Forecasting and optimization of service level in vague ... - Springer Link

Int J Adv Manuf Technol (2013) 68:1453–1470 DOI 10.1007/s00170-013-4934-9

ORIGINAL ARTICLE

Forecasting and optimization of service level in vague and complex SCM by a flexible neural network–fuzzy mathematical programming approach A. Azadeh & M. Sheikhalishahi & S. M. Asadzadeh & M. Saberi & A. E. Pirayesh Neghab

Received: 11 June 2011 / Accepted: 21 March 2013 / Published online: 18 April 2013 # Springer-Verlag London 2013

Abstract This study presents a flexible meta-modeling approach for modeling and optimization of service level (SL) in vague and complex supply chains. Service level is used as the dependent variable, and ten standard variables including lead time, forecast error, supplier service level, delay, stock coverage, backlog depth, number of deliverable product, and number of orders are used as independent variables. The proposed approach is composed of artificial neural network (ANN) and fuzzy linear regression (FLR) for optimum forecasting of SL in SCM. Moreover, it compares the efficiencies of FLR, RR, and ANN approaches by mean absolute percentage error (MAPE). The intelligent approach of this study is applied to an actual supply chain system. The case is an international firm, which its responsibility in the supply chain is to distribute electrical and automation products to local outlets. ANN is identified as the preferred model with lowest MAPE and a comprehensive sensitivity analysis. The proposed approach of this study is ideal for accurate forecasting of SL in supply chains with possible complexity, ambiguity, and uncertainty. This would help managers to identify the preferred policy with respect to performance of supply chain in vague and complex environments. This is the first study that presents a flexible approach for accurate prediction of SL in SCM with possible noise, nonlinearity, and uncertainty. A. Azadeh (*) : M. Sheikhalishahi : S. M. Asadzadeh : A. E. P. Neghab School of Industrial and Systems Engineering, College of Engineering, University of Tehran, Tehran, Iran e-mail: [email protected] M. Sheikhalishahi : S. M. Asadzadeh : A. E. P. Neghab Department of Industrial Engineering, University of Tafresh, Tafresh, Iran M. Saberi Institute for Digital Ecosystems & Business Intelligence, Curtin University of Technology, Perth, Australia

Keywords Supply chain . Service level . Fuzzy mathematical programming . Artificial neural network . Complexity . Uncertainty

1 Introduction Supply chain is a complex system consisting of different parts, which change resources and materials into a deliverable product for “end user” in a value adding process. The characteristics and complexities of this system have been surveyed from different aspects. Warehouses and plant locating, distribution and sale centers, and their capacity, production, and inventory planning, material handling planning, supplier selection, demand forecasting, and information flow are the decisions of this area and show the expansion of supply chain knowledge. Today, globalizing and companies’ competition, diversity of products, customers’ expectations, and reduction of product life cycle have caused the importance of supply guidelines coordination with competitive guidelines of the whole chain in the existing uncertainty space [10]. Thus, recognition of the effective factors that explain the supply chain behavior and its subsystems in different conditions is an important issue [40]. In the field of performance modeling of the system, simulation techniques have been utilized widely. These techniques are very time and cost consuming and require enough time to gather information about parts of the system and presumptions for the manner of system relation with external environment and also the function of internal parts, and the more complex systems required more presumptions, which increase the distance from the real system by itself (Fig. 1). In this study, a flexible meta-modeling approach for optimization of the performance of supply chain is presented. Service level is considered as one of the most important performance indicators of supply chain. The flexible approach

1454 Fig. 1 Comparison of information requirements in simulation and regression

Int J Adv Manuf Technol (2013) 68:1453–1470

a

b

Factors affecting the Sys. Performance

Factors affecting the Sys. Performance

System Performance

Inforamation requirements for Regression

System Performance

Information requirements for Simulation

Requires Data gathering and Data processing to consider the asumptions Requires Data gathering

is composed of fuzzy linear regression (FLR), ridge regression (RR), and artificial neural network (ANN). The most fitted FLR, RR, and ANN models are selected via mean absolute percentage error (MAPE). Furthermore, a sensitivity analysis is conducted to verify and validate the results. Furthermore, by changing the number of train and test data sets, the values of MAPE are calculated for ANN, FLR, and RR. According to the literature and to the best of our knowledge, this is the first study that presents a comprehensive flexible approach for identification of optimum performance of supply chain with possible noise, nonlinearity, and uncertainty. The significance of the proposed algorithm is flexibility, which identifies the best model based on the results of MAPE. The latest FLR models are utilized in the flexible approach to cover environmental uncertainty and noise. In addition, ANNs with different training algorithms are used to cover the data nonlinearity and complexity. The conventional regression approaches are applied to support the mechanism and results of this study. This remainder of the paper is organized as follow: In Section 2, the relevant literature in this area is reviewed. In Section 3, proposed approach is explained, and FLR, RR, and ANN models, Fuzzification method and error estimation methods are proposed. The case study is presented in Section 4. Computational results of FLR, RR, and ANN models according to the proposed flexible ANN-FLR-RR approach are shown by Sections 5. Section 6 presents sensitivity analysis, and conclusions are drawn in Section 7.

2 Literature review 2.1 Supply chain performance modeling Supply chain and supply chain management have been studied from different points of veiw [5, 17, 18, 22, 23, 35]. The most recent studies in the field of system performance modeling are in continuous and discrete simulation of the chain behavior. The most well-known dynamic

modeling is “beer game,” which inspired from the “Forrester model” [14]. Sterman [38] considered the application of this model in the business space with more details. Liang and Huang [27] presented an agent-based demand forecast in multiechelon supply chain. Angerhofer and Angelides [2] developed a performance measurement system for collaborative supply chains. The performance of SCM has also been investigated from general perspectives in addition to various alliances [6, 8, 13]. Kamath and Roy [21] provided an experiential method for designing the structure of a supply chain on the basis of dynamic modeling for short life-cycled product markets. Although a few work considered SCM performance from different points of view, to the best of our knowledge, none of them has considered a flexible approach for optimization of SCM performance in complex and vague environment. 2.2 Methods Regression analysis and other statistical tools such as factor analysis in supply chain area have been mostly applied to analyze data gathered from statistical information, opinion polls, and general rules deduction [26, 42]. Regression analysis has been criticized as being misused in many cases where the appropriate assumptions cannot be verified to hold [12]. One factor contributing to the misuse of regression is that it can take considerably more skill to critique a model than to fit a model [11]. The earliest form of regression was the method of least squares [16, 25]. The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average [29]. At the present time, the term “regression” is often synonymous with “least squares curve fitting.” Fuzzy sets have been applied for handling uncertainty and vagueness associated with data [45, 46]. FLR is an extension of the classical regression and is used to estimate the relationships among variables where the available data are very limited and imprecise and variables are interacting


in an uncertain, qualitative, and fuzzy way. Interest readers about differences between FLR and classical regression are referred to Azadeh et al. [4]. FLR models have been successfully applied to various problems such as forecasting [42] and engineering [24]. In general, there are two approaches of FLR due to different fitting criterions [33].The first approach is based on minimizing fuzziness as an optimal criterion, which was proposed by Tanaka et al. [39]. Different researchers used Tanaka’s approach to minimize the total spread of the output. As pointed out by Wang and Tsaur [43], the advantage of this approach is its simplicity in programming and computation, but it has been criticized to provide too wide ranges in estimation, which could not give much help in application and not to utilize the concept of least squares [7].The second approach uses least squares of errors as a fitting criterion to minimize the total square error of the output. Chang and Ayyub [7] discussed reliability issues of FLSRA, such as standard error and correlation coefficient. This approach, though providing narrower range, costs too much of computation time. Hojati et al. [19] introduced a goal programming-like approach to minimize the total deviation of upper values of H-certain estimated and corresponded observed intervals and deviation of lower values of H-certain estimated and related observed intervals. According to the literature review, it can be concluded that most recent studies do not introduce a flexible or consolidated approach or algorithm to face with both uncertainty and complexity. This is particularly true in the area of supply chain management. This study introduces a flexible approach to fill the gap that exists in previous studies.

3 Method: the flexible ANN-FLR approach Supply chain is a complex and nonlinear system consisting of different parts dealing with noisy, limited, and nonintegrated data. It therefore requires methods that can alleviate these problems. Thus, we proposed a neuro-fuzzyregression approach, namely, ANN-FLR-RR (according to Fig. 2) to alleviate these problems. According to the proposed approach, ANN, FLR, and RR models are applied for modeling and optimization of supply chain performance. After determining the input and output variables, related data are collected. Then, the data are preprocessed in order to decrease multiple correlations and to eliminate the noise. The optimum value of n (minimum number of observation for the test period) is computed using operational characteristic (OC) curve. Moreover, recent FLR models are compared, and the best one is selected with respect to MAPE. Furthermore, the proper regression model

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is identified. In addition, several ANNs are developed through multilayer perceptron (MLP), and the best one with minimum MAPE is selected for further considerations. The preferred model among the selected ANN, FLR, and RR is selected by the minimum value of MAPE. In addition, a sensitivity analysis is conducted to verify and validate the results. Furthermore, by changing the number of train and test data sets, the values of MAPE are calculated for neural network, fuzzy mathematical programming, and regression. Inputs that are used in order to predict service level are as follows: lead time, dispersion of lead time, forecast error, suppliers’ service level, suppliers delay, stock coverage, backlog depth, total number of deliverable products, express transportation rate, and number of total orders. The proposed approach will be applied with more details in Section 4. 3.1 Models description Aj x y n yi xij H yiU yi yiL byiU byiL diU diL l r U L h Ye i;L ðhÞ Ye i;R ðhÞ

Regression coefficient, a symmetric triangular fuzzy number with center αj and spreads cj Independent variable Dependent variable Sets of variables Response variable, a symmetric triangular fuzzy number with central value yi and spreads ei Independent variables, a symmetric triangular fuzzy number with a center xij and spreads fij Minimum acceptable degree of confidence Upper values of ith observed interval Center values of ith observed interval Lower values of ith observed interval Upper values of the ith estimated interval Lower values of the ith estimated interval Upper shift variables Lower shift variables Lower value for the intervals of the independent variable Upper value for the intervals of the independent variable Upper value of the observed and estimated intervals Lower value of the observed and estimated intervals Given threshold Lower points of the h-certain intervals of the Ye i and Ye i

Upper points of the h-certain intervals of the Ye i and Ye i

E2(h) Fdx fdxi

Difference from outliers,εi,L and εi,R Crisp data for ith prediction period of xth factor Fuzzy data for ith prediction period of xthfactor

1456


Fig. 2 The flexible ANN-FLR approach for optimum service level prediction in SCM with complexity and uncertainty

Selection of topology for ANN, FLR, and RR Output Servicelevel

Determination of standard inputs and output

Inputs

Preprocessing data

Lead time, Forecast error, Supplier service level, Delay, Dispersion of lead times Stock coverage, Backlog depth, Number of deliverable product Number of orders Express transportation rate

Identification of minimum number of observation (n) for test data by OC curve Development of ANN structure

Fuzzify crisp data Identification of RR model Selection of best FLR model by MAPE

Selection of best ANN based on MAPE

Sensitivity analysis: with different train and test data

Calculation of MAPE values

Selection of preferred model

3.1.1 Fuzzy linear regression models Fuzzy linear regression was introduced by Tanaka et al. [39] to determine a fuzzy linear relationship by Y ¼ A0 X0 þ A1 X1 þ . . . þ A10 X10 ; Where regression coefficients A j (j = 0, …,10)are supposed to be a symmetric triangular fuzzy number,with center αj, having membership function equal to one, and spreads cj(cj≥0). The independent variables (x) can be taken into consideration as crisp or fuzzy numbers. The dependent variable (y) is a fuzzy number. The input information are 30 sets of variables yi ; xi0 ; xi1 ; . . . ; xi10 ði ¼ 1; 2; . . . ; 30; n j þ 1Þ,where xi0 =1. The response variable yi is assumed to be a symmetric triangular fuzzy number with central value yi and spreads ei , where ei 0. Independent variables values xij, (i=1,2,…,30;j=1,2,…,10) is also supposing to be a symmetric triangular fuzzy number with a center xij and spreads fij (fij ≥0). The membership functions of both dependent and independent variables are linear. The interval ½yi ð1 H Þ ei yi þ ð1 H Þ ei should be used in the case that we are only interested in that part of yi, which has a membership value of at least H, where 0≤H≤1 and H stands for the minimum acceptable degree of confidence knowing as H-certain observed interval. Furthermore, suppose that the independent variables (xj) have exact values and regression coefficient (Aj) is assumed to be symmetric triangular fuzzy numbers. Thus, the estimated interval corresponding to a input set of independent variables X

ðxi0 ; xi1 ; . . . ; xi10 Þ having membership function value of at " 10 10 P P a j þ ð1 H Þ least H is aj ð1 H Þ cj xij j¼0

j¼0

cj Þ xij , this knowing as distance as H-certain estimated interval. The membership function of the fuzzy parameter Aj is represented by: ( jaj aj j for aj cj aj aj þ cj μAj aj ¼ 1 cj ð1Þ 0 otherwise In order to estimate Aj, Tanaka et al. [39] proposed the following linear programming formulation: c0 þ c1 þ c2 þ þ c10 10 P subject to : aj þ ð1 HÞ cj xij yi þ ð1 HÞ ei ; Minimize

j¼0

10 P

ði ¼ 1; ; 30Þ aj ð1 H Þ cj xij yi ð1 H Þ ei ; ði ¼ 1; ; 30Þ

j¼0

aj ¼ free; cj 0; ðj ¼ 0; ; 10Þ

ð2Þ One of the criticisms on the FLR model introduced by Tanaka et al. [39] is that the results are xj scale dependent, and many cj’s might equal to zero. Thus, sum of spreads of


1457

FLR model’s coefficients replace with sum of spreads of the estimated intervals as an objective function:

Minimize

30 X 10 X

ð3Þ

cj xij

i¼1 j¼0

Another comment of the model of Tanaka et al. [39] is that each H-certain estimated interval is needed to involve the corresponding H-certain observed interval.This result in large coefficient spreads cj if any dependent variable has large spreads ej or if there are outliers. Sakawa and Yano [36] categorize the independent variables into three classes depending upon the presumed range of values of coefficientsAj:

8 < J1 ¼ those variables j; j ¼ 0; ; 10; which will have aj ð1 HÞ cj 0; J2 ¼ those variables j; j ¼ 0; ; 10; which will have aj ð1 HÞ cj < 0; and aj þ ð1 HÞ cj 0; : J3 ¼ those variables j; j ¼ 0; ; 10; which will have aj þ ð1 HÞ cj < 0:

ð4Þ

Then, the FLR model of this approach will be formulated as follows [19]:

Minimize

30 P

ðb yiL Þ yiU b P aj þ ð1 HÞ cj xij þ ð1 HÞ fij þ aj þ ð1 HÞ cj xij ð1 HÞ fij ¼ b yiU ;

i¼1P

subject to :

j2J1 [J2

j2J3

ði ¼ 1; ; 30Þ; byP iU yi ð1 H Þ ei ; ði¼ 1; ; 30Þ P aj ð1 H Þ cj xij ð1 H Þ fij þ aj ð1 H Þ cj xij þ ð1 H Þ fij ¼ byiL ; j2J1

ði ¼ 1; ; 30Þ; byiL yi þ ð1 H Þ ei ; ði ¼ 1; ; 30Þ aj ¼ free; cj 0; j ¼ 0; ; 10:

j2J2 [J3

ð5Þ

Furthermore, the following problem was proposed bySakawa and Yano [36]: 30 P

ðb yiL Þ yiU b P P yiU ; aj þ ð1 HÞ cj xij þ ð1 HÞ fij þ aj þ ð1 HÞ cj xij ð1 HÞ fij ¼ b subject to : Minimize

i¼1

j2J1 [J2

j2J3

i ¼ 1; ; 30; byP iU yi H ei ; ði ¼ 1; ; nÞ P aj ð1 H Þ cj xij ð1 H Þ fij þ aj ð1 H Þ cj xij þ ð1 H Þ fij ¼ byiL ; j2J1

ði ¼ 1; ; 30Þ; byiL yi þ H ei ; ði ¼ 1; ; 30Þ aj ¼ free; cj 0; ðj ¼ 0; ; 10Þ:

j2J2 [J3

ð6Þ

Peters [33] assumed that yiU, yi , and yiL be the upper, center, and lower values of ith observed interval, and let byiU and byiL be the upper and lower values of the ith estimated interval. This model permits byiL to be greater than yiL but

smaller than yiU, and byiU to be smaller than yiU but greater than yiL. In fact, the mean of all deviations of byiU from yi , if byiU < yi , and byiL from yi , if byiL > yi , is minimized [19]. The formulation of Peters [33] model is:

1458


Maximize λ 10 P subject to : aj þ cj xij yi ð1 λi Þ ei ; ði ¼ 1; ; 30Þ 10 P

j¼0

aj cj xij yi þ ð1 λi Þ ei ; ði ¼ 1; ; 30Þ

Moreover, Wang and Tsaur [42] proposed a new formulation by minimizing the central values with eases the relations of constraint from Peters [33] formulation:

j¼0

λ ¼ ðλ1 þ λ1 þ λ1 Þ=n; 30 P 10 P cj xij P0 1 λ ;

Maximize λ subject to : ð1λÞ p

i¼1 j¼0

0 λi 1; ði ¼ 1; ; 30Þ; λ 0; aj ¼ free; cj 0; ðj ¼ 0; ; 10Þ:

ð1λÞ p þ

ð7Þ

ð1λÞ p

N P j¼0 N P

Maximize λ 30 P 10 P subject to : cj xij P0 ð1 λÞ i¼1 j¼0

aj xij þ aj xij þ

aj cj xij yi þ ð1 λÞ ei ; ði ¼ 1; . . . ; 30Þ ð8Þ

Ozelkan and Duckstein [32] introduced a similar model to Peters [33], which does not need the estimation intervals to divide the observed intervals. The formulation can be written as follows: ðdiU i¼1 10 P

þ diL Þ

aj þ ð1 HÞ cj xij yi þ ð1 HÞ ei diU

subject to :

j¼0

10 P j¼0

ði ¼ 1; ; 30Þ aj ð1 H Þ cj xij yi ð1 H Þ ei þ diL ði ¼ 1; ; 30Þ

30 P 10 P

cj xij yi þ ei ; ð8i ¼ 1; 2; M Þ

Hojati et al. [19] introduced a simple goal programminglike method to select the FLR coefficients such that the total deviation of upper values of H-certain estimated and corresponded observed intervals and deviation of lower values of H-certain estimated and related observed intervals are minimized. This can be obtained by using the following formulation: 30 P

ðdþ iU þ d iU þ dþ iL þ d iL Þ

aj þ ð1 HÞ cj xij d iU yi þ ð1 HÞ j¼0 ei dþ iU ði ¼ 1; ; 30Þ 10 P aj ð1 H Þ cj xij d iL yi ð1 H Þ ei þ d þ iL j¼0 ði ¼ 1; ; 30Þ 30 P 10 P cj xij u;

subject to :

0 λ 1;cj 0

30 P

j¼0

i¼1 10 P

j¼0

Minimize

j¼0 N P

cj xij yi þ ei ; ð8i ¼ 1; 2; M Þ

ð10Þ

Minimize

d0 ;

0 λ 1; ð8i ¼ 1; ; M Þ; xi0 ¼ 1; aj ¼ free; cj 0; j ¼ 0; ; 10:

10 P aj þ cj xij yi ð1 λÞ ei ; ði ¼ 1; . . . ; 30Þ j¼0 10 P

2 cj xij ei

j¼0 i¼1 N P

j¼0

It is difficult to determine a proper value for P0, and the result is sensitive to this parameter [19, 33]. Furthermore, in this formulation, only one membership function is considered, the result of which belongs to the set of good solution λ for all constraints. Therefore, a new fuzzy linear regression formulation is proposed by Peters [33]:

M N P P

cj xij u;

i¼1 j¼0

diL ; diU 0; ði ¼ 1; ; 30Þ aj ¼ free; cj 0; ðj ¼ 0; ; 10Þ

ð9Þ where υ is a parameter and which should be diversified over all possible amounts of total spreads of estimated intervals, and diU and diL (i = 1,…,30) are upper and lower shift variables.

i¼1 j¼0 d þ iU ; d iU ; d þ iL ; d iL

0; ði ¼ 1; ; 30Þ aj ¼ free; cj 0; ðj ¼ 0; ; 10Þ

ð11Þ The jd þ iU d iU j is the distance between upper value of H-certain estimation interval and the upper value of the H-certain observed interval; therefore, the jd þ iL d iL j is the distance between lower value of Hcertain estimated interval and the lower value of the Hcertain observed interval. The objective is to minimize the sum of these two intervals [19]. Furthermore, in case that independent and response variable are fuzzy, Hojati et al. [19] select the FLR coefficients so that the total difference between upper values of estimated and related observed intervals and distance among lower values of estimated and related observed intervals are minimized at both lower values and upper values of each of the independent variable (except x0). The following model is formulated for the


1459

condition that there is only one independent variable (in addition to x0)

Minimize subject to :

30 P

ðd þ ilU þ d ilU þ d þ ilL þ d ilL þ d þ irU þ d irU þ d þ irL þ d irL Þ

i¼1 l P

aj þ ð1 H Þ cj xij ð1 H Þ fij d ilU ¼ yi þ ð1 H Þ ei d þ ilU ;

j¼0

i ¼ 1; ; 30; l P aj þ ð1 H Þ cj xij þ ð1 H Þ fij d irU ¼ yi ð1 H Þ ei d þ irU ;

j¼0

i ¼ 1; ; 30; l P aj ð1 H Þ cj xij ð1 H Þ fij d ilL ¼ yi þ ð1 H Þ ei d þ ilL ;

ð12Þ

j¼0

i ¼ 1; ; 30; l P aj ð1 H Þ cj xij þ ð1 H Þ fij d irL ¼ yi ð1 H Þ ei d þ irL ;

j¼0

i ¼ 1; ; 30; 30 P 10 P cj xij u;

i¼1 j¼0 d þ ilU ; d ilU ; d þ ilL ; d ilL ; d þ irU ; d irU ; d þ irL ; d irL

aj ¼ free; cj 0; ðj ¼ 0; ; 10Þ

In the indices, l refers to the lower value, and r refers to the upper value for the intervals of the independent variable; moreover, U refers to the upper value, and L refers to the lower value of the observed and estimated intervals [19]. subject to :

m P 30 P

0; ði ¼ 1; ; 30Þ

Chen et al. [9] proposed a fuzzy least-squares regression model for symmetric membership functions Ã and . The formulation of this model is:

m P 30 h 2 P 2 2 i yi aj xij þ ei cj xij þ ei cj xij

i¼1 j¼0

subject to : yi ð1 H Þ ei yi þ ð 1 H Þ ei

30 P j¼0 30 P

i¼1 j¼0

aj xij cj xij ð i ¼ 1; ; mÞ

ð13Þ

aj xij þ cj xij ði ¼ 1; ; mÞ

j¼0

cj 0; xij 0: where H=[0, 1].

Wang and Tsaur [43] also considered least-squares approach to minimize the total vagueness of the given data, such that the membership degree of each observation is

Minimize subject to :

M N P P j¼0 N P

2 cj xij ei

i¼1

aj xij þ ð1 H Þ1=2

j¼0 N P j¼0

greater than a threshold h.This leads to the following nonlinear programming model:

N P

cj xij yi þ ð1 H Þ1=2 ei ð8i ¼ 1; ; M Þ

j¼0

aj xij þ ð1 H Þ1=2

N P j¼0

cj xij yi þ ð1 H Þ1=2 ei ð8i ¼ 1; ; M Þ

aj 2 R; cj 0; j ¼ 1; 2; ; N ; xi0 ¼ 1; 0 λ 1:

ð14Þ

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Chang and Ayyub [7] introduced a fuzzy linear regression model. The input and output data are crisp, but coefficients are triangular fuzzy numbers. The formulation of this model can be written as follows: 30 P 10 P

Minimize

cj xij

i¼1 j¼0 10 P

aj þ ð1 H Þ cj xij yi ði ¼ 1; ; 30Þ

subject to :

j¼0 10 P

aj ð1 H Þ cj xij yi ði ¼ 1; ; 30Þ

ei ¼ c0 þ

n X

cj xij þ fij aj ; ði ¼ 1; 2; : : :; mÞ

As mentioned by Peters [33] and Ozelkan and Duckstein [32], outliers can be formulated by considering soft constraints to the fuzzy linear regression model. To achieve the maximum degree of fitness of the predicted fuzzy linear regression model, h sets to be one. Then, the model can be constructed as follows:

j¼0

aj ¼ free; cj 0; j ¼ 0; ; 10:

Minimize

D2 ðhÞ ¼

Minimize

E 2 ðhÞ ¼

ð15Þ In Tanaka et al. [39] approach, the objective is minimizing the total spread of fuzzy coefficient Ãj. Thus, the relevance between response variable and independent variables is wanted to be as certain as possible.When studying the minimal total spread, such fuzziness or uncertainty should 30 P 10 P be considered as cj xij ei . The model that was i¼1 j¼0

proposed by Lai and Chang [24] is as follows: Minimize n n X X aj xij þ fleftð1 HÞ cj xij yi subject to : j¼0

j¼0

þ ð1 H Þ ei ð8i ¼ 1; ; mÞ n n X X aj xij þ ð1 H Þ cj xij yi j¼0

m P

ðei ei Þ2 ;

i¼1

m 2 P "2 i;L þ "2 i;R ; i¼1

subject to : yi yi "i;L ; ði ¼ 1; ; mÞ yi yi "i;R ; ði ¼ 1; ; mÞ ei 0; "i;L ; "i;R 0; ði ¼ 1; ; mÞ

m P

ðei ei Þ2 þ ð1 wÞ

ZðhÞ ¼ w

subject to :

yi yi "i;L ; ði ¼ 1; ; mÞ yi yi "i;R ; ði ¼ 1; ; mÞ ei 0; "i;L ; "i;R 0; ði ¼ 1; ; mÞ

i¼1

j¼0

As mentioned by Modarres et al. [28] and Nasrabadi et al. [31], a quadratic programming could be formulated as follows: m P Minimize DðhÞ ¼ ð ei ei Þ 2

i¼1

where ω is defined by decision makers.When 0