# Fuzzy regression model with fuzzy input and output data for manpower forecasting

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2 206 H.T. Lee, S.H. Chen / Fuzzy Sets and Systems 9 (200) size of a data set is small, error terms have small variability, and the relationships among variables are not well specied, fuzzy linear regression outperforms nonparametric linear regression with respect to descriptive capability which is concerned with how close the estimated model parameters are to the true parameter values. Although the fuzzy regression models assume that the residuals depend on the indeniteness of the model parameters that contracts the conventional least squares considered it to be measurement errors. From the view point of the system of relationship between input and output data sets, the deviations between the estimated and the observed values of output data may be caused not only by the vagueness of the parameters of the model, but also by the vagueness of the input data. This situation may occur when the input data are envisaged as facts or events of an observation which are uncontrollable or uninuenced by the observer rather than as the controllable levels of factors in an experiment. In this research, we concentrated on a fuzzy linear regression model that assumes the residuals are caused by the vagueness, both of the parameters of the model and of the input data simultaneously. In the next section, the original denition of the fuzzy regression by Tanaka [0] is described again for reading convenience. In Section 3, the membership function is conducted for the generalized fuzzy linear model that we are considering. In Section 4, a manpower forecasting example is proposed as an illustration of the generalized model. Finally, Section 5 is a conclusion. 2. The fuzzy linear model First of all, we assume that a fuzzy phenomenon can be presented as a fuzzy system of equations which, in turn, can be described by the fuzzy function Ỹ = X 2 X 2 n X n, in which the fuzzy parameters j with membership functions are as follows: ( j a j ); a j 6 j 6b j ; b j a j U j ( j )= ( j c j ); b j 6 j 6c j ; b j c j 0 otherwise: Then the membership function of the fuzzy linear model can be obtained from the following proposition. Proposition. Given fuzzy parameters j with membership functions as function (), the membership function of the fuzzy linear function Ỹ = X 2 X 2 n X n is obtained as the following: (y C 3 ); C 6y6C 3 ; C 3 C UY (y)= (2) (y C 2 ); C 3 6y6C 2 ; C 3 C 2 0 otherwise; in which C = n a jx j, C 2 = n c jx j, C 3 = n b jx j. Proof. By the extension principle of fuzzy sets [2,3] and the denition of the triangular fuzzy number [2], a triangular fuzzy number multiplied by a positive real number is still a triangular fuzzy number. For example, if a triangular fuzzy number A =(a; b; c) then A = A x =(ax; bx; cx). Another extension principle is that ()

3 H.T. Lee, S.H. Chen / Fuzzy Sets and Systems 9 (200) the summation of triangular fuzzy numbers is also a triangular fuzzy number. For example, if A =(a; b; c) and B =(d; e; f) then C = A B =(a + d; b + e; c + f). By the two extension principles, the membership function of the fuzzy linear function is available. From Proposition, the fuzzy linear regression model Y = X 2 X 2 n X n can be conceived as summation of the triangular fuzzy numbers multiplied by positive real numbers. In a fuzzy linear regression model, the bases of the triangular membership function of the fuzzy parameters are usually assumed to have the same spread width at both sides of their center values, such that b j = j, a j = b j l j, c j = b j + l j, where j is the center value of j, and l j is the spread width of parameter j. Consequently, the membership function of j is as follows: + ( j j ); l j j l j 6 j 6 j ; U j ( j )= ( j j ); l j j 6 j 6 j + l j ; 0 otherwise: By Proposition, the membership function of the fuzzy linear regression function Y = X 2 X 2 n X n will be + n l y jx j U Y (y)= n l y jx j j x j ; C 6y6C 3 ; j x j ; C 3 6y6C 2 ; 0 otherwise; in which C = n ( j l j )x j ; C 2 = n ( j + l j )x j ; C 3 = n jx j. The membership function of the output data y also should have the same special form of the previous fuzzy parameters such as + (y q); q w6y6q; w U Y (y)= (y q); q6y6q + w; w 0 otherwise; where q and w represent the center value and spread width of y, respectively. The degree of the tting of the estimated fuzzy linear regression model Y = X 2 X 2 n X n to the given data set Y =(q; w) is measured by the following index h which maximizes h subject to Y h Y h, where Y h = {y U Y (y) h}, Y h = {y U Y (y) h}, which are h-level sets. This index h is illustrated in Fig.. This degree of the tting of the fuzzy regression model to the data set Y =(q; w) is dened by minimizing h for each element of this data in set Y =(q; w). The vagueness of the fuzzy regression model Y = X 2 X 2 n X n is dened by V = n l j. The problem is explained as obtaining fuzzy parameters which minimize V subject to (3) (4) (5)

4 208 H.T. Lee, S.H. Chen / Fuzzy Sets and Systems 9 (200) Fig.. Degree of tting of y to a given fuzzy data y. h H for all data in Y =(q; w), where H is chosen as the degree of the tting of the fuzzy linear model by the decision maker. The following linear programming problem expresses the above situation: min j;l j s:t: V = l + l l n j x j +( H) l j x j q +( H)w; q6c 3 ; j x j +( H) l j x j q +( H)w; q C 3 ; l j 0 for j =;:::;n; w 0; the set of constraints must hold for every data y in set Y =(q; w). 3. The fuzzy linear model with fuzzy input and output data In the previous section, we discussed the original fuzzy linear function which assumes the deviation of the output data between its observed value and estimated value is only caused by the indeniteness of the model parameters. This interpretation can be extended if the fuzzy linear function represents the relationship between the input data and output data. From this point of view, the vagueness of the output data may be caused not only by the indeniteness of the model parameters but also by the vagueness of the input data. In this section, the general situation of the fuzzy linear function is discussed. Suppose the input data X j is a fuzzy number with the relationship function as following: (x j d j ); d j 6x j 6e j ; e j d j U Xj(x j) = (x j f j ); e j 6x j 6f j ; e j f j 0 otherwise: (6)

5 H.T. Lee, S.H. Chen / Fuzzy Sets and Systems 9 (200) The membership function for the generalized fuzzy linear function Ỹ = X 2 X 2 n X n is obtained from the following Proposition 2, Proposition 2. Given fuzzy parameters j with membership functions as function () and fuzzy input data X j with membership function as function (6), the membership function of the generalized fuzzy linear function Ỹ = X 2 X 2 n X n is obtained as the following: [ ( ) ] 2 =2 B B + C y ; C 6y6C 3 ; 2A 2A A U [ Y (y) = ( ) ] 2 =2 B 2 B2 C 2 y ; C 3 6y6C 2 ; 2A 2 2A 2 A 2 0 otherwise (7) in which, A = B = C = (b j a j )(e j d j ); A 2 = (a j (e j d j )+d j (b j a j )); B 2 = a j d j ; C 2 = (c j b j )(f j e j ); (c j (f j e j )+f j (c j b j )); c j f j ; C 3 = b j e j : Proof. From the relationship functions of the fuzzy number j and X j, the -cut interval sets for the two fuzzy numbers are j =[a j + (b j a j );c j + (b j c j )] and Xj =[d j + (e j d j );f j + (e j f j )], respectively. By the extension principle of fuzzy sets, the multiplication of two -cut interval sets causes a -cut interval set as Zj = j Xj = {[a j + (b j a j )] [d j + (e j d j )]; [c j + (b j c j )] [f j + (e j f j )]}. The membership function of the fuzzy number Zj is facile. Although the membership function of Zj is obtained from Proposition 2, it is not again a triangular fuzzy number. Hence, the summation of the n fuzzy number derived a fuzzy number with the approximated relationship function as function (7) above. Fig. 2 describes the shape of the approximated function. The membership function of the output data y is also supposed to have the same special form of triangular described as function (5) in Section 2 above. The degree of the tting of the estimated generalized fuzzy linear regression model Y = X 2 X 2 n Xn to the given data set Y =(q; w) is measured, as the same as the linear regression model in Section 2, by the index h which maximizes h subject to Y h Y h, where Y h = {y U Y (y) h}, Y h = {y U Y (y) h}. This index h is illustrated in Fig. 3. The degree of the tting of the fuzzy regression model to the data set Y =(q; w) is dened by minimizing h for each of this data in set Y =(q; w). The vagueness of the fuzzy regression model Y = X 2 X 2 n Xn is dened by V = n l j. The problem is explained as obtaining fuzzy parameters which minimizes V subject to h H for all data in Y =(q; w), where H is chosen as the degree of the tting of the fuzzy

6 20 H.T. Lee, S.H. Chen / Fuzzy Sets and Systems 9 (200) Fig. 2. Shape of the approximated membership Function 7. Fig. 3. Degree of tting of y to given fuzzy data y. linear model by the decision maker. The above situation is expressed by the following nonlinear programming problem: min j;l j s:t: V = l + l l n h = ( ) ( (wb ) 2 ( wb p w2 + p w2 p 2 wpb ) ) =2 + w2 C p H w 2A 2A 2A 2A A A if q6c 3 ; in which w = q p; and h = w if q C 3 ; in which w = r q: ( r wb ) 2 + w2 + 2A 2 2A 2 ( ( r wb ) 2 ( 2 + w2 r 2 wrb ) ) =2 2 + w2 C 2 2A 2 2A 2 A 2 A 2 r H The set of constraints must hold for every data y in set Y =(q; w). For simplicity, in practice the approximation formula U Y (y) = (C ;C 3 ;C 2 ) that represents triangular fuzzy number can be used. The above equation can be simplied as follows (see Fig. 4): min j;l j V = l + l l n s:t: h= C 3 C h = C 3 C 2 ( C (q p) p(c 3 C ) (q p)+(c 3 C ) ( C2 (q r) r(c 3 C 2 ) (q r) (C 3 C 2 ) C ) H if q C 3 ; C 2 ) H if q C 3 :

7 H.T. Lee, S.H. Chen / Fuzzy Sets and Systems 9 (200) Fig. 4. Degree of tting of y to given fuzzy data y for the simplied regression model. 4. Manpower forecasting problem Most businesses and government agencies are engaged in some form of manpower forecasting, indicating that to forecast the future demand for manpower is one important area of manpower policy for individual enterprises and government. Although the aggregate demand of an entire industry is essentially the sum of the demands of individual rms, it is hardly possible to acquire the demand of every rm. One method is to study a sample of rms and make inferences to the whole population. However, consideration must be given to two major sources which inuence the variance of the sample, making the forecasts unreliable. One factor is that most registered rms are considered as small businesses (in Taiwan, ROC approximately 98%) which will not last very long. Statistics indicate that in Taiwan, approximately 62% of these rms will close within ve years operation (Medium and Small Business Consulting Center []). The other factor is that, due to the large number of rms, only a small portion can be investigated. In addition to the above two sources of variance of sample, the demand for manpower is typically concentrated in very large companies. To deal with these factors, Kao and Lee [4] proposed the idea of sampling from the more important companies to acquire reliable data than making suitable extrapolations to the less important small companies and using regression analysis by selecting approximate explanatory variables to make forecasts. In this research, the method of Kao and Lee [4] is followed. A sample of 00 rms of the top 000 rms according to their sales volume in Taiwan is extracted. To ensure that the correct data is obtained for analysis, site visits to the rms were undertaken. Each rm investigated is requested to provide the number of employees in the areas of industrial engineering and management (IE&M) planned to be hired in 998. Table shows the prole of the sample 00 rms, grouping the rms into the classes by size, there are 2 rms of the top 00 rms, and 7; ; 8; 4; ; 0; ; 8 and 8 rms are in the subsequent classes of 00 rms, respectively. The means and variances for the ranks and IE&M employees planned to be hired corresponding to the classed rms in the sample are described in Table, too. These two values indicate that both of the employees with IE&M profession and ranks are variable with some percentage of vagueness. It can be seen, from Table that there is a clear descending trend along the rank-classes in an average sense. Most of the employees in IE&M profession are needed in the 00 largest rms and its variance is relatively large compared to the other smaller rms. Fig. 5 also depicts this relationship in an exponential form. Consequently, we assume the fuzzy regression model is Y = X 2, in which, variables Y and X represent the number of IE&M employees planned to be hired and the rank of the rm, respectively. After implementing the natural log operation on both sides of the regression model, the equation ln(y +)=ln( )+ 2 ln(x ) is derived. The reason for using Y + instead of Y in the previous model is because some original data for Y are zero, and adding to the original data Y make the logarithm operations meaningful. According to the property of the collected data set, the generalized fuzzy linear model is applied to identify the regression line. After transforming the original data to natural log form, the mean and one standard deviation value of the data are taken as the central and spread range values of variables ln(y ) and ln(x ), respectively. The decision maker

8 22 H.T. Lee, S.H. Chen / Fuzzy Sets and Systems 9 (200) Table The prole for the 00 observations in the 000 largest companies Rank Number Mean Standard error Mean of Standard error of rms of the ranks of the ranks IE&M employees of IE&M employees Fig. 5. The relationship between the IE employees and rank of the rms. Table 2 Center values and spread widths for the fuzzy parameters Fuzzy parameters ln( ) 2 Center Width parameter H is adopted as value 0.5. A generalized reduced gradient algorithm [8], which has been found [7] to be an ecient method, is applied to solve the non-linear programming problem presented in Section 3. Finally, the central and spread ranges of the parameters ln( ) are and 0.0, of 2 are and , respectively. All of these are described in Table 2. We then substitute the central value of the above parameters into the original exponential model and integrate the area dened by the regression line, the y-axis and the x-axis. The estimated total manpower demand for profession of IE&M is identied as 362 persons, which is obtained by calculating the following integration: X [e4: : ln(x ) ] dx. The regression line approximately intersects the x-axis when the rank reaches 228, indicating that the manpower demand for IE&M employees in the rms with ranks larger than 228 are negligible.

9 H.T. Lee, S.H. Chen / Fuzzy Sets and Systems 9 (200) Conclusion In this research, a generalized fuzzy linear function is discussed. Considering a fuzzy linear function as a model for the fuzzy structure of a system, a fuzzy linear regression function analysis is formulated. From this point of view, the vagueness of the output data is caused not only by the indeniteness of the model parameters, but also by the vagueness of the input data. This is in distinction to the conventional regression model which always assumes that the deviation of the dependent variables from their estimated value to their observed values is caused only by the measurement error and regards the independent variables as being controllable. During the implementation of this approach, such as in the above manpower forecasting example, no assumption such as the normality of the distribution of error terms in the conventional regression analysis are made. Consequently, the robustness of the generalized fuzzy linear model is apparent. References [] Business Consulting Center, Medium and Small Business Annual Financial Report, Taipei, Taiwan, ROC, 984. [2] D. Dubois, H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci. 9 (978) [3] B. Heshmaty, A. Kandel, Fuzzy linear regression and its applications to forecasting in uncertain environment, Fuzzy Sets and Systems 5 (985) [4] C. Kao, H.T. Lee, An integration model for manpower forecasting, J. Forecasting 5 (996) [5] K.J. Kim, H. Moskowitz, M. Koksalan, Fuzzy versus statistical linear regression, European J. Oper. Res. 92 (996) [6] K.J. Kim, H.R. Chen, A comparison of fuzzy and nonparameteric linear regression, Comput. Oper. Res. 24 (997) [7] H.T. Lee, A study of generalized reduced gradient method for solving nonlinear programming problems, Thesis, National Cheng Kung University, Taiwan, ROC, 986. [8] D.G. Luenberger, Linear and Nonlinear Programming, Addison-Wesley, Reading, MA, 984. [9] H. Moskowitz, K.J. Kim, On assessing the H value in fuzzy linear regression, Fuzzy Sets and Systems 58 (993) [0] H. Tanaka, S. Uejima, K. Asai, Fuzzy linear regression model, IEEE Trans. Systems Man Cybernet. 0 (980) [] H. Tanaka, S. Uejima, K. Asai, Linear regression analysis with fuzzy model, IEEE Trans. Systems Man Cybernet. 2 (982) [2] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part, 2 and 3, Inform. Sci. 8 (975) , ; 9 (976) [3] H.J. Zimmermann, Fuzzy Set Theory and its Application, Kluwer Academic Publishers, Reading, 99.

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