6.4) 1 29 15 (71.99,0.709) 0.89973 (104.3,9.4) 11 56 8 (106.25,0.709) 0.93644 (87.6,7.8) 11 31 8 (88.96,0.709) 0.97485 (95.5,8.6) 7 52 6 (96.30,0.709) 0.99266 (109.2,9.9) 11 55 9 (105.75,0.709) 0.89977 (102.7,9.3) 3 71 17 (104.81,0.709) 0.95658 (72.5,6.2) 1 31 22 (74.75,0.709) 0.89949 (93.1,8.3) 2 54 18 (91.56,0.709) 0.971210 (115.9,10.6) 21 47 4 (116.20,0.709) 0.999311 (83.8,7.4) 1 40 23 (81.16,0.709) 0.899412 (113.3,10.6) 11 66 9 (113.35,0.709) 0.999913 (109.4,9.9) 10 68 8 (112.85,0.709) 0.8996Table 3. Performance of the proposed model in Example 2.By using the proposed method, we have1 2 3(47.299,0.7093) (1.6963,0) (0.6914,0) (0.196,0)w w w w w wY x x x= ⊕ ⊗ ⊕ ⊗ ⊕ ⊗%with h=0.675 (shown in Fig.3)
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 201381Figure 3. Fuzzy regression model with crisp multiple input fuzzy output in Example 2.Furthermore, we calculate the goodness of fit using the fitted equation and the observed values.The results are given in table3, the goodness of fit of every iy% and iY% are all greater than 0.9,which indicates that the fitted result of the model is very good.EXAMPLE-3Consider the following crisp input-fuzzy output data given by Tanaka et al. [27]( ; ) ((8.0,1.8) ;1), ((6.4,2.2) ;2), ((9.5,2.6) ;3), ((13.5,2.6) ;4), ((13.0,2.4) ;5),T T T T Ty x =%By applying the proposed approach described in section 3, the fuzzy regression model is derivedas (4.079,2.1) (1.8458,0)w wY x= ⊕ ⊗% with h = 0.468. A Summary of results of someother techniques, including their models as well as their performances, are given in Table 4.To show the performance of fuzzy regression models we considered these five pairs ofobservations listed above. The dataset do not have the level of detail and complexity thanthose used in other studies, but in the literature these data have been considered by manyresearchers for the experimental evaluation and comparison of their proposed methodology. Table4 lists the regression models obtained by the methods proposed by other authors based on the fivepairs of observations considered above. The first fuzzy regression model based on fuzzyobservations was proposed by Tanaka et al.[27] (THW) (1982). Several authors pointed out thatthis method has several disadvantages and modified it or developed their own methodologies toprevent the problems. THW[26] regression model estimates the fuzzy regression coefficients bylinear programming. This model has a numeric slope and a fuzzy intercept. KB[28], DM[29],WT[30] and HBS[31] models have a fuzzy slope and intercept. If the explanatory variables arenumeric values and dependent variables are symmetric fuzzy numbers, WT[30] model is the sameas DM[29] approach. NN[32] and CH[33]model have a numeric slope. In this case the proposedregression model is given in Figure 4.
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 201382Figure 4. Fuzzy regression model for the data set in Example 3.MethodsGoodness of fitmeasureSimilaritymeasureProposed method using Tw-norm based operations(4.079,2.1) (1.8458,0)w wY x= ⊕ ⊗% withh=0.4680.9068 0.61116Tanaka et al. method (THW)[27](0.385,7.7) 2.1Y x= ⊕ ⊗% 0.9424 0.43625Kim and bishu model 1988 (KB)[28](3.11,4.95,6.84) (1.55,1.71,1.82)Y x= ⊕ ⊗% 0.9125 0.4987Nasrabedi and Nasrabedi (2006) method (NN)[32](2.36,4.86,7) 1.73Y x= ⊕ ⊗% 0.9149 0.51554Diamond (1988)(DM)[29](3.11,4.95,6.79) (1.55,1.71,1.87)Y x= ⊕ ⊗% 0.9135 0.4865Wu and Tseng method (WT)[30](3.11,4.95,6.79) (1.55,1.71,1.87)Y x= ⊕ ⊗% 0.9135 0.4865Hojati et al 2005(HBS)[31](5.1,6.75,8.4) (1.1,1.25,1.4)Y x= ⊕ ⊗% 0.8915 0.62467Chen and Hseuh method 2009(CH)[33]1.71 (2.63,4.95,7.27)Y x= ⊗ ⊕% 0.9175 0.4865Table 4.Performance of different models for crisp input and fuzzy output givenin Example 3.Analyzing these results, we can observe that our technique is very easy to compute in practiceand gives a first and easy approach for the problem of analyzing regression problems with crispinput and triangular fuzzy output data.5. CONCLUSIONBased on the distance between the centers and spreads, a new method is proposed for fuzzysimple regression using Tw -norms. The models which are used here have the input and the
    • International Journal of Fuzzy Logic Systems (IJFLS) Vol.3, No2, April 201383output data as well as the coefficients are assumed to be fuzzy. The arithmetic operations basedon the weakest t-norm are employed to derive the exact results for estimation of parameters. Theefficiency of the proposed approach is studied by similarity measure based on the graded meanintegration representation distance of fuzzy numbers. In addition the effect of the outlier isdiscussed for the proposed approach. By comparing the proposed approach with some wellknown methods, applied to three data sets, it is shown that the proposed approach using shapepreserving operations was more effective. Studying the effect of outlier in center value ofresponse variable using proposed method with TW- norm operation is our future work.REFERENCES[1] Chang P.T. & Lee E.S, (1994) “fuzzy least absolute deviations regression based on the ranking offuzzy numbers”, in Proc. IEEE world congress on computational intelligence, pp.1365-1369.[2] Kim K.J, Kim D.H & Choi S.H, (2005) “Least absolute deviation estimator in fuzzy regression”, J.Appl. Math. Comput. Vol.18, pp. 649-656.[3] Torabi H, & Behboodian J, (2007) “Fuzzy least absolutes estimates in linear regression models”,Communi. Stat. – Theory methods, Vol. 36, pp. 1935-1944.[4] Chen L. H. & Hsueh C.C, (2007) “A mathematical programming method for formuation a fuzzyregression model based on distance criterion”, IEEE Trans. Syst. Man Cybernet. Vol.B, pp. 37 705-712.[5] Choi S. H. & Buckley J. J, (2008) “Fuzzy regression using least absolute deviation estimators”, SoftComput. Vol. 12, pp. 257-263.[6] Taheri S. M. & Kelkinnama M, (2008) “Least absolute regression”, in Proc. 4thinternational IEEEconference on intelligent systems, varna Bulgaria, , Vol. 11, pp. 55-58.[7] Taheri S. M. & Kelkinnama M, (2012) “Fuzzy linear regression based on least absolute deviations”,Iranian J. fuzzy system Vol. 9, pp. 121-140.[8] Hong D.H., Lee S. & Do D.Y, (2001) “Fuzzy linear regression analysis for fuzzy input-output datausing shape preserving operations”, Fuzzy sets Syst. Vol.122, pp. 513-526.[9] Ling C.H, (1965) “Representation of associative functions”, Publications Mathematicae – Debrecen,Vol. 12, pp. 189-212.[10] Zadeh L.A, (1978) “Fuzzy sets as a basis for a theory of possibility”, Fuzzy sets syst. Vol. 1, pp. 3-28.[11] Hong D.H, (2001) “Shape preserving multiplication of fuzzy intervals”, Fuzzy sets and Syst. Vol.123, pp. 81-84.[12] Hong D.H,(2002) “On shape preserving addition of fuzzy intervals”, Journal of MathematicalAnalysis and applications, Vol. 267, pp. 369-376.[13] Kolsevera A, (1994) “Additive preserving the linearity of fuzzy intervals”, Tata mountains Math.Publi. Vol. 6 pp.75-81.[14] Mesiar R, (1997) “Shape preserving addition of fuzzy intervals”, Fuzzy sets and Syst. Vol. 86 pp. 73-78.
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