Chaos-Based Fuzzy Regression Approach to Modeling ... - IEEE Xplore

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 21, NO. 5, OCTOBER 2013

Chaos-Based Fuzzy Regression Approach to Modeling Customer Satisfaction for Product Design Huimin Jiang, C. K. Kwong, W. H. Ip, and Zengqiang Chen

Abstract—The success of a new product is very much related to the customer satisfaction level of the product. Therefore, it is important to estimate the customer satisfaction level of a new product in its design stage. Quality function deployment is commonly used to develop customer satisfaction models for product design. Relationships between customer satisfaction and design attributes are highly fuzzy and nonlinear, but these relationship characteristics cannot be captured by existing customer satisfaction models. In this paper, we propose a novel chaos-based fuzzy regression (FR) approach with which fuzzy customer satisfaction models with second- and/or higher order terms, and interaction terms can be developed. The proposed approach uses a chaos optimization algorithm to generate the polynomial structures of customer satisfaction models. Thereafter, it employs an FR method to determine the fuzzy coefficients of the individual terms of models. To illustrate and validate the proposed approach, it is applied in the development of a customer satisfaction model for a mobile phone design. Five validation tests are conducted to compare modeling results from the chaos-based FR with those from statistical regression, FR, and fuzzy least-squares regression. Results of the validation tests show that the proposed approach outperforms the other three approaches in terms of mean relative errors and variance of errors and customer satisfaction models with second- and/or higher order terms, and interaction terms can be developed effectively using the proposed chaos-based FR approach. Index Terms—Chaos optimization algorithms (COAs), customer satisfaction models, fuzzy regression (FR), product design.

I. INTRODUCTION VER the past few decades, there has been an increasing emphasis on a company’s ability to produce high-quality products. Such products can be identified by the measurement of their associated customer satisfaction level. The success of products is heavily dependent on the associated customer satisfaction level. If customers were satisfied with a new product, the chance of that product being successful in marketplaces would be higher. Customer satisfaction also has a direct influence on customer retention [30], sustained customer loyalty [31], [34],

O

Manuscript received January 16, 2012; revised July 23, 2012; accepted November 20, 2012. Date of publication January 3, 2013; date of current version October 2, 2013. This work was supported by a grant from The Hong Kong Polytechnic University. H. Jiang, C. K. Kwong, and W. H. Ip are with the Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Kowloon 80234, Hong Kong (e-mail: [email protected]; mfckkong@ polyu.edu.hk; [email protected]). Z. Chen is with the Department of Automation, NanKai University, Tianjin 300071, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TFUZZ.2012.2236841

and competitiveness for a company [23]. A product is usually associated with a number of design attributes, such as size, weight, and power consumption. Settings of the design attributes can affect customer satisfaction on the product. For example, customer satisfaction on notebook computer A in terms of “portability” is better than notebook computer B, because the size of computer A is smaller and its weight is lighter. In this regard, it is crucial to model the relationships between customer satisfaction and design attributes for product design. The developed models, namely customer satisfaction models, can be employed to formulate an optimization model to determine an optimal setting of design attributes that leads to the maximum customer satisfaction. To develop the customer satisfaction models, datasets are required, which are commonly obtained from lead user surveys, customer surveys, or market surveys. In the surveys, respondents are always asked to rate various dimensions of customer satisfaction of competitive products, such as quality and user friendliness [33]. Unavoidably, respondents’ rating on the products involves their subjective judgments that would lead to a high degree of fuzziness of the survey data. On the other hand, quite a number of previous studies [24], [27] already showed that settings of design attributes would have substantial nonlinear effects on customer satisfaction. For example, suppose the customer satisfaction level of “portability” of a notebook computer is 3 (the range of customer satisfaction level is 1 to 5. “5” is the highest satisfaction level, while “1” is the lowest level), and the weight of the notebook computer is 1.5 kg. If the weight of the computer is reduced to 1.25 kg, the customer satisfaction level is found to increase to 3.5 after a user survey. If the weight is further reduced to 1.0 kg, the customer satisfaction level increases largely to 4.5 as customers are excited with the lightweight computer. Therefore, the major challenge of modeling customer satisfaction is to generate models that are able to capture both the fuzziness of survey data and nonlinearity of the modeling. Previous studies have attempted to model the relationships between customer satisfaction and design attributes/elements that are based on market/customer survey data. They have used modeling techniques that require a large number of survey datasets. Chen et al. [1] employed Kohonen’s self-organizing map neural network to model the relationships between design attributes and customer satisfaction in a product design prototype system. Park and Han [2] developed models to relate customer satisfaction to design attributes using a fuzzy rule-based approach. Liu et al. [4] proposed fuzzy techniques to calculate an e-commerce customer satisfaction index that is based on a five-level quantity table. Lin et al. [5] proposed a fuzzy logic model to assess the form design of mobile phones. Their experimental results show that the fuzzy

1063-6706 © 2013 IEEE

JIANG et al.: CHAOS-BASED FUZZY REGRESSION APPROACH TO MODELING CUSTOMER SATISFACTION FOR PRODUCT DESIGN

model outperforms neural network-based models in terms of the root of mean square errors. However, only implicit customer satisfaction models were generated in those studies by which no explicit information of product design and no clear relationships between customer satisfaction and design attributes could be obtained. Grigoroudis and Siskos [6] developed the multicriteria satisfaction analysis (MUSA) method to measure and analyze customer satisfaction. MUSA is a preference disaggregation model that is based on the working principles of ordinal regression analysis. Using survey data, MUSA aggregates individual judgments to develop a collective value function and thereby quantify customer satisfaction. Grigoroudis et al. [7] further applied MUSA to measure user-perceived web quality. However, those previous studies assumed that the overall customer satisfaction was measured solely with respect to several customer attributes, and customer attributes were considered dependently. Using quantification I analysis, You et al. [8] developed customer satisfaction models to examine the related design variables and preferred design features of automotive interior materials. They identified design variables (including their corresponding values) that affect customer satisfaction. Han et al. [9] developed various usability dimensions, including both subjective and objective dimensions, to evaluate product usability based on statistically regressed models, which relate design attributes and customer satisfaction. However, the statistical regression (SR) approaches can only be used to generate linear customer satisfaction models. In addition, the fuzzy relationships cannot be addressed. Genetic programming (GP) was proposed by Chan et al. [16] to model customer satisfaction for product development. The developed models contain interaction terms and higher order terms and are explicit, by which the relationships between customer satisfaction and design attributes can be clearly recognized and sensitivity analysis of the design attributes can be done easily. However, GP fails to capture the fuzziness of modeling. In order to address both the fuzziness and nonlinearity, Kwong et al. [3] introduced a neuro-fuzzy network approach to developing customer satisfaction models for new product development. However, this approach cannot effectively handle the problem that involves a number of attributes, and the conventional learning algorithms are gradient descent. The calculation of gradients in each step is difficult and the use of chain rule may cause a local minimum. Quality function deployment (QFD) is commonly used to develop customer satisfaction models, where normally only a small number of datasets are available for the modeling. Dawson and Askin [10] used a multiple linear regression method, which considers nonlinear terms, to model customer satisfaction in QFD. However, the order of the generated polynomials is user defined; thus, an optimal model may not be generated. The fuzzy set theory has been adopted by few studies to address the fuzziness of QFD-based customer satisfaction models. Fung et al. [11] introduced fuzzy logic to develop fuzzy rule-based models to relate customer satisfaction to design attributes. Kim and Park [12] suggested a fuzzy regression (FR) approach to generating customer satisfaction models. Chen et al. [13] proposed another FR approach, which was based on asymmetric triangular fuzzy coefficients, to modeling customer satisfaction

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in QFD. Customer satisfaction was modeled by Chen [14] using nonlinear programming and FR models. Kwong et al. [15] proposed a generalized fuzzy least-squares regression approach to modeling customer satisfaction. The aforementioned approaches can only be used to generate linear fuzzy models, which contain only first-order terms, to relate customer satisfaction to design attributes. In conclusion, explicit customer satisfaction models, which can capture both the fuzziness and nonlinearity, have not yet been developed in previous studies. To fill the existing research gap, a novel FR approach, namely chaos-based FR is proposed to model the relationships between customer satisfaction and design attributes. The proposed approach employs a chaos optimization algorithm (COA) to generate polynomial structures of customer satisfaction models with second- and/or higher order terms and interaction terms. COA employs chaotic dynamics to solve an optimization problem, which does not rely on learning factors, and has been shown to have faster convergence and can search for more accurate solutions compared with the conventional optimization methods [26]. COA also has better capacity in searching for the global optimal solution of an optimization problem and can escape from a local minimum easier than the conventional optimization algorithms. However, it cannot address the fuzziness of survey data. Tanaka’s FR method [20] is introduced to determine the fuzzy coefficients of the models. FR is effective for the modeling problems, where the degree of fuzziness of datasets for modeling is high, and only a small amount of datasets are available for the modeling. However, the FR method can only yield linear-type models. The proposed chaos-based FR approach combines the advantages of COA and FR to generate customer satisfaction models, which are able to capture both the fuzziness and nonlinearity of modeling. This paper is organized as follows. Section II describes the chaos-based FR method. Section III describes how the proposed approach is applied in modeling customer satisfaction for mobile phone products. The validation of the proposed approach is presented in Section IV. Finally, conclusions are given in Section V. II. CHAOS-BASED FUZZY REGRESSION METHOD To solve the problems of modeling customer satisfaction, a chaos-based FR approach to developing fuzzy polynomial models that are based on a small number of datasets is proposed. Fig. 1 shows a flowchart of the proposed approach to modeling customer satisfaction for product design. The generated fuzzy polynomial models can contain second- and/or higher order terms and interaction terms, such that the nonlinearity of modeling can be better captured. In the proposed approach, a COA is introduced to generate the optimal polynomial structure of fuzzy models. Tanaka’s FR analysis is then adopted to determine fuzzy coefficients for each term of the developed structure. A. Fuzzy Polynomial Models With Second- and/or Higher Order Terms and Interaction Terms The Kolmogorov–Gabor polynomial has been used widely to evolve general nonlinear models, but it is incapable of

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The fuzzy polynomial model (1) can be rewritten as follows: y˜ = A˜0 + A˜1 ∗x1 + A˜2 ∗x2 . . . AÑ N R ∗xN N R

(2)

s c s c s y˜ = (ac 0 , a0 ) + (a1 , a1 ) ∗x1 + (a2 , a2 ) ∗x2 c + · · · + aN N R , as N N R ∗xN N R

(3)

or

where A˜0 = A˜0 , A˜1 = A˜1 , A˜2 = A˜2 , . . . , AÑ N R = AÑ ...N ; A˜0 = c s c s s ˜ ˜ (ac 0 , a0 ) , A1 = (a1 , a1 ) , . . . , AN N R = aN N R , aN N R , and x0 = 1, x1 = x1 , x2 = x1 x2 , . . . , xN N R = x1 x2 . . . xd . Furthermore, xj and A˜j (j = 0, 1, 2 . . . , NN R ) are transformed variables and fuzzy coefficients, respectively. The vector of the fuzzy coefficients can be defined as follows: A˜ = A˜ , A˜ , . . . A˜ 0

Fig. 1.

addressing the fuzziness of modeling data [25]. In the fuzzy polynomial models that are developed based on the proposed approach, nonlinear terms and interaction terms between independent variables are represented in a form of a higher order Kolmogorov–Gabor polynomial. Fuzzy coefficients of the models are determined using Tanaka’s fuzzy regression method. The proposed models can overcome the limitation of conventional fuzzy regression models where only first-order terms are generated. A fuzzy polynomial model that is based on the chaos-based FR approach can be expressed as follows: y˜ = fÑ R (x) = A˜0 +

N

A˜i 1 xi 1 +

i 1 =1

+

N N N

N N

A˜i 1 i 2 xi 1 xi 2

i 1 =1 i 2 =1

A˜i 1 i 2 i 3 xi 1 xi 2 xi 3

i 1 =1 i 2 =1 i 3 =1

+ ...

N i 1 =1

...

N i d =1

A˜i 1 ...i d

d

1

NN R

c s c s s = (ac 0 , a0 ) , (a1 , a1 ) , . . . , aN N R , aN N R

Flowchart of the proposed approach.

xi j

(1)

j =1

where y˜ is the dependent variable, xi j is the ij th independent variable, ij = 1, . . . , N and j = 1, 2, . . . , d. N and d are the number of design variables. A˜ are the fuzzy coefficients of the linear, second-order, and/or higher order terms and interaction terms of the model A˜0 = (ac0 , as0 ) , A˜1 = (ac1 , as1 ) A˜2 = (ac2 , as2 ) , . . . , AÑ = (acN , asN ) A˜11 = (ac11 , as11 ) , A˜12 = (ac12 , as12 ) A˜13 = (ac13 , as13 ) , . . . , AÑ N = (acN N , asN N ) A˜111 = (ac111 , as111 ) , A˜112 = (ac112 , as112 ) A˜113 = (ac113 , as113 ) , . . . , AÑ N N = (acN N N , asN N N ) . . . . . . . . . AÑ ...N = (acN ...N , asN ...N ) where ac and as are the central value and the spread of the fuzzy numbers, respectively.

c

a =

c c (ac 0 , a1 , . . . , aN N R

)

s s as = (as 0 , a1 , . . . , aN N R ).

(4) (5) (6)

B. Determination of Model Structures Using Chaos Optimization Algorithm In this research, a COA was introduced to determine the polynomial structure of fuzzy models. COA is a stochastic search algorithm in which chaos is introduced into the optimization strategy to accelerate the optimum seeking operation and find the global optimal solution [28]. There are two phases in searching for an optimal solution in the chaos optimization process. The first phase, which is called wide search, is for the whole solution space according to an ergodic track. When the end condition is satisfied, the current optimal state becomes close to the optimal solution and the second phase starts. The second phase is based on the results of the first phase. It is a narrow search that focuses on a local region, and it adds a small disturbance term until the final requirement is met. The added disturbance can be a chaos variable, random variable that is based on Gaussian distribution or uniform distribution, or bias that is generated by the mechanism of gradient descent. Current COAs use the carrier wave method in mapping chosen chaos variables to the space of optimization variables linearly, and then searching for optimal solutions that are based on the ergodicity of chaos variables. COA employs chaotic dynamics to solve optimization problems [17], and it has been successfully applied in various areas, such as robot optimization control, function optimization, supply chain optimization [18], and manufacture process and fuzzy logic [19]. Most current COAs focus on chaos variable-based optimization rather than on the introduction of chaos variables as a small disturbance in search optimization. The logistic model that is used in chaos optimization is shown in (7). The logistic mapping can generate chaos variables by iteration cn +1 = f (cn ) = μcn (1 − cn )

(7)

where μ = 4, and cn ∈ [0, 1] is the nth iteration value of the chaos variable c.


The optimization process uses the chaos variables that are generated from logistic mapping to conduct the search through its own locomotion law. Chaos has the dynamic properties that include ergodicity, intrinsic stochastic property, and the sensitive dependence on initial conditions. The characteristic of randomness ensures the possibility of large-scale search. Ergodicity allows COA to traverse all possible states without repetition and to overcome the limitations that are caused by ergodic search in general random methods. The linear mapping to convert chaos variables into optimization variables is formulated as follows: qn = a + (b − a) · cn

(8)

where qn is the optimization variable; and a and b are the lower and upper limits of the optimization variable q, respectively. Based on the iteration, the chaos variables traverse in [0, 1], and the corresponding optimization variables traverse in the corresponding range [a, b]. Therefore, the optimal solution can be found in the area of feasible solution. Each optimization variable represents the polynomial structure of a fuzzy model. It is described by the input variables [x1 , x2 , . . . and xm ] and the arithmetic operations. Referring to (2), there are two arithmetic operations, addition (+) and multiplication (∗), in the fuzzy polynomial model. The optimization variable at the nth generation is defined as follows: (9) qn = qn1 , qn2 , . . . , qnN c where Nc is an odd number that represents the number of elements in a chaos variable. For example, if there are four design variables in the model, the value of Nc is first selected as 7 in which four elements represent the four design variables, and another three elements in the middle of every two adjacent design variables represent arithmetic operations to guarantee that the optimization variable qn has the chance to include all four variables. Thereafter, if the error requirement is not satisfied, the Nc value is adjusted until a satisfactory modeling error, which is close to zero and is smaller than the modeling errors that are based on the previous studies, is achieved. The elements in odd numbers (qn1 , qn3 , . . . , qnN c ) are used to represent variables in a nonlinear structure. For odd number k, if (l − 1)/m < qnk ≤ l/m (m is the number of variables in a nonlinear fuzzy model, 1 ≤ l ≤ m), the position of qnk is represented by the lth variable xl . The elements in even numbers (qn2 , qn4 , . . . , qnN c −1 ) are used to determine arithmetic operations. For even number k, if 0 < qnk ≤ 1/2 and 1/2 < qnk ≤ 1, the arithmetic operations are chosen to be addition (+) and multiplication (∗), respectively. For example, an optimization variable with nine elements is used to represent the structure of a fuzzy polynomial model with four dependent variables. If the optimal variable is obtained as q = [x2 , +, x3 , ∗, x4 , ∗, x4 , +, x1 ], the polynomial structure can be expressed as x2 + x3 x24 + x1 . The transformed variables are x0 = 1, x1 = x2 , x2 = x3 x24 , x3 = x1 . Therefore, the fuzzy polynomial model with fuzzy coefficients can be represented as y˜ = s ˜ A˜0 + A˜1 x2 + A˜2 x3 x24 + A˜3 x1 , where A˜0 = (ac 0 , a0 ) , A1 =

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s c s c s ˜ ˜ (ac 1 , a1 ) , A2 = (a2 , a2 ) and A3 = (a3 , a3 ). The central valc s ues aj and spread values aj (j = 0, 1, . . . , 4) of the fuzzy coefficients are determined using Tanaka’s fuzzy regression analysis [20].

C. Determination of Fuzzy Coefficients Using Fuzzy Regression Analysis Tanaka’s fuzzy regression aims to use the fuzzy functions that are defined by Zadeh’s extension principle [21], [32] in describing a phenomenon that is imprecise and vague in nature. All the input and output variables as well as the coefficients of the relationships are considered as fuzzy numbers. Two different criteria, the least absolute deviation and the minimum spread, are used for the evaluation of the fuzziness of the output. Deviations between observed and estimated values are supposed to depend on the indefiniteness of system structures and are regarded as the fuzziness of system parameters. The fuzzy parameters of fuzzy regression models suggest possibility distribution and are obtained as fuzzy sets that represent the fuzziness of models. The objective of fuzzy regression analysis is to minimize the fuzziness of the model commonly by minimizing the total spread of the fuzzy coefficients, which leads to a minimum uncertainty of the output. Based on chaos optimization, a fuzzy model that contains second- and/or higher order terms and interaction terms is represented in a polynomial structure. Fuzzy regression analysis is used to determine fuzzy coefficients for each term of the fuzzy polynomial model. In this research, Tanaka’s fuzzy regression [22] is applied to determine fuzzy coefficients. Fuzzy coefficients with the central point ac and the spread value as are determined by solving the following linear programming (LP) problem:

N M NR xj (i) (10) as Min J = j j =0

i=1

where J is the objective function that represents the total fuzziness of the system, 1 + NN R is the number of terms of the fuzzy polynomial model, M is the number of datasets, xj (i) is the jth transformed variable of the ith dataset in the fuzzy polynomial model, and | · | refers to absolute value of transformed variable. The constraints can be formulated as follows: N NR

ac j xj (i) + (1 − h)

j =0

N NR

as j xj (i) ≥ yi

j =0

i = 1, 2, . . . , M (11) N NR

ac j xj (i) − (1 − h)

j =0

N NR

as j xj (i) ≤ yi

j =0

i = 1, 2, . . . , M (12) as j

≥ 0,

ac j

∈ R, j = 0, 1, 2, . . . NN R

x0 (i) = 1 for all i and 0 ≤ h ≤ 1 where h, which refers to the degree to which the fuzzy model fits the given data, is between 0 and 1; and yi is the value

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of the ith dependent variable. Constraints (11) and (12) ensure that each objective yi has at least h degree of belonging to y˜i as μy˜i (yi ) ≥ h, i = 1, 2, . . . , M . The last constraints for the c variables ensure that as j and aj are nonnegative. Therefore, the fuzzy parameters A˜j (j = 0, 1, 2, . . . NN R ) can be determined by solving the LP problem subject to μy˜i (yi ) ≥ h. D. Algorithms of Chaos-Based Fuzzy Regression The algorithms of the proposed chaos-based FR method are described as follows. Step 1: The number of iterations and the number of elements Nc in a chaos variable are initialized. Nc is an odd number, and Nc values are chosen randomly in range of [0, 1] to decide the value of the initialized chaos variable. Step 2: The iteration starts from n = 1. The chaos variables cn are generated based on the logistic model in (7) and are transformed into the optimization variables qn using (8). Step 3: Based on the value of the optimization variable qn , the polynomial structure of the fuzzy model is determined. According to the rules that are described in Section B, the elements in odd numbers and even numbers are substituted by the input variables xk (k = 1, . . . , N and N is the number of variables) and arithmetic operations “+” and “∗,” respectively. Subsequently, the transformed variables xj with linear, second-order, and/or higher order terms and interaction terms are generated based on arithmetic operations. If the operation is “∗,” the second- and/or higher order terms and interaction terms are determined by the multiplication of the variables on both sides of “∗.” The arithmetic operation “+” is used to add all terms, which include the linear terms, to generate the final polynomial structure of the fuzzy model. Step 4: According to the generated polynomial structure, s are assigned to all trans, a the fuzzy coefficients A˜j = ac j j formed variables of the developed structure. The values of the fuzzy coefficients are calculated by solving the LP problem as shown in (10)–(12). Step 5: Based on the developed fuzzy polynomial model, the predicted variable y˜i can be calculated. The fitness value RE can then be obtained by the calculation of the relative error between y˜i and the actual data yi . Step 6: Step 2 is repeated. The algorithms are again executed another training using next experimental data until all the training datasets are used. The mean fitness value MRE(n) for all the training datasets is calculated. Step 7: The iteration continues by n + 1 → n and stops after the number of iterations reaches the predefined value. The values of MRE(n) are recorded for each iteration and the solution with the smallest mean fitness value is selected. The fuzzy polynomial model with the smallest training error is then found. Finally, the chaos-based FR model is generated. III. MODELING CUSTOMER SATISFACTION FOR MOBILE PHONE PRODUCTS USING THE CHAOS-BASED FUZZY REGRESSION METHOD In the product development stage for a new mobile phone, QFD was introduced to translate customer preferences into de-


sign attributes of the new product and thereafter perform competitive benchmarking. Thus, the house of quality (HoQ) was developed as shown in Fig. 2. Ten major competitive mobile phones were identified and denoted as Brands A–J. Lead users of mobile phones were invited to assess the ten mobile phones in terms of five dimensions of customer satisfaction: “easy to use,” “easy to read display,” “good sound quality,” “long operation time,” and “comfortable to hold.” A five-point scale labeled “very bad,” “bad,” “moderate,” “good,” and “very good” (scored from 1 to 5, respectively) was used to describe the degree of customer satisfaction. Assessment results are shown in Region A of Fig. 2. There are 11 design attributes, namely, menu layers, screen size, display font size, display brightness, maximum sound volume, power consumption, battery capacity, weight, length, width, and thickness. The value settings of the design attributes of the ten competitive products are shown in Region B of the HoQ. The middle of the HoQ shows a relationship matrix, which is defined by the product development team based on their experience and engineering judgments. It represents how design attributes relate to the dimensions of customer satisfaction: “” means strong relationship, “O” means moderate relationship, “Δ” means weak relationship, and blank means no relationship at all. For illustrative purposes, the development of a customer satisfaction model for the dimension “comfortable to hold” based on the proposed chaos-based FR approach is described in this paper. The identified design attributes related to this dimension are weight, length, width, and thickness. Therefore, four variables xk (k = 1, 2, 3, 4) are involved in the modeling (m = 4). In the proposed approach, the number of iterations for chaotic search is set as 200 to make sure that the least number of the iterations and the smallest errors are obtained. The number of elements in the chaos variable and the value of h are determined based on a trial and error method. The value setting of h was determined by using different values within a range of [0, 1]. After a number of trials, h was set as 0.5 since it yielded the smallest training errors of the fuzzy regression models. The number of elements in the chaos variable increases by two each time from seven and the proper number is set as 9. The range [a, b] for the optimization variable q is [0, 1]. The proposed approach to modeling the customer satisfaction dimension “comfortable to hold” was implemented using MATLAB programming software based on the datasets contained in the HoQ. The first element q 1 of the optimization variable q is used as an example, and search results based on COA are shown in Fig. 3. After the 20th iteration, the optimal values for q 1 become stable in the range of [2/4, 3/4]. Thus, l = 3 and the optimal solution for q 1 is x3 . Following the same process, the optimal solution for q is obtained as follows: q = [x3 , 0, x4 , 0, x1 , 0, x1 , 1, x2 ] .

(13)

Therefore, the polynomial structure of the customer satisfaction model for “comfortable to hold” can be represented as x3 + x4 + x1 + x1 x2 . The transformed variables xj are expressed as x0 = 1, x1 = x3 , x2 = x4 , x3 = x1 , x4 = x1 x2 . Using the fuzzy regression method, the fuzzy coefficients s ˜ A˜j = ac j , aj , j = 0, 1, . . . , 4, are determined as A0 =


Fig. 2.

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House of quality (HoQ) for mobile phone products.

14.9991, 2.8422 ∗ 10−14 , A˜1 = −1.5036, 2.8422 ∗ 10−14 , A˜2 = (−0.2890, 0) , A˜3 = −0.3634, 5.6843 ∗ 10−14 , and A˜4 = (0.0045, 0.0077). The customer satisfaction model for “comfortable to hold” can then be obtained as follows: y˜ = 14.9991, 2.8422 ∗ 10−14 + −1.5036, 2.8422 ∗ 10−14 x3 + (−0.2890, 0) x4 + −0.3634, 5.6843 ∗ 10−14 x1 + (0.0045, 0.0077)x1 x2 .

(14)

The output of the customer satisfaction model is a fuzzy number that represents a range of predicted values. The central value of the fuzzy number of y˜ is used as the crisp output of the chaos-based fuzzy regression model. For example, if the values for the four variables x1 , x2 , x3 , and x4 are 12.5, 10.3, 5.6, and 1.8, respectively, based on (14), the predicted fuzzy output of y˜ can be calculated as follows:

Fig. 3.

Searching results of q 1 based on COA.

y˜ = (14.9991 − 1.5036 ∗ 5.6 − 0.2890 ∗ 1.8 − 0.3634 ∗ 12.5 + 0.0045 ∗ 12.5 ∗ 10.3, 2.8422 ∗ 10−14

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+ 2.8422 ∗ 10−14 ∗ 5.6 + 0 + 5.6843 ∗ 10−14 ∗ 12.5 + 0.0077 ∗ 12.5 ∗ 10.3) = (2.0956, 0.9914). Thus, the predicted crisp output of y˜ is 2.0956. IV. VALIDATION OF THE PROPOSED APPROACH To evaluate the effectiveness of the proposed approach, five validation tests were conducted in this study. The major purpose of the validation tests is to investigate how effective the proposed approach to modeling customer satisfaction is in comparison with the other three approaches, SR, FR, and fuzzy least-squares regression (FLSR), which all were employed in the previous studies to model customer satisfaction in recent years. For each validation test, two datasets of the ten are selected randomly as the validation datasets. The remaining eight datasets are used as training datasets to develop customer satisfaction models. No repeat of datasets is allowed for the study of prediction performance. In this study, the SR, FR, FLSR, and chaos-based FR approaches to modeling customer satisfaction were all implemented using the MATLAB software programming language. For the SR, the confidence interval was set as 95% which is a common setting for regressing models statistically [29]. To determine the h value for FR, different h values within the range of [0,1] were used to generate FR models. Then, their modeling errors were derived and compared. The h value corresponding to the smallest error was selected. In this research, the h value was set as 0.5, as it leads to the smallest modeling errors. The degree of fitness of the estimated FLSR model h(0 ≤ h