*2.1. Fuzzy Theory*

Fuzzy theory is a subject of science that deals with the study and processing of fuzzy phenomena. It was proposed in 1965 by L.A. Zadeh, an expert in control theory at the University of California, Berkeley, USA. It has been a subject for more than 40 years and has been used for processing imprecise data with vagueness. With rigorous mathematical methods for calculation, the strategic problems under a fuzzy environment can be resolved by this approach [14]. During design assessment, there are many targets, such as aesthetics (overall e ffect, shape, color, decoration, etc.), amenity, safety, machinability, etc. These targets are di fficult to determine by conventional quantitative analysis. Therefore, linguistic variables must be introduced for description in order to acquire a solution. After that, the fuzzy mathematical approach can be used to numerate the fuzzy information for quantitative evaluation.

Hsiao utilized fuzzy theory and hierarchical analysis to carry out strategic decisions [25] and utilized fuzzy theory to evaluate monochromatic color schemes during the car appearance design phase [27,28]. In addition, he also applied fuzzy semantics to the strategic decision of automobile shape designs [21,22]. Moreover, the semantic transformation from adjectives was applied to the design of product forms in order to carry out computer-aided industrial design and assess form images [29,30].

In addition, a number of scholars have also proposed di fferent computer-aided systems to assist designers and engineers in reducing the time for research and development and document processing. In 1997, Moskowitz and Kim proposed a decision support system with optimal product designs [31]. Temponi et al. proposed a deduction framework in 1999 in order to deduce the relationship between customer requirements (CRs) and design requirements (DRs) [32]. However, the knowledge and experience of experts are required to construct the rules when developing this type of system. At the same time, designers also encounter problems whether a system is in good operation or not. According to fuzzy set theory, Kim et al. utilized data, such as the analysis of the competitiveness of companies, to construct a correlation function between CRs and DRs. They proposed a fuzzy multi-objective model. However, it is di fficult to build this correlation function. This is especially true when developing a brand-new product without any data from competitors for analysis [33]. Other scholars have applied fuzzy sets, fuzzy algorithms, or defuzzification techniques to process complex and imprecise quality function deployment problems. However, these methods do not take into account the relationships between engineering design demands. Other scholars have emphasized that the quality function deployment needs to not only determine the performance based on customer satisfaction but also consider organizational criteria such as cost factors and technical di fficulties so as to determine the optimal decision that is economical and satisfactory for customers.

Among various targets of a design, some required concepts are definite while others are vague. It is also required to consider several targets and comprehensively consider various relevant factors and carry out comprehensive assessments. This is the so-called fuzzy comprehensive assessment. The process of implementing fuzzy assessment includes constructing the a ffecting factor set, determining the factor weight set, determining the parameter evaluation set, creating the single-factor assessment matrix, and conducting the fuzzy evaluation. These procedures are described as follows:

#### 2.1.1. Constructing the A ffecting Factor Set

When conducting a fuzzy evaluation, first confirm the factors a ffecting the values of the evaluation parameters. If it is known that the a ffecting factors are *u*1, *u*2, ... , *u*m, the factor set that is composed of these parameters is *U* = {*<sup>u</sup>*1, *u*2, ... , *u*m}. This factor set is a common set.

#### 2.1.2. Determining the Factor Weight Set

The degree of influence and importance of each parameter for the factors are di fferent; that is, the weight of each parameter is di fferent for each factor. The factor weight set is the set that is composed of the degree of influence of each parameter for each factor and can be expressed as *A* = {*a1, a2,* ... *, an*}. If ai indicates the *i*th factor weight, the weights of all of the factors need to satisfy Equation (1). A weight set is a fuzzy subset of the factor set and can be expressed as Equation (2):

$$\sum\_{i=1}^{n} a\_i = 1, \ a\_i \ge 0 \ (\mathbf{i} = 1, 2, 3, \dots, a\_{\text{ll}}),\tag{1}$$

$$\begin{array}{rcl} A &=& \frac{a\_1}{u\_1} + \frac{a\_2}{u\_2} + \frac{a\_3}{u\_3} + \dots + \frac{a\_n}{u\_n} \\ &=& \{a\_1, a\_2, a\_3, \dots, a\_n\} \end{array} \tag{2}$$

The weight of each factor can be determined by the weight coefficient method, analytic hierarchical process, paired comparison method, Likert scale, or any subjective determination according to the practical problem. It does not matter which approach is used, and human factors are involved, with the only difference being in credibility. Here, A is the fuzzy set of each factor weight. For the same factor, if different data are taken, the assessment result is different. The method of the determinant table is to carry out pairwise comparison of the importance of these evaluation targets. Scores are given after further calculation and are entered into the table. The equation for calculating *ai* is as follows:

$$a\_i = k\_i / \sum\_{i=1}^{n} k\_i \tag{3}$$

where *ki* is the total score of each evaluation target and n is the number of evaluation targets.

$$\sum\_{i=1}^{n} k\_i = \frac{n^2 - n}{2} \times 4 = 2(n^2 - n). \tag{4}$$

#### 2.1.3. Determining the Parameter Evaluation Set

The evaluation set is composed of the various evaluation results that an assessor might obtain for the evaluation target. It is represented as *V*, and *V* = {*<sup>v</sup>*1, *v*2, ... , *v*n} and *vi* (*i* = 1, 2, 3, ... , *n*) represent the possible total evaluation results. The purpose of fuzzy evaluation is to comprehensively consider all of the affecting factors and obtain an optimal evaluation result from the evaluation set. The relationship between *vi* and *V* is also a common set relationship. Therefore, the evaluation set is also a common set. For the evaluation in this study, the evaluation set is *V* = {*very satisfied, satisfied, neither satisfied nor dissatisfied, dissatisfied, very dissatisfied*}.

#### 2.1.4. Creating the Single-Factor Assessment Matrix

A single-factor fuzzy evaluation is conducted to judge one factor separately and confirm the degree of membership for the target of evaluation toward evaluation-set elements. The evaluation target is carried out by the *i*th factor Ui and the membership grade of the *j*th element *Vi* in the evaluation set is *rij*. Therefore, the evaluation result of the *i*th factor *Ui* can be determined as follows:

$$R\_i = \frac{r\_{i1}}{V\_1} + \frac{r\_{i2}}{V\_2} + \dots + \frac{r\_{in}}{V\_n} \tag{5}$$

where *Ri* is called the single factor evaluation set, which is the fuzzy subset of the evaluation set. It can be expressed as *Ri* = (ri1, ri2, ..., rin). Similarly, the single factor evaluation set of each factor can be determined as follows:

$$\begin{array}{l} \mathcal{R}\_1 = (r\_{11}, r\_{12}, \dots, r\_{1n})\\ \mathcal{R}\_2 = (r\_{21}, r\_{22}, \dots, r\_{2n})\\ \vdots\\ \mathcal{R}\_{\text{ill}} = (r\_{m1}, r\_{m2}, \dots, r\_{\text{min}}) \end{array} \tag{6}$$

The fuzzy matrix that is composed of the membership grade of each single factor evaluation set is called the single-factor assessment matrix, *R*, as shown in Equation (7). R is a fuzzy matrix and can also be viewed as the fuzzy relational matrix from U to V, or so-called fuzzy mapping.

$$R = \begin{bmatrix} R\_1 \\ R\_2 \\ \vdots \\ R\_i \\ \vdots \\ R\_n \end{bmatrix} = \begin{bmatrix} r\_{11} & r\_{12} & \dots & r\_{1j} & \dots & r\_{1m} \\ r\_{21} & r\_{22} & \dots & r\_{2j} & \dots & r\_{2m} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ r\_{i1} & r\_{i2} & \dots & r\_{ij} & \dots & r\_{im} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ r\_{n1} & r\_{n2} & \dots & r\_{nj} & \dots & r\_{nm} \end{bmatrix} \tag{7}$$

However, in this study, there are numerous factors to be considered and each factor can have di fferent levels. It is di fficult to resolve a problem by using single-factor fuzzy evaluation. It is also di fficult to obtain reasonable assessment results. As a result, this study is based on fuzzy comprehensive evaluation, and then the fuzzy comprehensive evaluation is implemented. This is because there are too many factors to consider when making a selection from complicated schemes. There are levels between factors, therefore we must adopt the multifactor assessment matrix. The procedure is to divide the factor set into several levels according to its characteristics, carry out comprehensive assessment on each level, and then conduct in-depth combined evaluations on the evaluation results.

#### 2.1.5. Conducting the Fuzzy Evaluation

If the fuzzy evaluation matrix of a certain scheme on the evaluation target is the R in Equation (7), the weighted comprehensive fuzzy evaluation and the product of fuzzy matrices to be considered is as follows:

$$B = A \bullet R = \left[ b\_1, b\_2, \dots, b\_{j\_1}, \dots, b\_m \right] \tag{8}$$

where • indicates the fuzzy synthetic operation.

There are several synthetic approaches of fuzzy weight matrix *A* and factor judgment matrix *R*. In this study, four synthetic approaches were implemented for the analysis and comparison of the evaluation results. These four models are described as follows:

Model 1: *M* (<sup>∧</sup>,<sup>∨</sup>) synthetic operation, in which

$$b\_j = \bigvee\_{i=1}^{m} (a\_i \wedge r\_{ij}); j = 1, 2, \dots, n,\tag{9}$$

where ∨ and ∧ indicate taking the maximum or minimum. When taking the minimum, the *rij* of all *rij* > *wij* is not taken into consideration. Therefore, when there are various factors, the values are bound to be large after the weighted coe fficients are normalized. As a result, a lot of the single-factor evaluation information will be lost. When there are fewer factors, *wij* might be larger and therefore the *wij* of all *wij* > *rij* will not be taken into consideration. The influence of the main factors might be lost.

Model 2: *M* (•,<sup>∨</sup>) synthetic operation, in which

$$b\_j = \bigvee\_{i=1}^{m} (a\_i r\_{i\bar{j}}); j = 1, 2, \dots, n. \tag{10}$$

The feature of this model deals with no loss of any useful information. However, the operation of taking the maximum ∨ could possibly lose a lot of useful information. It can still reflect the single-factor evaluation result and the degree of importance of each factor.

Model 3: *M* (<sup>∧</sup>, +◦) synthetic operation, in which

$$b\_{\vec{j}} = \min\{1, \sum\_{i=1}^{m} (a\_i \wedge r\_{\vec{i}\vec{j}})\}; \vec{j} = 1, 2, \dots, n,\tag{11}$$

where +◦ takes the summation with an upper limit of 1. The feature of this model deals with the loss of a lot of valuable information during the operation of taking the minimum. Therefore, it might not achieve the expected evaluation result. When the values of *wi* and *rij* are larger, the resulting *bj* could

be equal to the upper limit of 1. When the values of *wi* and *rij* are smaller, the resulting *bj* could be equal to the summation of all *wi* As a result, it could be more difficult to obtain the expected evaluation result.

Model 4: *M* (•, +) synthetic operation, in which

$$b\_{\vec{j}} = \min\{1, \sum\_{i=1}^{m} a\_i r\_{i\vec{j}}\}; j = 1, 2, \dots, n. \tag{12}$$

This model is also called the weighted-average model and it deals with the situation when *wi* is equipped with normalization, i.e., m i=1 *ai* = 1, m i=1 *airij* << 1.This model can be restructured as *M* (•, +), in which

$$b\_j = \sum\_{i=1}^{m} a\_i r\_{ij}; j = 1, 2, \dots, n; \sum\_{i=1}^{m} a\_i = 1,\tag{13}$$

where *m i*=1 *ai* = 1. This model not only considers the influence of all factors, but also keeps all the information of the single-factor evaluation. During its operation, it does not apply the upper limit to *ai* and *rij* (*i* = 1, 2, ··· , m; *j* = 1, 2, ··· , *<sup>n</sup>*). However, *ai* should be normalized. These are the significant characteristics and advantages of this model. When carrying out the fuzzy comprehensive evaluation and fuzzy optimization design on engineering design parameters, this model is implemented, since typically it can obtain better effects. This model not only considers the influence of all factors, but also keeps all information of the single-factor evaluation. During the operation, there is no upper limit on *wi* and *rij*. It is only required to carry out normalization on *wi*. This is the distinguishing feature and the main advantage of this model.

The goal of Models 1–3 is to obtain individual evaluation results based on certain limitations and by taking the extreme value. Therefore, a lot of useful information could be lost to various degrees during the evaluation process. As a result, these three models are applied to the scenarios that only care about the extreme values of objects in order to highlight certain main factors. Based on this, Model 4 is used for the synthetic operation in this study.

#### 2.1.6. Processing the Evaluation Indices

After the evaluation indices *bj* (*j* = 1, 2, ... , *n*) are determined, the results of the evaluation target can be determined by the methods of maximum degree of membership and weighted average, as follows.

#### (a) Maximum degree of membership

Based on the principle of maximum degree of membership, the evaluation element *vi* corresponding to the maximum evaluation index *bj* is selected. This approach considers only the contribution of the maximum evaluation index; information supplied by other indices is neglected. Moreover, when there are more than one maximum evaluation indices, it will be difficult for maximum degree of membership to determine a concrete result. In this case, weighted average is usually used.
