A fuzzy ARIMA model by using quadratic programming approach for time series data

International Journal of Information Systems for Logistics and Management Vol. 5, No. 1 (2009) 41-51

http://www.knu.edu.tw/academe/englishweb/web/ijislmweb/index.html

A Fuzzy ARIMA Model by Using Quadratic Programming Approach for Time Series Data Ming-Jyh Wang1, Gwo-Hshiung Tzeng2 and Ten-Der Jane3 1Department

of Insurance, Chaoyang University of Technology, Taiwan 168 Gifeng E. Rd., Wufeng, Taichung 41300, Taiwan 2Department of Business and Entrepreneurial Management, Kainan University, Taiwan No. 1, Kainan Road, Luchu, Taoyuan 338, Taiwan 3Institute of Business and Management, National Chiao Tung University,Taiwan 4F, 114 Chung-Hsiao W. Road, Sec. 1, Taipei 100, Taiwan Received 1 February 2009; received in revised form 23 October 2009; accepted 25 November 2009

ABSTRACT Considering the time-series ARIMA(p, d, q) model in realistic fuzzy environments, we proposed a fuzzy ARIMA (FARIMA) model by using Quadratic Programming (QP) which was developed to be applied to deal with time series data. The characteristics of the QP-FARIMA model include the intervalization of parameters and the provision of possibility distribution for forecasting data. To illustrate the suitability and usability of this model, a fuzzy numerical example will be provided. Compared with the LP-ARIMA model, this model makes it possible for decision makers to forecast the best- and worst-possible situations based on fewer observations. Keywords: ARIMA, foreign exchange market, fuzzy regression, fuzzy ARIMA, time series, quadratic programming (QP).

1. INTRODUCTION The ARIMA model, proposed by Box and Jenkins (1976), has been widely applied in forecasting social, economic, engineering, foreign exchange, and stock problems. It assumes that the future values of a time series model have a clear and definite functional relationship with current, past values and white noise. This model has the advantage of accurate forecasting in a short time period; however, it also has the limitation that at least 50 observations (over 100 observations are preferred) are required to verify the model. In addition, this model uses the concept of the measurement error to deal with the differences between forecasting values and observations, but these data are precise values that do not include the measurement errors. Tanaka et al. have suggested that the fuzzy regression model can be used to avoid the modeling error under

the fuzzy environment (Tanaka, 1987; Tanaka and Ishibuchi, 1992; Tanaka et al., 1982). However, the disadvantage of the fuzzy regression model is that the prediction interval could be very wide while some extreme values exist. Song and Chissom (1993a, 1993b, 1994) presented the definition of fuzzy time series and introduced the corresponding model by means of fuzzy relational equations and approximate reasoning. Chen (1996) presented a fuzzy time series method based on the concept of Song and Chissom (1993a, 1993b, 1994). An application of fuzzy regression to fuzzy time series analysis was first found by Watada (1992), and Tseng et al. (2001) used Linear Programming (LP) to formulate the fuzzy regression model based on the concept of the Box-Jenkins model. In this paper, Quadratic Programming (QP), instead of LP, is used to formulate QP-FARIMA, which is based on the concept of Tanaka and Lee (1998), for dealing

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International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 5, No. 1 (2009)

with time series data under fuzzy environments. In order to show the applicability and effectiveness of the proposed method in practical applications, an illustration performed by simulation will be given. From the results, it can be found that the QP-FARIMA model seems to be better than other fuzzy regression models in dealing with fuzzy time series data. The advantages of the proposed model can be listed as follows: (1) Provide decision makers a more flexible way to model fuzzy time series data, compared with Tseng’s Fuzzy ARIMA model. (2) The number of observations required in the proposed model is less than that of the ARIMA model which claims at least 50 observations and more than 100 observations are preferred. The structure of this paper is organized as follows: the concepts of the time series ARIMA model and interval regression analysis by using QP are presented in Section 2. In Sectons 3, the QP-FARIMA model is formulated and proposed. The QP-FARIMA model applied to simulated data is addressed in Section 4 and conclusions are in the last section. 2. THE REVIEW OF THE ARIMA AND FUZZY REGRESSION MODELS A time series {Zt} with the mean µ can be presented by Box-Jenkins’ ARIMA (p, d, q) process (Box and Jenkins, 1976), if

ϕ (B)(1 – B)d(Zt – m) = q(B)at,

(1)

where ϕ (B) = 1 – ϕ 1B – ϕ 2B2 – … – ϕPBP and θ(B) = 1 – θ1B – θ2B2 – … – θqBq are polynomials in B of degree p and q, B is the backward shift operator, parameters p, d and q are integers, Zt denotes the observed value of time series data, where t = 1, 2, …, k. The formulation of the ARIMA model includes the following steps: (1) Identification of the ARIMA(p, d, q) structure. Use the autocorrelation function (ACF) and partial autocorrelation function (PACF) to develop the temporary function; (2) Estimation of the unknown model parameters; (3) Diagnostic checks are applied to the object of possible uncovering; (4) Lack of fit and cause diagnosis; (5) Forecast from the selection model. It is assumed by the ARIMA model that the random errors {at} are independent and identically distributed (iid) and followed by the process of N(0, σ 2) and both the roots of ϕ (Z) = 0 and θ (Z) = 0 lie outside the unit circle. At

least 50 observations and if possible, more than 100 observations is preferred, should be used in the ARIMA model. In the real world, however, the environment is full of uncertainty and fluctuations, we usually forecast the future situations by using small data in a short time span, and it is hard to verify if the data is a normal distribution. Therefore, this assumption restricts the applications of using the ARIMA model. This model uses the concept of the measurement error to deal with the difference between the forecasting values and observations, but these data are precise values and do not include the measurement errors. The basic concept of the fuzzy regression model as suggested by Tanaka et al. (1982) is identical to the model mentioned above. The following equation presents the generalized form of fuzzy linear regression:

Y t = β 0 + β 1x t + n

=

+ β n xt

(2)

n

Σ β i xt = iΣ= 1 β i xt = x′t β , i=1

where xt is the vector of independent variables, in time period t superscript ′ denotes the transposition operation, n is the number of variables and βi represents the ith fuzzy parameter of the model. Instead of using crisp parameters, fuzzy parameters βi in this paper employ the L-type fuzzy numbers which proposed by Dubois and Prade (1980), (αi, ci) and L, possibility distribution is

µβi(βi) = L{(xi – βi)/c},

(3)

where L is a function type. In the paper, the fuzzy parameters are represented by the triangular fuzzy numbers and can be written as:

1–

µβ i (β i ) =

αi – β i , αi – c i ≤ β iαi + c i , ci 0

(4)

otherwise ,

where µβi(βi) is the membership function of the fuzzy parameter βi, αi is the center value of the fuzzy number and ci is the width or spread around the center of the fuzzy number. Through the extension principle, the membership function of the fuzzy number yt = x′tβ can be defined by using pyramidal fuzzy parameter vector β as follows:

1–

µβ i (β i ) =

1 0

y t – x tα c′ x t

for x t ≠ 0, for x t = 0, y t = 0, for x t = 0, y t ≠ 0,

(5)

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M. J. Wang et al.: A Fuzzy ARIMA Model by Using Quadratic Programming Approach for Time Series Data

where α and c denote the vectors of model values and spreads for all model parameters, respectively and t is the number of observations, where t =1, 2, …, k. Finally, the method uses the criterion to minimize the total vagueness, S, defined as the sum of individual spreads of the fuzzy parameters of the model. k

minimum S =

Σ (c′ t=1

xt )

(6)

Meanwhile, this approach emphasizes the condition that the membership value of each observation yt is greater than an imposed threshold, h ∈ [0, 1]. This condition simply expresses the fact that the fuzzy output of the model should be higher than all data points y1, y2, …, yk under a certain h-level. That is, the choice of the value of h-level will influences the widths, c, of the fuzzy parameters:

µy(yt) ≥ h for t = 1, 2, …, k.

(7)

where the index t refers to the number of non-fuzzy data to construct the model. Then the issue of finding the fuzzy regression parameters can be formulated by the following linear programming problem, as suggested by Tanaka and Ishibuchi:

Σ (c′

t=1

k

Σ (yj – a tx j )2, j=1

(11)

which is similar to the concept of the least squares. Therefore, Tanaka and Lee (1998) suggested a new objective function by combining Eqs. (9) and (11), which reflect both the properties of the least squares and possibility approaches as k

k

j=1

t=1

J = k 1 Σ (y j – a tx j )2 + k 2 Σ (c′ x t )2,

(12)

where k1 and k2 are weight coefficients. According to the meaning of Eq. (12), Eq. (10) can be written as follows: k

k

j=1 ′ x t α + (1 – h) c′ x ′t α – (1 – h) c′

t=1

Minimize J = k 1 Σ (y j – a tx j )2 + k 2 Σ (c′ x t )2

k

Minimize S =

Tanaka and Lee (1998) proposed interval regression analysis by using the QP-based approach that integrates the central tendency of least squares and possibility property of fuzzy regression. In the basic formulation of QP, the sum of squared spreads of estimated outputs are used as an objection function. In addition, to minimize the sum of squared distances between the estimated output centers and the observed outputs is also considered, which can be formulated as

xt )

subject to x ′t α + (1 – h) c′ x t ≥ y t , t = 1, 2,

, k,

x ′t α – (1 – h) c′ x t ≤ y t , t = 1, 2, c ≥ 0,

, k,

subject to

x t ≥ y t , t = 1, 2,

, k,

x t ≤ y t , t = 1, 2,

, k,

c ≥ 0.

(8)

(13)

where α′ = (α1, α2, …, αn) and c′ = (c1, c2, …, cn) are vectors of unknown variables and S, as previously defined, denotes the total vagueness. Tanaka and Lee (1998) proposed internal regression analysis by using QP which objective function is defined as

Tseng et al. (1982) suggested another fuzzy time series model based on Box and Jenkins (1976) model, which used LP programming. In this paper, we will combine Tseng and Tanaka’s approaches to develop the ARIMA model.

k

J=

Σ (c′ t=1

x t )2 = c′

k

Σ (c′ t=1

xt ) c

(9)

where J is the sum of square of the estimated outputs and Σ kj = 1 x j x j t is an (n + 1) × (n + 1) symmetric positive definite matrix. Thus the Eq. (8) can be modified as the following QP program: k

Minimize J = c′

Σ (c′ t=1

xt ) c

subject to x ′t α + (1 – h) c′ x t ≥ y t , t = 1, 2, x ′t α – c≥0

(1 – h) c′ x t ≤ y t , t = 1, 2,

, k, , k, (10)

3. MODEL FORMULATION The ARIMA model is a precise forecasting model for short time periods, but the limitation of this model is that it requires a large amount of historical data (at least 50 and preferably 100 or more). However, in the real world, due to the factors of the uncertain environment and rapid development of new technology, we usually have to forecast future situations based on few data in a short time span. The historical data are usually less than the requirement of the ARIMA model and this limitation actually restricts its application. The fuzzy regression model is an interval forecasting model suitable for the condition of few attainable historical data. In order to include all possible conditions in this model, the spread of the model should be considered. However, the spread

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International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 5, No. 1 (2009)

of the model will be too wide if data include a significant difference of bias. Therefore, the purpose of this paper is to combine the advantages of the fuzzy regression and ARIMA models to formulate the fuzzy ARIMA model and to overcome the limitations of the previous models. The parameters of traditional ARIMA (p, d, q), including ϕ 1, …, ϕ p and θ1, …, θq, are crisp numbers. Instead of using crisp numbers, fuzzy parameters, ϕ~1, …, ϕ~p ~ ~ and θ1, …, θq, respresented by the triangular fuzzy form, are used in this paper. By using the fuzzy parameters, the required amount of historical data in this paper will be significantly reduced (since at should be obtained from observations, at are crisp numbers). In addition, this study employs the integrated methodology combined by the conditions of Tanaka and Tseng’s models with a wide spread of the forecasted interval. The fuzzy parameters of the fuzzy ARIMA(p, d, q) model in this paper are presented by the following fuzzy functions:

Using fuzzy parameters βi in the form of triangular fuzzy numbers and applying the extension principle, the membership of W in Eq. (12) is given as Simultaneously, Zt represents the tth observation, and h-level is the threshold value representing the degree to which the model should be satisfied by all data points Z1, Z2, …, ZK. A choice of the h value influences the widths ci of the fuzzy parameters: ZZ(Zt) ≥ h for t = 1, 2, …, k,

where the index t refers to the number of nonfuzzy data used to construct the model. On the other hand, the fuzziness S included in the model is defined as p

J=

(14)

Wt = (1 – B)d(Zt – µ),

(15)

~ Wt = ϕ~1Wt – 1 + ϕ~2Wt – 2 + … + ϕ~pWt – p + at ~ ~ ~ – θ 1at – 1 – θ 2at – 2 – … – θqat – q,

(16)

~ ~ where Zt are observations and ϕ~1, …, ϕ~p and θ 1, …, θp are fuzzy numbers. In addition, Eq. (11) is modified as ~ ~ ~ ~ ~ Wt = β1Wt – 1 + β2Wt – 2 + … + βpWt – p + at – βp + 1at – 1

p+q

p

p

Minimuze J =

q

(17)

In this paper, fuzzy parameters with the triangular fuzzy forms are used:

where µβ~i(βi) is the membership function of the fuzzy parameter βi, αi is the center value of the fuzzy number, and ci is the width around the center of the fuzzy number.

Wt Σ i = 1 αiWt – i – a t + Σ i = p + 1 αi a t + p – i p

1–

ρi – p a t + p – i ) , 2

Σ pi = 1 ci

ϕii Wt – i )2

k

Σ Σ (ci i=p+1 t=1

ρ i – p a t + p – i )2

p

subject to

p+q

Σ αiWt – i + a t – i =Σp + 1 αi a t + p – i i=1 p

+ (1 – h) t = 1, 2,

αi – β i 1– , αi – c i ≤ β i ≤ αi + c i , ci (18) 0 otherwise,

k

Σ Σ (ci i=1 t=1 p+q

Σ β iWt – i + a t – i =Σp + 1 β iWt – i .

µW (Wt) =

(21)

k

where ρi – p is the autocorrelation coefficient of time lag i – p, ϕ ii is the partial autocorrelation coefficient of time lag i. The weight of ci depends on the relation of time lag i and the present observations, where the parameter p of AR(p) is derived from PACF and the parameter q of MA(q) is derived from ACF. Next, the problem of finding the fuzzy ARIMA parameters can be formulated by the following QP problem:

+

i=1

µβ (β i ) = i

ϕii Wt – i )2

Σ Σ (ci

+

~ ~ – βp + 2at – 2 – … – βp + qat – q,

=

k

Σ Σ (ci i=1 t=1

i=p+1 t=1

~ ~ Φp(B)Wt = Θq(B)at,

(20)

Σ ci i=1

p+q

Wt – i +

Σ

i=p+1

ci a t + p – i

≥ Wt ,

, k,

p

p+q

Σ αiWt – i + a t – i =Σp + 1 αi a t + p – i i=1 p

+ (1 – h)

Σ ci i=1

p+q

Wt – i +

t = 1, 2, , k, c i ≥ 0 ∀ i = 1, 2,

Σ

i=p+1

ci a t + p – i

≤ Wt ,

, p + q.

(22)

p+q

Wt – i + Σ i = p + 1 c i a t + p – i p+q

0

for Wt ≠ 0, a t ≠ 0, otherwise

(19)

M. J. Wang et al.: A Fuzzy ARIMA Model by Using Quadratic Programming Approach for Time Series Data

Then, we consider interval regression analysis by using the QP-based approach that integrates the central tendency of least squares and possibility property of fuzzy ARIMA. In the basic formulation of QP, the sum of squared spreads of estimated outputs are used as an objection function. In addition, we consider minimizing the sum of squared distances between the centers of the estimated output and the observed outputs from Eq. (16) which can be formulated as k

p

Σ

Wt – ( Σ β iWt – i –

t=1

i=1

2

q

Σ

i=p+1

β iWt – i) ,

(23)

which is corresponding to the concept of least squares. Here, we propose a new objective function, combining Eqs. (21) and (23) which reflect both the properties of the least squares and possibility approaches such that p

k

Σ Σ (ci

J = k1

i=1 t=1 p+q

ϕii Wt – i )

k

Σ Σ (ci

+

k

p

t=1

i=1

+ k 2 Σ Wt – ( Σ β iWt – i –

q

Σ

i=p+1

2

β iWt – i )

4. AN NUMERICAL EXAMPLE

where k1 and k2 are weight coefficients. The new objective function, i.e., Eq. (24), minimizes the objective function J and subjects to the linear constraint Eq. (22) and can be represented as the following QP problem: p

p+q

+

k

Σ Σ (ci i=1 t=1 k

Σ Σ ci i=p+1 t=1

ϕii Wt – i )2

ρ i – p a t + p – i )2

k

p

t=1

i=1

+ k 2 Σ Wt – ( Σ β iWt – i – p

subject to

q

2

(1.84656)

Σ ci i=1

p+q

Wt – i +

(27)

Σ

i=p+1

ci a t + p – i

≥ Wt ,

where the values inside the brackets denoted as the t-values. Phase II: determining the minimal fuzziness numbers: the fuzzy parameter obtained by using Eq. (22) (with h=0) are shown in Eq. (27). The results are plotted as shown in Fig. 2.

≤ Wt ,

~ Zt = (α0, c0) + (α1, c1)Zt – 1 + at = (2.367650, 0.5350053) + (0.5820088, 0)Zt – 1 + at (28)

, k,

p

p+q

Σ αiWt – i + a t – i =Σp + 1 αi a t + p – i i=1 p

+ (1 – h)

(2.02039)

p+q

p

t = 1, 2,

In this section, we use a simulated time series data set, as shown in Table 1, for verifying the performance of the proposed model. The proposed QP approache is applied to the data set to illustrate the effectiveness of the proposed model. In addition, we will compare the difference between Eqs. (22) and (25). Phase I: fitting ARIMA(p, d, q) model: by using the Scientific Computing Associates (SCA) package software, the best-fitted model is assigned as ARIMA(1, 0, 0) and the residuals follow the white noise process. The results are plotted as shown in Fig. 1 and the corresponding model is presented as

Z t = 2.835573 + 0.522411Z t – 1 + a t

Σ β iWt – i ) i=p+1

Σ αiWt – i + a t – i =Σp + 1 αi a t + p – i i=1 + (1 – h)

Wt = Wt – 1 + … + Wt – p + at

where Wt = (1 – B)d(Zt – µ), αi is the center value of the fuzzy number and ci is the width around the center value of the fuzzy number.

2

(24)

Minimize J = k 1

ming is described as follows: Phase I: fitting the ARIMA(p, d, q) by using the available information of observations, i.e., input data are considered as non-fuzzy numbers. The result of Phase I derives the optimum solutions of the parameters, α* = (α*1, α*2, …, α*p + q) and the residuals at. The results of Phase I will be used as the input data sets of Phase II (the concept is derived from Savic and Pedrycz (1991)). Phase II: determining the minimal fuzziness by using the same criterion as shown in Eqs. (22), (25) and α* = (α*1, α*2, …, α*p + q). The number of constraint functions is equal to the number of observations (the concept is derived from Savic and Pedrycz (1991)). The fuzzy ARIMA model can be presented as

– at – 1 – … – at – q, (26)

2

ρi – p a t + p – i )

i=p+1 t=1

45

Σ ci

i=1

p+q

Wt – i +

t = 1, 2, , k c i ≥ 0 ∀ i = 1, 2,

Σ

i=p+1

, p + q.

ci a t + p – i

(25)

The procedure of FARIMA by using QP-program-

The FARIMA method provides possible intervals. From Fig. 2, we know the actual values are located in the fuzzy intervals.

46

International Journal of the Information Systems for Logistics and Management (IJISLM), Vol. 5, No. 1 (2009)

Table 1. Obs. Zt

1

2

3

4

5

6

7

8

9

10

11

12

13

14

5.66021 6.28327 5.95332 5.55001 5.0628 5.84925 5.75944 5.63032 5.94266 5.87116 6.17395 6.3551 6.60138 6.08102

Table 2. Optimal coefficients by QP, i.e., Eq. (25), with various weights Weights

Case A B C D E F

Optimal coefficient vectors

k1

k2

0.2 0.4 0.6 0.7 0.8 1.0

0.8 0.6 0.4 0.3 0.2 0.0

α*

= (α0, α1)

Objective value

c* = (c0, c1)

(2.367650, 0.5820088) (2.367650, 0.5820088) (2.367650, 0.5820088) (2.302086, 0.5943645) (2.072967, 0.6375424) (2.835506, 0.5224221)

(0.5350053, 0) (0.5350053, 0) (0.5350053, 0) (0.5380152, 0) (0.5485335, 0) (1.2415130, 1.265142)

3.282660 2.844321 2.405983 2.186046 1.950257 1.351728

Table 3. FARIMA form various weights Case

Series of Wt

A B C D E F

7 6.5 6 5.5 5 4.5 4

Weights k1

k2

0.2 0.4 0.6 0.7 0.8 1.0

0.8 0.6 0.4 0.3 0.2 0.0

~ Zt = (α0, c0) + (α1, c1)Zt – 1 ~ Zt = (2.367650, 0.5350053) + (0.5820088, 0)Zt – 1 ~ Zt = (2.367650, 0.5350053) + (0.5820088, 0)Zt – 1 ~ Zt = (2.367650, 0.5350053) + (0.5820088, 0)Zt – 1 ~ Zt = (2.302086, 0.5380152) + (0.5943645, 0)Zt – 1 ~ Zt = (2.072967, 0.5488335) + (0.6375424, 0)Zt – 1 ~ Zt = (2.835506, 1.2425130) + (0.5224221, 1.265142)Zt – 1

7 6.5 6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5.5 Fig. 1.

5

Next, we apply the data of Table 1 again to Eq. (25) in our proposed QP method, which can reflect both the property of central tendency in least squares and the possibility property in the fuzzy ARIMA model. Here, the influence of weight coefficients k1 and k2 of the QP method in Eq. (25) is examined. By appling the Eq. (25) in the QP method, six sets of the optimal coefficients (by changing weights k1 and k2 as shown in Table 2) are derived from the procedure of the FARIMA model, as shown in Table 3. In this approach, the weight coefficients and play an important role in FARIMA models. For example, if we put a large value compared with , the more central

Zt actual value fuzzy ARIMA

4.5 4

1

2

3

4

5

6

upper interval value lower interval value 7

8

9

10 11 12 13

Fig. 2. Results of FARIMA historical data

tendency would be expected, that is, the obtained center line tend to be identical to the regression line obtained by least squares regression. On the contrary, if we put a large value compared with , it means that we focus on reducing the fuzziness of the model and the result of Eq. (25) will be quite similar to Eq. (22).

M. J. Wang et al.: A Fuzzy ARIMA Model by Using Quadratic Programming Approach for Time Series Data

5. CONCLUSIONS In this paper, we introduced interval fuzzy ARIMA by using QP approach. Even though QP is more complicated than Tseng's LP method, interval fuzzy ARIMA by using QP can provide the more diverse coefficients. In addition, the proposed method can effectively reflect the property of central tendency in least squares regression and the possibility property in the fuzzy ARIMA model. From the result of the numerical example, by changing the weight coefficients of the objective function in QP, we can explain the results from different viewpoints. An analyst can flexibly assign the weight coefficients k1 and k2 by considering a trade-off between k

Σ

p

t=1

Wt – ( Σ β tWt – i – i=1

2

q

Σ

i=p+1

β tWt – i)

analysis with fuzzy model. IEEE Transactions Systems, Man and Cybernetucs, 12(6), 903- 907. Watada, J. (1992) Fuzzy time series analysis and forecasting of sales volume, in: J. Kacprzyk, M. Fedrizzi (Eds.), Fuzzy Regression Analysis, Omnitech Press, Warsaw and Physica-Verlag, Heidelberg, pp. 211- 227. Tseng, F. M., Tzeng, G. H., Yu, H. C. and Yuan, B. J. C. (2001) Fuzzy ARIMA model for forecasting the foreign exchange market. Fuzzy Sets and Systems, 118(1), 9-19. Tanaka, H. and Lee, H. (1998) Internal regression Analysis by quadratic programming approach. IEEE Transactions on Fuzzy Systems, 6(4), 473-481.

Appendix LINGO Program and Summed Output Formula 1: (QP) p

Minimuze J =

and

k

Σ Σ (ci i=1 t=1 p+q

+ p

k

Σ Σ (ci

i=1 t=1

ϕii Wt – i )2 +

p+q

k

Σ Σ (ci

i=p+1 t=1

ρ i – p a t + p – i )2

Since the proposed method is developed from the concept of QP, the formulation can be clearly expressed in mathematical ways.

ϕii Wt – i )2

k

Σ Σ (ci i=p+1 t=1

ρ i – p a t + p – i )2

p

subject to

p+q

Σ αiWt – i + a t – i =Σp + 1 αi a t + p – i i=1 p

+ (1 – h)

Σ ci i=1

t = 1, 2,

p+q

Wt – i +

ci a t + p – i

≥ Wt ,

p+q

Σ αiWt – i + a t – i =Σp + 1 αi a t + p – i i=1 p

Box, G. P. and Jenkins, G. M. (1976) Time Series Analysis: Forecasting and Control, Holden-day Inc., San Francisco, CA, 1976. Chen, S. M. (1996) Forecasting enrollments based on fuzzy time series. Fuzzy Sets and Systems, 81(3), 311-319. Dubois, D. and Prade, H. (1980) Theory and Applications, Fuzzy Sets and Systems, Academic Press, New York. Ishibuchi, H. and Tanaka, H. (1988) Interval regression analysis based on mixed 0-1 integer programming problem. J.Journal of Japan Soc. Ind. Eng., 40(5), 312- 319. Savic, D. A. and Pedrycz, W. (1991) Evaluation of fuzzy linear regression models. Fuzzy Sets and Systems, 39(1), 5163. Song, Q. and Chissom, B. S. (1993a) Fuzzy time series and its models. Fuzzy Sets and Systems, 54(3), 269- 277. Song, Q. and Chissom, B. S. (1993b) Forecasting enrollments with fuzzy time series-part I. Fuzzy Sets and Systems, 54(1), 1- 9. Song, Q. and Chissom, B. S. (1994) Forecasting enrollments with fuzzy time series-part II. Fuzzy Sets and Systems, 62(1), 1-8. Tanaka, H. (1987) Fuzzy data analysis by possibility linear models. Fuzzy Sets and Systems, 24(3), 363- 375. Tanaka, H. and Ishibuchi, H. (1992) Possibility regression analysis based on linear programming, in: J. Kacprzyk, M. Fedrizzi (Eds.), Fuzzy Regression Analysis, Omnitech Press, Warsaw and Physica-Verlag, Heidelberg, pp. 4760. Tanaka, H., Uejima, S. and Asai, K. (1982) Linear regression

Σ

i=p+1

, k,

p

REFERENCES

47

+ (1 – h)

Σ ci i=1

p+q

Wt – i +

t = 1, 2, , k, c i ≥ 0 ∀ i = 1, 2,

Σ

i=p+1

ci a t + p – i

≤ Wt ,

, p + q.

LINGO Program: min = 13*c1^2 + c2*c1*2*77.04899 + c2^2* 77.04899*77.04899; a1 + c1 + a2*6.01633 + c2*6.01633 >= 6.28327; a1 + c1 + a2*6.28327 + c2*6.28327 >= 5.95332; a1 + c1 + a2*5.95332 + c2*5.95332 >= 5.55001; a1 + c1 + a2*5.55001 + c2*5.55001 >= 5.0628; a1 + c1 + a2*5.0628 + c2*5.0628 >= 5.84925; a1 + c1 + a2*5.84925+ c2*5.84925 >= 5.75944; a1 + c1 + a2*5.75944 + c2*5.75944 >= 5.63032; a1 + c1 + a2*5.63032 + c2*5.63032 >= 5.94266; a1 + c1 + a2*5.94266 + c2*5.94622 >= 5.87116; a1 + c1 + a2*5.87116 + c2*5.87116 >= 6.17395; a1 + c1 + a2*6.17395 + c2*6.17395 >= 6.3551; a1 + c1 + a2*6.3551 + c2*6.3551 >= 6.60138; a1 + c1 + a2*6.60138 + c2*6.60138 >= 6.08102; a1 – c1 + a2*6.01633 – c2*6.01633 = 6.3551; a1 + c1 + a2*6.3551 + c2*6.3551 >= 6.60138; a1 + c1 + a2*6.60138 + c2*6.60138 >= 6.08102; a1 – c1 + a2*6.01633 – c2*6.01633 = 6.3551; a1 + c1 + a2*6.3551 + c2*6.3551 >= 6.60138; a1 + c1 + a2*6.60138 + c2*6.60138 >= 6.08102; a1 – c1 + a2*6.01633 – c2*6.01633 = 6.3551; a1 + c1 + a2*6.3551 + c2*6.3551 >= 6.60138; a1 + c1 + a2*6.60138 + c2*6.60138 >= 6.08102; a1 – c1 + a2*6.01633 – c2*6.01633 = 6.3551; a1 + c1 + a2*6.3551 + c2*6.3551 >= 6.60138; a1 + c1 + a2*6.60138 + c2*6.60138 >= 6.08102; a1 – c1 + a2*6.01633 – c2*6.01633 = 6.3551; a1 + c1 + a2*6.3551 + c2*6.3551 >= 6.60138; a1 + c1 + a2*6.60138 + c2*6.60138 >= 6.08102; a1 – c1 + a2*6.01633 – c2*6.01633 = 6.3551; a1 + c1 + a2*6.3551 + c2*6.3551 >= 6.60138; a1 + c1 + a2*6.60138 + c2*6.60138 >= 6.08102; a1 – c1 + a2*6.01633 – c2*6.01633