Rating TAs in fuzzy QFD by objective penalty function and fuzzy TOPSIS based on weighted Hamming distance

Abstract

Quality function deployment (QFD) is an useful tool to solve Multi-criteria decision making, which can translate customer requirements (CRs) into the technical attributes (TAs) of a product and helps maintain a correct focus on true requirements and minimizes misinterpreting customer needs. In applying quality function deployment, rating technical attributes from input variables is a crucial step in fuzzy environments. In this paper, a new approach is developed, which rates technical attributes by objective penalty function and fuzzy technique for order preference by similarity to an ideal solution (TOPSIS) based on weighted Hamming distance under the case of uncertain preference characteristics of decision makers in fuzzy quality function deployment. A pair of nonlinear programming models with constraints and a relevant pair of nonlinear programming models with unconstraints called objective penalty function models are proposed to gain the fuzzy important numbers of technical attributes. Then, this paper compares the fuzzy numbers by fuzzy technique for order preference by similarity to an ideal solution (TOPSIS) method based on weighted Hamming distance in consideration of the uncertain preference characteristics of decision makers. To end with, the developed method is examined with the numerical examples.

Keywords

Quality function deployment objective penalty function fuzzy TOPSIS weighted Hamming distance

1 Introduction

Multi-criteria decision making (MCDM) [1 –3] refers to the decision to choose between a set of solutions that are mutually conflicting and cannot be shared, which is an important management issue to rate these solutions and find the optimal alternative. A MCDM problem is formatted as following. (C1C2 . . . Cn) $B = (\begin{matrix} A 1 \\ A 2 \\ . \\ . \\ . \\ Am \end{matrix}) [\begin{matrix} B 11 & B 12 & . & . & . & B 1 n \\ B 21 & B 22 & . & . & . & B 2 n \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ . & . & . & . & . & . \\ Bm 1 & Bm 2 & . & . & . & Bmn \end{matrix}]$ (1)

and W = (W1, W2, . . . , Wm). Where (A1, A2, . . . , An) are alternatives, (C1, C2, . . . , Cn) are criteria, Bij is the score of Ai on Cj and Wi is the weight of Ai.

Recently, in order to fully comply with the people-oriented principle, more and more MCDM problems are considered based on decision-making requirement. Therefore, quality function deployment (QFD) [4, 5] is developed. QFD, which is a method based on reasoning and deduction, an effective tool to couple customer requirements (CRs) with technical attributes (TAs). Over the past 20 years, the application and field of QFD have been continuously expanded, not only in manufacturing [6], but also in supply chain [7, 8], healthcare service [9], transportation [10, 11] and other fields. In the above MCDM problem, A and C represents customer requirements and technical attributes respectively in QFD; W represents the customer requirement priorities in QFD, which is an important level matrix; and B represents the relationships between customer requirements and technical attributes in QFD, which is a relation matrix. The rating of technical attributes is a key result of QFD since it guides the design team in decision-making, resource allocation, and the subsequent QFD analyses [12]. What’s more, it makes a company achieve higher customer satisfaction and thus more competitive advantages. Therefore, how to derive the importance of TAs from the important level matrix and the relation matrix is a key step for the success of QFD. However, the important level matrix and the relation matrix need the opinions of experts and customers, which is generally vague and uncertain. The traditional QFD method processes the input information by a definite numerical method, which often cannot accurately express the input information, resulting in a lot of information being lost or distorted. Fuzzy theory [13] is a powerful tool to deal with fuzzy information, which can quantify subjective and uncertain information. Each input of the traditional QFD can be taken as a language variable or fuzzy number, and then fuzzy reasoning, fuzzy evaluation, fuzzy decision-making and other methods can be applied to carry out QFD. So, the fuzzy QFD has been developed. The fuzzy QFD is shown by the various input information, which reflects people’s subjective judgments, understandings and evaluations in many cases [16 –20].

In a word, in order to follow the people-oriented principle and gain higher innovation efficiency in new product development, based on fuzzy QFD, this paper aims to develop a new method to rate the importance of technical attributes by the important level matrix and the relation matrix in fuzzy environment, which becomes an inevitable trend of sustainable development in different fields.

A vital process in fuzzy QFD is the merge TAs with a product by design experts. To meet specific evaluation requirements, the combination, integration, mixing of MCDM approaches have become increasingly important for decision makers. Researchers have developed a series of hybrid methods in fuzzy QFD. The combination of fuzzy analytical hierarchy process (AHP) and fuzzy QFD [21 –23] focuses on the importance weights of customer requirements to prioritize and infer the qualitative, vague and imprecise Voice of Customer; The mixture of fuzzy analytic network process (ANP) and fuzzy QFD [24, 25] laies emphasis on addressing the inner-relationship and inter-relationship among QFD components; The integrations of grey relational analysis (GRA) and fuzzy QFD [26, 27], possibility theory and fuzzy arithmetic in fuzzy QFD [28, 29], which pay attention to the effectiveness of QFD in handling the vague, subjective and limited information. Furthermore, the linkage of fuzzy technique for order preference by similarity to an ideal solution (TOPSIS) and fuzzy QFD [30, 31] attaches importance to determine the priority of technical attributes in QFD. Therefore, this paper applies to the method of fuzzy TOPSIS in fuzzy QFD to rate TAs.

However, most of these methods are calculated by the weighted sum of the relationship measure in the important level matrix and the relation matrix. In practice, calculating the importance of each TA falls under the category of fuzzy weighted average in fuzzy situation. Therefore, Vangeas and Labib [32] first used fuzzy weighted average in QFD, and they developed a model to obtain the optimal solution of TAs by the fuzzy weighted average. Afterwards, the fuzzy weighted average is widely combined with many different methods, such as Chen et al. [34] integrate fuzzy weighted average method and fuzzy expected value operator in QFD; Wang and Chin [37] integrate fuzzy normalization and fuzzy weighted average in QFD. In these papers, a pair of nonlinear programming (NLP) models are obtained by fuzzy weighting method, and then, NLP model with constraints is converted to LP model with constraints by introducing parameters. What’s more, the fuzzy importance of TAs are obtained by solving the pair of LP models with constraints. Since it is hard to get the membership function of TAs accurately, the defuzzification method of fuzzy expected value operator is developed and is used widely [33, 34]. But in these traditional methods, there are exists some problems (for example: [34]): (1) The traditional defuzzification method of fuzzy expected value operator need the h-level set to derive the membership functions of TAs, which need to take a large amount of h; What’s more, the defuzzification method loses some information; (2) The preference of decision makers is not taken into account in these methods.

Therefore, in order to rate TAs in fuzzy QFD and solve above questions, a new approach is developed in this paper, which is combining objective penalty function method and the fuzzy TOPSIS method based on the weighted Hamming distance. Firstly, the objective penalty function method, which can convert NLP model with constraints into NLP model with unconstraints. It is the most common approach to NLP models with constraints, which can decrease the complexity of calculations. Secondly, the fuzzy TOPSIS method is developed to make a comparison between fuzzy numbers rather than crisp numbers obtained by the frequently-used defuzzification method in fuzzy QFD, which can make the utmost of the original data to rate TAs, accurately reflect the gap between TAs, and make the data calculation simple and easy. Finally, the parameter β is defined to represent the degree of decision makers’ preference in the weighted Hamming distance, which can fully takes into account the characteristics of decision makers preferences for uncertainty.

As above discussed, the paper is structured in Fig. 1 and as follows: In Section 2, we show definitions of fuzzy set theory, objective penalty function and weighted Hamming distance. In Section 3, we proposed the method of objective penalty function and fuzzy TOPSIS based on weighted Hamming distance to rate TAs. In Section 4, two examples are given to illustrate the feasibility of the proposed method. Finally, conclusions are made in Section 5.

Fig. 1

Flowchat of the proposed method.

2 Preliminaries

In this section, some basic knowledge and necessary concepts are introduced which related to fuzzy set theory (Kong [39]), objective penalty function (Meng [38]) and weighted Hamming distance (Kong [39]).

2.1 Fuzzy set theory

Definition 1. (Fuzzy set [39]) Let U denote a universal set. Then a fuzzy subset A of U is defined by its membership function $A (x) : U \to [0, 1]$ which assigns to each element x ∈ U a real number in the interval [0, 1], where the value of $A (x)$ at x represents the grade of membership of x in A. Thus, the nearer the value of $A (x)$ is unity, the higher the grade of membership of x in A.

Definition 2. (α-level of fuzzy set [39]) Let B be a fuzzy set with membership function $B (x)$ . Then the set $B_{α} = {x \in R | B (x) \geq α}$ is called the α-level of B. Especially, the set ${x \in R | B (x) > 0}$ is called the support of B.

Definition 3. (Fuzzy number [39]) Fuzzy numbers are special cases of fuzzy sets, described by given intervals of crisp numbers. Let N is a trapezoidal fuzzy number, denoted by N = (a, b, c, d), whose membership functions is defined as: $N (x) = {\begin{matrix} \frac{x - a}{b - a}, & ifa \leq x \leq b, \\ 1, & if b \leq x \leq c, \\ \frac{c - x}{c - b}, & ifc \leq x \leq d, \\ 0, & ifelse . \end{matrix}$ Interval fuzzy numbers and triangular fuzzy numbers can be regarded as special cases of trapezoidal fuzzy numbers. Such as, N is a triangular fuzzy number when b = c, N is an interval fuzzy number when a = b and c = d. In particular, N is a crisp number when a = b = c = d, crisp numbers can also be considered as a special case of fuzzy numbers.

Definition 4. (Extension principle [39]) Let C and D be two fuzzy sets with membership functions $C (x)$ and $D (x)$ , and let $F : R^{2} \to R$ be a function. Then $F (C, D)$ is also a fuzzy set whose membership function is $E (x) = \sup {C (x) \land D (x) | x = F (y, z)} .$

Example: The membership function of $C + D$ is $E (x) = \sup {C (x) \land D (x) | x = y + z}$ .

2.2 Objective penalty function

The objective penalty function method is to convert a constraint optimization problem into an unconstrained optimization problem in order to easy to solve the problem. We consider the following nonlinear constrained optimization problem:

$(P^{'}) \begin{matrix} min f_{0} (x) \\ s . t . f_{i} (x) \leq 0, i = 1, 2, \dots, h, \\ x \in R^{n}, \end{matrix}$ where f_i : Rⁿ → R, i ∈ I₀ = {1, 2, . . . , h}. The feasible set of (P^′) is denoted by X = {x ∈ Rⁿ|f_i (x) ≤0, i ∈ I}, where I = {1, 2, . . . , h}.

Meng et al. [38], studied the following objective penalty function $F^{'} (x, M) = (f (x) - M)^{2} + Σ_{i \in I} \max {0, g i (x)}^{p}$ where p ≥ 1 and M ∈ R is an objective penalty parameter. The corresponding unconstrained optimization problem is as follows: $(P (M)^{'}) \min_{x \in R^{n}} F^{'} (x, M)$

Theorem 1. [38] Let x^* be an optimal solution to (P^′). For some M, let $x_{M}^{*}$ be an optimal solution to (P(M)^′) and a feasible solution to (P^′) with $F^{'} (x_{*}^{M}, M) > 0$ . If M < f₀ (x^*), $x_{M}^{*}$ is an optimal solution to (P^′).

Theorem 2. [38] Let the feasible set X be connected and compact, f₀ a continuous function in Rⁿ, $M_{*} = min_{x \in X} f_{0} (x)$ , and $M^{*} = max_{x \in X} f_{0} (x)$ . For some M, let $x_{M}^{*}$ be an optimal solution to (P^′). Then

(i) M_* ≤ M ≤ M^*, if $F^{'} (x_{*}^{M}, M) = 0$ .

(ii) M ≤ M_*, if $F^{'} (x_{*}^{M}, M) > 0$ and M ≤ M^*. Furthermore, $x_{M}^{*}$ is an optimal solution to (P^′) if $x_{M}^{*}$ is a feasible solution to (P^′).

Therefore, the objective penalty function is exactness. That is to say, in some conditions, the optimal solution of the original problem is the optimal solution of the objective penalty function problem, and the optimal solution of the objective penalty function problem is the optimal solution of the original problem.

2.3 Weighted Hamming distance

Definition 5. [39] Let M = (a, b, c) be a triangular fuzzy number. Then, the left and right fuzzy set are denoted by M_L and M_R respectively. And their membership functions are follows: $M_{L} (x) = sup_{x = y + z, z \leq 0} M (y),$ $M_{R} (x) = sup_{x = y + z, z \geq 0} M (y) .$ The figures as shown in Fig. 2 and Fig. 3.

Fig. 2

The left set.

Fig. 3

The right set.

Therefore, $M = M_{L} \cap M_{R} .$

Definition 6. (Hamming distance [39]) Let N₁, N₂ are two fuzzy numbers. The Hamming distance of N₁, N₂ is defined as: $d_{H} (N_{1}, N_{2}) = \int [N_{1} (x) - N_{2} (x)] dx .$

Definition 7. (Weighted Hamming distance [39]) Let N₁, N₂ are two fuzzy numbers. The weighted Hamming distance of N₁, N₂ is defined as:

$d (N_{1}, N_{2}) = β d_{H} (N_{1 L}, N_{2 L}) + (1 - β) d_{H} (N_{1 R}, N_{2 R})$ $= β \int [N_{1 L} (x) - N_{2 L} (x)] dx$ $+ (1 - β) \int [N_{1 R} (x) - N_{2 R} (x)] dx .$

where β represents the uncertain preference factor of decision makers. Such as, β = 0.5 denotes that the decision maker is neutral in respect of uncertain; β > 0.5 indicates that the decision maker is nauseous about uncertain; and β < 0.5 indicates that the decision maker is preferential concerning uncertain.

3 The method of rating technical attributes

3.1 Date acquisition and review

QFD is a method to translate CRs into product TAs. In this process, the weights of CRs is as important as the importance of TAs.

In fuzzy QFD, the relative weights of customer requirements, the relationships expressed between the CRs and TAs, and even the correlations among TAs could be represented in fuzzy numbers (Chen and Fung [34]).

Assuming that during a product design, there are m CRs denoted by CR_i, i = 1, 2, . . . , m, and n TAs denoted by TA_j, j = 1, 2, . . . , n . Let p be the number of customers investigated in the target market, what’s more, the pth individual preference on the ith CR is denoted by $W_{i}^{s}, s = 1, 2, . . ., p$ . A pre-defined set of fuzzy weights ${W_{1}^{*}, . . ., W_{r}^{*}, . . ., W_{l}^{*}}, r = 1, 2, . . ., l,$ which is used to represent the individual weight $W_{i}^{s}$ , such as, $W_{i}^{s} \in {W_{1}^{*}, . . ., W_{r}^{*}, . . ., W_{l}^{*}}, s = 1, 2, . . ., p, r = 1, 2, . . ., l,$ where r is the number of linguistic terms of relative weights for the fuzzy set, and $W_{r}^{*}$ is defined as a triangular fuzzy number.

By integrating the weights of p customers, the relative weights of each W_i, the final fuzzy weight of the ith CR, i = 1, 2, . . . , m, can be given as: $W_{i} = \frac{1}{p} \sum_{s = 1}^{p} W_{i}^{s}, i = 1, 2, . . ., m .$ (3.1) where $W_{i} = {w_{i}, W_{i} (w_{i}) | w_{i} \in {\tilde{W}}_{i}}, i = 1, 2, . . ., m,$ W_i (w_i) is the membership functions of the fuzzy numbers W_i, and ${\tilde{W}}_{i}$ is the crisp universal set.

Analogous to the fuzzy weights of CRs, let q be the number of expertise, and their individual assessment on the degree between the ith CR and jth TA is denoted by $U_{ij}^{b}, b = 1, 2, . . ., q .$ And a pre-defined fuzzy weight set ${U_{1}^{*}, . . ., U_{t}^{*}, . . ., U_{v}^{*}}$ , is adopted to express the interaction measure between the CRs and the TAs, i.e., $U_{ij}^{b} \in {U_{1}^{*}, . . ., U_{t}^{*}, . . ., U_{v}^{*}}, t = 1, 2, . . ., v, b = 1, 2, . . ., q, .$ By synthesizing the weights of q experts, the final fuzzy relationship measure between the ith CR and the jth TA as follows: $U_{ij} = \frac{1}{q} \sum_{b = 1}^{q} U_{ij}^{b}, i = 1, 2, . . ., m, j = 1, 2, . . ., n .$ (3.2) where $U_{ij} = {u_{ij}, U_{ij} (u_{ij}) | u_{ij} \in {\tilde{U}}_{ij}}, i = 1, 2, . . ., m, j = 1, 2, . . ., n,$ $U_{ij} (u_{ij})$ is the membership functions of the fuzzy numbers U_ij, and ${\tilde{U}}_{ij}$ is the crisp universal set.

Therefore, let Y_j denote the fuzzy weight of TA_j, can be obtained as: $Y_{j} = \frac{\sum_{i = 1}^{m} W_{i} U_{ij}}{\sum_{i = 1}^{m} W_{i}}, j = 1, 2, . . ., n .$ (3.3)

Obviously, the fuzzy weighted average Y_j is a fuzzy number.

3.2 Determining the fuzzy importance of TAs

Then, the fuzzy weighted average Y_j is defined as the following membership function $Y_{j} (y_{j})$ , with respected to the fuzzy extension principle (Zadeh [35]): $Y_{j} (y_{j}) = sup_{w, u} min {W_{i} (w_{i}), U_{ij} (u_{ij}),$ $\forall i, j | y_{j} = \sum_{i = 1}^{m} w_{i} u_{ij} / \sum_{i = 1}^{m} w_{i}} .$ (3.4) Clearly, the equation above can be converted into the equivalent NLP model as follows: $(P) \begin{matrix} Y_{j} (y_{j}) = max z \\ s . t . z \leq W_{i} (w_{i}), i = 1, 2, . . ., m, \\ z \leq U_{ij} (u_{ij}), i = 1, 2, . . ., m, \\ y_{j} = \sum_{i = 1}^{m} w_{i} u_{ij} / \sum_{i = 1}^{m} w_{i}, i = 1, 2, . . ., m, \\ w_{i} \in {\tilde{W}}_{i}, u_{ij} \in {\tilde{U}}_{ij}, i = 1, 2, . . ., m . \end{matrix}$

Generally, the functions $W_{i} (w_{i})$ and $U_{ij} (u_{ij})$ are nondifferentiable, so it is difficult to get the optimal solution in this model. In order to solve this problem, by handling the h-cuts of Y_j to replace the above model (Kao and Liu [36]).

Let (W_i) _h and (U_ij) _h indicate the h-cuts sets of W_i and U_ij respectively, (W_i) _h and (U_ij) _h can be shown as:

$(W_{i})_{h} = {w_{i} \in {\tilde{W}}_{i}, \forall i | W_{i} (w_{i}) \geq h},$ (3.5a) $(U_{ij})_{h} = {u_{ij} \in {\tilde{U}}_{ij}, \forall i, j | U_{ij} (u_{ij}) \geq h} .$ (3.5b) They can be shown in the following alternative form: $(W_{i})_{h} = [(W_{i})_{h}^{L}, (W_{i})_{h}^{R}] = [min_{w_{i}} {w_{i} \in {\tilde{W}}_{i}, \forall i | W_{i} (w_{i}) \geq h}, max_{w_{i}} {w_{i} \in {\tilde{W}}_{i}, \forall i | W_{i} (w_{i}) \geq h}] .$ (3.6a) $(U_{ij})_{h} = [(U_{ij})_{h}^{L}, (U_{ij})_{h}^{R}] = [min_{u_{ij}} {u_{ij} \in {\tilde{U}}_{ij}, \forall i, j | U_{ij} (u_{ij}) \geq h}, max_{u_{ij}} {u_{ij} \in {\tilde{U}}_{ij}, \forall i, j | U_{ij} (u_{ij}) \geq h}] .$ (3.6b)

Let W_i and U_ij are triangular fuzzy numbers, When h = 1, $(W_{i})_{h = 1}^{L} = (W_{i})_{h = 1}^{R}$ and $(U_{ij})_{h = 1}^{L} = (U_{ij})_{h = 1}^{R}$ , which denoted by (W_i) _h=1 and (U_ij) _h=1, respectively. Therefore, W_i and U_ij can be expressed as follows:

W_i = (a_i, b_i, c_i) $= ((W_{i})_{h = 0}^{L}, (W_{i})_{h = 1}, (W_{i})_{h = 0}^{R}),$ (3.7a)U_ij = (a_ij, b_ij, c_ij) $= ((U_{ij})_{h = 0}^{L}, (U_{ij})_{h = 1}, (U_{ij})_{h = 0}^{R}) .$ (3.7b) Therefore, the accordingly membership function as (3.8a) and (3.8b):

Next, according to the model of (P), $(Y_{j})_{h} = [(Y_{j})_{h}^{L}, (Y_{j})_{h}^{R}]$ can be obtained by the following models: $(P_{1}) \begin{matrix} (Y_{j})_{h}^{L} = min \sum_{i = 1}^{m} w_{i} u_{ij} / \sum_{i = 1}^{m} w_{i} \\ s . t . (W_{i})_{h}^{L} \leq w_{i} \leq (W_{i})_{h}^{R}, i = 1, 2, . . ., m, \\ (U_{ij})_{h}^{L} \leq u_{ij} \leq (U_{ij})_{h}^{R}, i = 1, 2, . . ., m, \\ j = 1, 2, . . ., n . \end{matrix}$

$W_{i} (w_{i}) = {\begin{matrix} \frac{w_{i} - (W_{i})_{h = 0}^{L}}{(W_{i})_{h = 1} - (W_{i})_{h = 0}^{L}}, & if (W_{i})_{h = 0}^{L} \leq w_{i} < (W_{i})_{h = 1}, \\ 1, & if w_{i} = (W_{i})_{h = 1}, \\ \frac{(W_{i})_{h = 0}^{R} - w_{i}}{(W_{i})_{h = 0}^{R} - (W_{i})_{h = 1}}, & if (W_{i})_{h = 1} < w_{i} \leq (W_{i})_{h = 0}^{R}, \\ 0, & if else . \end{matrix}$ (3.8a)

$U_{ij} (u_{ij}) = {\begin{matrix} \frac{u_{ij} - (U_{ij})_{h = 0}^{L}}{(U_{ij})_{h = 1} - (U_{ij})_{h = 0}^{L}}, & if (U_{ij})_{h = 0}^{L} \leq u_{ij} < (U_{ij})_{h = 1}, \\ 1, & if u_{ij} = (U_{ij})_{h = 1}, \\ \frac{(U_{ij})_{h = 0}^{R} - u_{ij}}{(U_{ij})_{h = 0}^{R} - (U_{ij})_{h = 1}}, & if (U_{ij})_{h = 1} < u_{ij} \leq (U_{ij})_{h = 0}^{R}, \\ 0, & ifelse . \end{matrix}$ (3.8b) $(P_{2}) \begin{matrix} (Y_{j})_{h}^{R} = max \sum_{i = 1}^{m} w_{i} u_{ij} / \sum_{i = 1}^{m} w_{i} \\ s . t . (W_{i})_{h}^{L} \leq w_{i} \leq (W_{i})_{h}^{R}, i = 1, 2, . . ., m, \\ (U_{ij})_{h}^{L} \leq u_{ij} \leq (U_{ij})_{h}^{R}, i = 1, 2, . . ., m, \\ j = 1, 2, . . ., n . \end{matrix}$ As can be seen from above parts, the models of (P₁) and (P₂) should be solved to obtain the importance of TAs. Evidently, they are the nonlinear programming (NLP) with constraints, in order to get the optimal solution, Chen and Fung [34] transformed them into two linear programs. The more commonly method for solving NLP is the penalty function (e.g. Zangwill [40], Zaslavski [41]). In these traditional exact penalty function methods, the constraint penalty parameter need to be continuously increased, which makes it not efficient in practical computing of Matlab. In order to deal with this situation and simplify the calculation process, this paper uses the penalty function with objective penalty parameter called objective penalty function (Meng and Hu [38]) to get the optimal solution of the models of (P₁) and (P₂). The specific models are as follows:

For the model of (P₁), which is converted into: $(P_{1}^{'}) \begin{matrix} (Y_{j})_{h}^{L} = min \sum_{i = 1}^{m} w_{i} (U_{ij})_{h}^{L} / \sum_{i = 1}^{m} w_{i} \\ s . t . (W_{i})_{h}^{L} - w_{i} \leq 0, i = 1, 2, . . ., m . \\ w_{i} - (W_{i})_{h}^{R} \leq 0, i = 1, 2, . . ., m . \end{matrix}$ the corresponding objective penalty function model as follows: $F_{1} (x, M) = [\sum_{i = 1}^{m} w_{i} (U_{ij})_{h}^{L} / \sum_{i = 1}^{m} w_{i} - M]^{2} + (\max {0, (W_{i})_{h}^{L} - w_{i}}^{2} + \max {0, w_{i} - (W_{i})_{h}^{R}}^{2})$ (3.9a) and then, corresponding unconstrained optimization problem as follows: $(P_{1} (M)) \begin{matrix} min_{x \in R^{n}} F_{1} (x, M) \end{matrix}$

For the model of (P₂), which is converted into: $(P_{2}^{'}) \begin{matrix} - (Y_{j})_{h}^{R} = min - \sum_{i = 1}^{m} w_{i} (U_{ij})_{h}^{R} / \sum_{i = 1}^{m} w_{i} \\ s . t . (W_{i})_{h}^{L} - w_{i} \leq 0, i = 1, 2, . . ., m . \\ w_{i} - (W_{i})_{h}^{R} \leq 0, i = 1, 2, . . ., m . \end{matrix}$ the corresponding objective penalty function model as follows: $F_{2} (x, M) = [- \sum_{i = 1}^{m} w_{i} (U_{ij})_{h}^{R} / \sum_{i = 1}^{m} w_{i} - M]^{2} + (\max {0, (W_{i})_{h}^{L} - w_{i}}^{2} + \max {0, w_{i} - (W_{i})_{h}^{U}}^{2})$ (3.9b) and then, the corresponding unconstrained optimization problem as follows: $(P_{2} (M)) \begin{matrix} min_{x \in R^{n}} F_{2} (x, M) \end{matrix}$

But it is hard to construct the membership function $Y_{j} (y_{j})$ accurately. In the existing research, most researchers propose a defuzzification method of fuzzy expected value operator to get a crisp number of Y_j in order to rate TAs, (e.g. Chen and Fung [34]. However, this method loses a lot of information of TAs and does not care about the uncertain preference of decision makers. In addition, some discrete points can only be obtained by taking different possibility level h. So this paper develops an approach which Y_j is defined as an approximate fuzzy number, and then rate TAs by comparing fuzzy numbers. Let h = 0 and h = 1, $(Y_{j})_{h = 0}^{L}, (Y_{j})_{h = 1}$ and $(Y_{j})_{h = 0}^{R}$ can be obtained by the models given in (P₁ (M)) and (P₂ (M)). Then, if W_j and U_ij are triangular fuzzy numbers, Y_j be defined by triangular fuzzy numbers. Therefore, Y_j can be seen as $Y_{j} = ((Y_{j})_{h = 0}^{L}, (Y_{j})_{h = 1}, (Y_{j})_{h = 0}^{R})$ , and the membership function is obtained as (3.10):

3.3 Rating the fuzzy importance of TAs

Now, the problem is how to rate the fuzzy importance of TAs by the obtained fuzzy number Y_j of Section 3.2.

$Y_{j} (y_{j}) = {\begin{matrix} \frac{y_{j} - (Y_{j})_{h = 0}^{L}}{(Y_{j})_{h = 1} - (Y_{j})_{h = 0}^{L}}, & if (Y_{j})_{h = 0}^{L} \leq y_{j} < (Y_{j})_{h = 1}, \\ 1, & if y_{j} = (Y_{j})_{h = 1}, \\ \frac{(Y_{j})_{h = 0}^{R} - y_{j}}{(Y_{j})_{h = 0}^{R} - (Y_{j})_{h = 1}}, & if (Y_{j})_{h = 1} < y_{j} \leq (Y_{j})_{h = 0}^{R}, \\ 0, & ifelse . \end{matrix}$ (3.10)

There are many methods which can be used to rate fuzzy numbers mentioned in literature (Li and Lee [42]), Kim and Park [43], Li [45]. But the first method is only applicable to the situation where the decision maker is an uncertain hater (Lee and Li [42]); The second method will be affected by the maximum and minimum values (Kim and Park [43]); and the third method does not take into account the uncertain preferences of decision makers.

In this paper, the fuzzy TOPSIS method (Chen [44]) is used. In order to better consider the preferences of decision makers, the fuzzy TOPSIS method based on weighted Hamming distance is developed as follows:

Step 1. Determine ideal solution Y⁺ and negative ideal solution Y^-

Determine the ideal solution Y⁺ and negative ideal solution Y^- by the triangular fuzzy numbers $Y_{j} = (a_{j}^{'}, b_{j}^{'}, c_{j}^{'}), j = 1, 2, . . ., n .$ $Y^{+} = 1 = (1, 1, 1) = (a^{+}, b^{+}, c^{+}),$ (3.11a) $Y^{-} = 0 = (0, 0, 0) = (a^{-}, b^{-}, c^{-}) .$ (3.11b)

Step 2. Calculate the weighted Hamming distance from Y_j to the ideal solution $D_{j}^{+}$ and from Y_j to the negative ideal solution $D_{j}^{-}$ $D_{j}^{+} = d (Y_{j}, Y^{+}) = β d_{H} (Y_{jL}, Y_{L}^{+}) + (1 - β) d_{H} (Y_{jR}, Y_{R}^{+}) = β \int [Y_{jL} (x) - Y_{L}^{+} (x)] dx + (1 - β) \int [Y_{jR} (x) - Y_{R}^{+} (x)] dx,$ (3.12a) $D_{j}^{-} = d (Y_{j}, Y^{-}) = β d_{H} (Y_{jL}, Y_{L}^{-}) + (1 - β) d_{H} (Y_{jR}, Y_{R}^{-}) = β \int [Y_{jL} (x) - Y_{L}^{-} (x)] dx + (1 - β) \int [Y_{jR} (x) - Y_{R}^{-} (x)] dx .$ (3.12b) Where β denotes the uncertain preference factor of decision makers.

Step 3. Calculate the relative closeness C_j of the fuzzy number Y_j to the ideal solution $C_{j} = \frac{D_{j}^{+}}{D_{j}^{+} + D_{j}^{-}}$ (3.13) where 0 ≤ C_j ≤ 1, j = 1, 2, . . . , n . C_j = 1 means Y_j is an ideal solution, C_j = 0 means Y_j is a negative ideal solution. The closer Y_j gets to the ideal solution, the closer C_j gets to 1. On the contrary, the closer Y_j gets to the negative ideal solution, the closer C_j gets to 0. That is to say, bigger the relative closeness C_j is the better.

TOPSIS method makes the best use of the information of the original data, not only can it rank the technical attributes, but also the results can accurately reflect the gap between the technical attributes. Most importantly, TOPSIS method makes the data calculation simple and easy. What’s more, TOPSIS method based on weighted Hamming distance fully takes into account the characteristics of decision makers’ preferences for uncertainty. It is not only suitable for the comparison between fuzzy numbers and fuzzy numbers, but also suitable for the comparison between fuzzy numbers and crisp numbers. Therefore, the ideal solution can be determined as the crisp number 1 and the negative ideal solution can be set as the crisp number 0.

4 Numerical examples

In this section, two fuzzy QFD examples are examined to illustrate the performance of the method developed in this paper.

Example 1. Chen and Fung [34] proposed the traditional method of fuzzy expected value operator to rate the TAs. Consider their example, the feasibility of the proposed method is verified.

In that example, eight major customer requirements are identified to represent the biggest concerns of the customers, which are defined by CR_i (i = 1, 2, . . . , 8). And based on the design team’s experience and expert knowledge, 10 technical attributes which are defined by TA_j (j = 1, 2, . . . , 10) are identified responding to the eight major CRs. The CRs and the TAs are shown in Table 1.

Table 1
CRs and TAs considered for the example study

CRs Customer requirements TAs Technical attributes

CR ₁ High production volume TA ₁ Automatic gauging

CR ₂ Short setup time TA ₂ The tool change system

CR ₃ Load carrying capacity TA ₃ Tool monitoring system

CR ₄ User friendliness TA ₄ Coordinate measuring machine

CR ₅ Associated functions TA ₅ Automated guided vehicle

CR ₆ Modularity TA ₆ Conveyor

CR ₇ Wide tool variety TA ₇ Programmable logic controller

CR ₈ Wide product variety TA ₈ Storage and retrieval system

TA ₉ Modular fixturing

TA ₁₀ Robots

CRs	Customer requirements	TAs	Technical attributes
CR ₁	High production volume	TA ₁	Automatic gauging
CR ₂	Short setup time	TA ₂	The tool change system
CR ₃	Load carrying capacity	TA ₃	Tool monitoring system
CR ₄	User friendliness	TA ₄	Coordinate measuring machine
CR ₅	Associated functions	TA ₅	Automated guided vehicle
CR ₆	Modularity	TA ₆	Conveyor
CR ₇	Wide tool variety	TA ₇	Programmable logic controller
CR ₈	Wide product variety	TA ₈	Storage and retrieval system
		TA ₉	Modular fixturing
		TA ₁₀	Robots

Part 1. Date acquisition and review. The important level matrix and the relation matrix will be shown. The an important level matrix, which represents the importance of CRs, is expressed as the linguistic data at seven levels, which are shown in Table 2. And the membership functions of these triangular fuzzy numbers are shown in Fig. 4. What’s more, assume that 10 customers are surveyed in the target market, use those seven fuzzy numbers to express their individual assessments on each CR. After integrating weights of 10 customers on the CRs, the important level matrix can be shown in Table 3. Furthermore, the relation matrix, which represents the relationships between the CRs and the TAs, can be expressed by triangular fuzzy numbers. They are shown in Table 4, and their membership functions are shown in Fig. 5. Assume that 7 experts are involved in evaluating the relationships between the CRs and the TAs, after synthesizing the evaluations of 7 experts, the final relation matrix can be obtained as shown in Table 5.

Table 2

The pre-defined triangular fuzzy weights $W_{r}^{*}$

$W_{1}^{*}$	very unimportant	(0, 0, 0.2)
$W_{2}^{*}$	quite unimportant	(0, 0.2, 0.4)
$W_{3}^{*}$	unimportant	(0.2, 0.35, 0.5)
$W_{4}^{*}$	some important	(0.3, 0.5, 0.7)
$W_{5}^{*}$	moderately important	(0.5, 0.65, 0.8)
$W_{6}^{*}$	important	(0.6, 0.8, 1)
$W_{7}^{*}$	very important	(0.8, 1, 1)

Fig. 4

Membership functions for the weight to CAs.

Fig. 5

Membership functions for the relationship strength.

Table 3

Weight W_i of 10 customers on CR_i
W ₁	(0.71, 0.91, 0.98)
W ₂	(0.40, 0.58, 0.75)
W ₃	(0.61, 0.80, 0.94)
W ₄	(0.57, 0.76, 0.88)
W ₅	(0.30, 0.47, 0.64)
W ₆	(0.47, 0.65, 0.83)
W ₇	(0.52, 0.70, 0.86)
W ₈	(0.63, 0.82, 0.93)

Table 4

The pre-defined triangular fuzzy weights $U_{t}^{*}$
$U_{1}^{*}$	none	(0, 0, 0.3)
$U_{2}^{*}$	weak	(0, 0.25, 0.5)
$U_{3}^{*}$	moderate	(0.3, 0.5, 0.7)
$U_{4}^{*}$	strong	(0.5, 0.75, 1)
$U_{5}^{*}$	very strong	(0.7, 1, 1)

Table 5

Weight W_i of the 10 experts on CR_i

U _ij	CR ₁	CR ₂	CR ₃	CR ₄
TA ₁	(0.31,0.54,0.76)	(0.09,0.25,0.50)	(0.56,0.82,0.96)	(0.39,0.64,0.90)
TA ₂	(0.16,0.36,0.60)	(0.47,0.71,0.91)	(0.27,0.50,0.73)	(0.27,0.50,0.73)
TA ₃	(0.27,0.50,0.73)	(0.56,0.82,0.96)	(0.61,0.89,1.00)	(0.23,0.46,0.70)
TA ₄	(0.37,0.61,0.80)	(0.13,0.32,0.56)	(0.13,0.36,0.63)	(0.24,0.46,0.69)
TA ₅	(0.13,0.29,0.53)	(0.13,0.29,0.53)	(0.17,0.36,0.59)	(0.13,0.32,0.56)
TA ₆	(0.17,0.39,0.61)	(0.24,0.46,0.69)	(0.13,0.36,0.59)	(0.17,0.36,0.59)
TA ₇	(0.36,0.57,0.79)	(0.34,0.57,0.80)	(0.53,0.79,0.91)	(0.24,0.46,0.70)
TA ₈	(0.17,0.39,0.61)	(0.27,0.50,0.73)	(0.17,0.39,0.61)	(0.20,0.43,0.66)
TA ₉	(0.13,0.29,0.53)	(0.47,0.71,0.87)	(0.09,0.25,0.50)	(0.23,0.46,0.70)
TA ₁₀	(0.17,0.32,0.56)	(0.04,0.18,0.44)	(0.09,0.29,0.53)	(0.09,0.25,0.50)
U _ij	CR ₅	CR ₆	CR ₇	CR ₈
TA ₁	(0.56,0.82,0.96)	(0.13,0.36,0.59)	(0.09,0.32,0.56)	(0.21,0.43,0.64)
TA ₂	(0.59,0.86,1.00)	(0.13,0.32,0.56)	(0.33,0.54,0.74)	(0.09,0.29,0.53)
TA ₃	(0.39,0.61,0.83)	(0.28,0.50,0.71)	(0.36,0.57,0.79)	(0.39,0.61,0.79)
TA ₄	(0.59,0.86,0.96)	(0.39,0.61,0.83)	(0.04,0.25,0.50)	(0.31,0.54,0.76)
TA ₅	(0.09,0.25,0.50)	(0.09,0.32,0.56)	(0.29,0.50,0.71)	(0.04,0.25,0.50)
TA ₆	(0.13,0.36,0.59)	(0.17,0.39,0.61)	(0.31,0.61,0.83)	(0.17,0.39,0.61)
TA ₇	(0.27,0.50,0.73)	(0.09,0.29,0.53)	(0.56,0.82,1.00)	(0.50,0.75,0.96)
TA ₈	(0.31,0.54,0.76)	(0.09,0.32,0.56)	(0.36,0.57,0.79)	(0.36,0.57,0.79)
TA ₉	(0.34,0.57,0.76)	(0.13,0.36,0.59)	(0.09,0.32,0.56)	(0.09,0.32,0.56)
TA ₁₀	(0.41,0.64,0.83)	(0.13,0.32,0.56)	(0.04,0.29,0.53)	(0.61,0.39,0.63)

Part 2. Determining the fuzzy importance of TAs. The NLP models of P₁ and P₂, which obtained by fuzzy weighted average, are converted into the objective penalty function models given in P₁ (M) and P₂ (M) to solve, and the results of $(Y_{j})_{h = 0}^{L}$ , (Y_j) _h=1 and $(Y_{j})_{h = 0}^{R}$ are shown in Table 6. What’s more, the values of $(Y_{j})_{h = 0}^{L}$ , (Y_j) _h=1 and $(Y_{j})_{h = 0}^{R}$ are also given in Table 6 by the traditional method in [34]. Due to the value of $(Y_{j})_{h = 0}^{L}$ is the lower bound and the value of $(Y_{1})_{h = 0}^{U}$ is the upper bound, the smaller $(Y_{j})_{h = 0}^{L}$ or the bigger $(Y_{j})_{h = 0}^{U}$ , the more exact the result is. Where $(Y_{j})_{h = 1}^{L}$ = $(Y_{j})_{h = 1}^{U}$ =(Y_j) _h=1. In Table 6, bold numbers represent more exact values when compared with the two methods. Therefore, it is clear shown in Table 6 that the results of two methods are very close (the error of each value is not over 0.15 between two methods), and even the new method is slightly better than the traditional one (the new method has more bold numbers than the traditional one). Finally, the fuzzy importance of TAs are obtained in Table 7. Where the fuzzy importance of TA_j is ( $(Y_{j})_{h = 0}^{L}, (Y_{j})_{h = 1}, (Y_{j})_{h = 0}^{R}$ ).

Table 6

The exactness of objective penalty function method and traditional method

TA _j	The new method			The traditional method in [34]
	$(Y_{j})_{h = 0}^{L}$	(Y_j) _h=1	$(Y_{j})_{h = 0}^{U}$	$(Y_{j})_{h = 0}^{L}$	(Y_j) _h=1	$(Y_{j})_{h = 0}^{U}$
TA ₁	0.2518	0.5218	0.7746	0.2507	0.5225	0.7730
TA ₂	0.2342	0.4844	0.7335	0.2456	0.4868	0.7390
TA ₃	0.3583	0.6165	0.8320	0.3584	0.6163	0.8307
TA ₄	0.2312	0.4910	0.7401	0.2316	0.4902	0.7382
TA ₅	0.1233	0.3246	0.5740	0.1218	0.3229	0.5729
TA ₆	0.1761	0.4139	0.6548	0.1832	0.4138	0.6520
TA ₇	0.3409	0.6029	0.8410	0.3259	0.5909	0.8294
TA ₈	0.2175	0.4592	0.7026	0.2229	0.4647	0.7042
TA ₉	0.1553	0.3901	0.6444	0.1529	0.3895	0.6431
TA ₁₀	0.1144	0.3252	0.5851	0.1138	0.3248	0.5839

Table 7

The exactness of objective penalty function method and traditional method

TA _j	The new method											The traditional method
fuzzy importance		C _j										E (Y_j)	rank
		β = 0	rank	β = 0.25	rank	β = 0.5	rank	β = 0.75	rank	β = 1	rank
TA ₁	(0.2518,0.5228,0.7746)	0.3513	3	0.4166	3	0.4820	3	0.5473	3	0.6127	3	0.5172	3
TA ₂	(0.2342,0.4828,0.7335)	0.3918	5	0.4543	5	0.5167	5	0.5791	5	0.6415	5	0.4896	4
TA ₃	(0.3599,0.6157,0.8320)	0.2762	1	0.3352	1	0.3942	1	0.4532	1	0.5122	1	0.6057	1
TA ₄	(0.2312,0.4915,0.7401)	0.3842	4	0.4478	4	0.5114	4	0.5750	4	0.6386	4	0.4880	5
TA ₅	(0.1233,0.3240,0.5740)	0.5510	10	0.6073	10	0.6637	10	0.7200	9	0.7763	9	0.3352	10
TA ₆	(0.1762,0.4135,0.6548)	0.4658	7	0.5257	7	0.5855	7	0.6453	7	0.7051	7	0.4160	7
TA ₇	(0.3409,0.6052,0.8410)	0.2769	2	0.3394	2	0.4020	2	0.4645	2	0.5270	2	0.5845	2
TA ₈	(0.2175,0.4590,0.7026)	0.4192	6	0.4798	6	0.5404	6	0.6011	6	0.6617	6	0.4643	6
TA ₉	(0.1556,0.3890,0.6444)	0.4833	8	0.5444	8	0.6055	8	0.6666	8	0.7277	8	0.3938	8
TA ₁₀	(0.1144,0.3250,0.5851)	0.5449	9	0.6038	9	0.6626	9	0.7215	10	0.7803	10	0.3367	9

Part 3. Rating the fuzzy importance of TAs. The ranking of TAs with the fuzzy number will be obtained by TOPSIS method based on the weighted Hamming distance. In the new method, taking into account the preferences of decision makers, the value of β will be defined as 0, 0.25, 0.5, 0.75, 1. The relative closeness of each TA and the ranking in different β as shown respectively in Table 7. To comparison, the results obtained in traditional method are also shown in Table 7.

In the traditional method, the rating of TAs is TA₃≻ TA₇≻TA₁≻TA₂≻TA₄≻TA₈≻TA₆≻TA₉≻TA₁₀≻TA₅ (where ≻ means “at least as preferred as”). In fact, this rating should be the same as that obtained by the new method when the preference factor β = 0.5, which denotes a neutral attitude towards uncertain, that is to say, decision makers do not consider the uncertain preference. However, as can be seen from Table 7, the ranking of TAs is TA₃≻ TA₇≻TA₁≻TA₄≻TA₂≻TA₈≻TA₆≻TA₉≻TA₁₀≻TA₅ when β = 0.5. The reason they are different as follows: By using the linear programming model, when h = 0 and h = 1, the fuzzy number of TA₂ and TA₄ can be calculated respectively as (0.2456, 0.4868, 0.7390) and (0.2316,0.4880,0.7382). By using the objective penalty function method, the fuzzy number of TA₂ and TA₄ can be obtained respectively as (0.2342, 0.4828, 0.7335) and (0.2312, 0.4915, 0.7401). Comparing them in pairs, it is easy to see that the error is over 0.01 only between the value of $(Y_{2})_{h = 0}^{L}$ . Because the value of $(Y_{2})_{h = 0}^{L}$ is the lower bound, the result is better by the new method. And it is this reason that leads to the difference in ranking between TA₂ and TA₄ due to the great proximity.

In addition, from Table 7, the different values of β are considered, the rating of TAs is different. When β = 0, 0.25, 0.5, TA₁₀ ≻ TA₅ (where ≻ means "at least as preferred as"); and when β = 0.75, 1, TA₅ ≻ TA₁₀. Assume that C₅ = C₁₀, we can calculate that β ≈ 0.6040. Therefore, when β > 0.6040, the rating of TAs is TA₃≻ TA₇≻TA₁≻TA₄≻TA₂≻TA₈≻TA₆≻TA₉≻TA₁₀≻TA₅; and when β > 0.6040, the rating of TAs is TA₃≻ TA₇≻TA₁≻TA₄≻TA₂≻TA₈≻TA₆≻TA₉≻TA₅≻TA₁₀.

Finally, taking computing time of this task into account in Matlab, we get that the computing time is about 0.9959s in the new method and about 1.2777s in the traditional method.

Example 2. Taking the design of the refrigerator as an example: For customers, besides product price, product quality is also very important. Therefore, the quality requirements that customers care about are very vital to enterprises. However, there is a contradiction between customer requirements and quality, which requires comprehensive trade-offs and specific analysis.

By investigating the customer’s thoughts, we summarized eight customer requirements, which are defined by CR_i (i = 1, 2, . . . , 8) as follows: fast refrigeration (CR₁), low power consumption (CR₂), low noise (CR₃), environmental protection (CR₄), beautiful appearance (CR₅), multi-function (CR₆), low price (CR₇) and simple operation (CR₈). Simultaneously, 10 product qualities which are known as technical attributes are defined by TR_j (j = 1, 2, . . . , 10) as follows: refrigeration capacity (TA₁), power consumption (TA₂), noise level (TA₃), insulation layer thickness (TA₄), box size (TA₅), temperature control mode (TA₆), cooling mode (TA₇), additional function (TA₈) and input power (TA₉).

Part 1. Date acquisition and review. Similar Example 1, the importance of CRs are expressed as the linguistic data at seven levels, a pre-defined triangular fuzzy weight set ${W_{1}^{*}, W_{2}^{*}, W_{3}^{*}, W_{4}^{*}, W_{5}^{*}, W_{6}^{*}, W_{7}^{*}}$ is used to express the individual preference, (Table 2 and Fig. 3 in Example 1). Using these seven fuzzy numbers, market research was conducted on 10 customers, and the importance of each requirement was assessed. The weights of these 10 customers for requirements were integrated, and the final weights of 8 customer requirements were obtained as shown in Table 8.

Table 8

Weight W_i of 10 customers on CR_i

W ₁	(0.74, 0.94, 1.00)
W ₂	(0.69, 0.89, 0.98)
W ₃	(0.64, 0.78, 0.91)
W ₄	(0.75, 0.95, 0.98)
W ₅	(0.32, 0.50, 0.68)
W ₆	(0.45, 0.56, 0.87)
W ₇	(0.65, 0.85, 0.97)
W ₈	(0.51, 0.70, 0.87)

Similarly, the relationships between CRs and TAs are linguistically judged as five levels, which can be expressed by a pre-defined triangular fuzzy set ${U_{1}^{*}, U_{2}^{*}, U_{3}^{*}, U_{4}^{*}, U_{5}^{*}}$ , (Table 4 and Fig.

4 in Example 1). Ten experts participated in the assessment of the relationships between CRs and TAs, and by integrating the individual assessments of experts using the fuzzy numbers, the final relationships between the CRs and the TAs can be obtained as shown in Table 9.

Table 9

Weight W_i of the 10 experts on CR_i

U _ij	CR ₁	CR ₂	CR ₃	CR ₄
TA ₁	(0.70,1.00,1.00)	(0.06,0.18,0.44)	(0.00,0.00,0.30)	(0.00,0.00,0.30)
TA ₂	(0.06,0.18,0.44)	(0.70,1.00,1.00)	(0.60,0.88,1.00)	(0.62,0.90,1.00)
TA ₃	(0.00,0.00,0.30)	(0.06,0.15,0.42)	(0.70,1.00,1.00)	(0.54,0.80,0.94)
TA ₄	(0.00,0.00,0.30)	(0.60,0.88,1.00)	(0.54,0.80,0.94)	(0.00,0.10,0.38)
TA ₅	(0.00,0.00,0.30)	(0.00,0.13,0.40)	(0.00,0.00,0.30)	(0.00,0.13,0.40)
TA ₆	(0.00,0.00,0.30)	(0.54,0.80,0.94)	(0.00,0.00,0.30)	(0.00,0.05,0.34)
TA ₇	(0.66,0.95,1.00)	(0.54,0.80,0.94)	(0.54,0.80,0.94)	(0.66,0.95,1.00)
TA ₈	(0.00,0.05,0.34)	(0.00,0.13,0.40)	(0.06,0.15,0.42)	(0.00,0.00,0.30)
TA ₉	(0.00,0.10,0.38)	(0.70,1.00,1.00)	(0.00,0.00,0.30)	(0.00,0.00,0.30)
U _ij	CR ₅	CR ₆	CR ₇	CR ₈
TA ₁	(0.00,0.00,0.30)	(0.00,0.05,0.34)	(0.06,0.18,0.44)	(0.00,0.05,0.34)
TA ₂	(0.00,0.00,0.30)	(0.30,0.58,0.75)	(0.33,0.54,0.74)	(0.00,0.00,0.30)
TA ₃	(0.00,0.00,0.30)	(0.30,0.58,0.75)	(0.06,0.18,0.44)	(0.00,0.00,0.30)
TA ₄	(0.30,0.58,0.75)	(0.06,0.15,0.42)	(0.30,0.58,0.75)	(0.00,0.00,0.30)
TA ₅	(0.66,0.95,1.00)	(0.00,0.10,0.38)	(0.06,0.15,0.42)	(0.00,0.00,0.30)
TA ₆	(0.00,0.00,0.30)	(0.00,0.00,0.30)	(0.62,0.90,1.00)	(0.66,0.95,1.00)
TA ₇	(0.00,0.00,0.30)	(0.00,0.05,0.34)	(0.62,0.90,1.00)	(0.62,0.90,1.00)
TA ₈	(0.00,0.00,0.30)	(0.70,1.00,1.00)	(0.32,0.55,0.85)	(0.06,0.15,0.42)
TA ₉	(0.00,0.00,0.30)	(0.66,0.95,1.00)	(0.70,1.00,1.00)	(0.00,0.15,0.42)

Part 2. Determining the fuzzy importance of TAs. By the objective penalty function, the results of $(Y_{j})_{h = 0}^{L}$ , (Y_j) _h=1 and $(Y_{j})_{h = 0}^{R}$ are shown in Table 10. What’s more, the values of $(Y_{j})_{h = 0}^{L}$ , (Y_j) _h=1 and $(Y_{j})_{h = 0}^{R}$ are also given in Table 10 by the traditional method in [34]. Similarly, bold numbers represent more exact values when compared with the two methods. Therefore, it is clear shown in Table 10 that the results of two methods are very close (the error of each value is not over 0.1 between two methods), and even the new method is slightly better than the traditional one (the new method has more bold numbers than the traditional one). Finally, the fuzzy importance of TAs are obtained in Table 11.Where the fuzzy importance of TA_j is ( $(Y_{j})_{h = 0}^{L}, (Y_{j})_{h = 1}, (Y_{j})_{h = 0}^{R}$ ).

Table 10

The exactness of objective penalty function method and traditional method

TA _j	The new method			The traditional method in [34]
	$(Y_{j})_{h = 0}^{L}$	(Y_j) _h=1	$(Y_{j})_{h = 0}^{U}$	$(Y_{j})_{h = 0}^{L}$	(Y_j) _h=1	$(Y_{j})_{h = 0}^{U}$
TA ₁	0.0959	0.2245	0.4986	0.0907	0.2133	0.4848
TA ₂	0.3090	0.5646	0.7864	0.3037	0.5485	0.7695
TA ₃	0.1708	0.3527	0.6306	0.1743	0.3487	0.6227
TA ₄	0.1807	0.3888	0.6620	0.1832	0.3840	0.6547
TA ₅	0.0303	0.1376	0.4403	0.0380	0.1455	0.4436
TA ₆	0.1725	0.3554	0.6498	0.1768	0.3549	0.6401
TA ₇	0.4474	0.7584	0.9181	0.4378	0.7382	0.9021
TA ₈	0.0879	0.2182	0.5506	0.0966	0.2289	0.5512
TA ₉	0.1952	0.4007	0.6712	0.1982	0.4005	0.6624

Table 11

The exactness of objective penalty function method and traditional method

TA _j	The new method											The traditional method
fuzzy importance		C _j										E (Y_j)	Rank
		β = 0	Rank	β = 0.25	Rank	β = 0.5	Rank	β = 0.75	Rank	β = 1	Rank
TA ₁	(0.0959,0.2245,0.4986)	0.6385	8	0.6888	8	0.7391	8	0.7895	8	0.8398	7	0.2497	8
TA ₂	(0.3090,0.5646,0.7864)	0.3245	2	0.3842	2	0.4439	2	0.5035	2	0.5632	2	0.5421	2
TA ₃	(0.1708,0.3527,0.6306)	0.5083	6	0.5658	6	0.6233	6	0.6808	6	0.7383	6	0.3730	6
TA ₄	(0.1807,0.3888,0.6620)	0.4746	4	0.5347	4	0.5114	4	0.6551	4	0.7152	4	0.4007	4
TA ₅	(0.0303,0.1376,0.4403)	0.7111	9	0.7623	9	0.8136	9	0.8648	9	0.9160	9	0.1927	9
TA ₆	(0.1725,0.3554,0.6498)	0.4974	5	0.5571	5	0.6167	5	0.6764	5	0.7360	5	0.3805	5
TA ₇	(0.4474,0.7584,0.9181)	0.1618	1	0.2206	1	0.2794	1	0.3383	1	0.3971	1	0.7035	1
TA ₈	(0.0879,0.2182,0.5506)	0.6156	7	0.6734	7	0.7313	7	0.7892	7	0.8470	8	0.2762	7
TA ₉	(0.1952,0.4007,0.6712)	0.4640	3	0.5235	3	0.5830	3	0.6425	3	0.7020	3	0.4156	3

Part 3. Rating the fuzzy importance of TAs.By these fuzzy numbers of TAs in Table 11, the TOPSIS method based on the weighted Hamming distance is used to rank these nine TAs. When the preference value of decision makers is equal to 0, 0.25, 0.5, 0.75 and 1, respectively, based on the relative closeness C_j, the rankings as shown in Table 11. To make a comparison with the traditional method, the rating by which is also given.

The results show that the ranking of TA₁ and TA₈ is different when the uncertain preference β of decision makers are different. By calculation, β ≈ 0.7608 when C₁ = C₈. Therefore, when β < 0.7608, the rating of TAs is TA₇≻ TA₂≻TA₉≻TA₄≻TA₆≻TA₃≻TA₈≻TA₁≻TA₅; and when β > 0.7608, the rating of TAs is TA₇≻ TA₂≻TA₉≻TA₄≻TA₆≻TA₃≻TA₁≻TA₈≻TA₅. What’s more, compared with the traditional method, the new method take the preference of decision makers in consideration and get the different rating by different preference factor. And when decision makers do not consider the uncertain preference, that is β = 0.5, the rating of TAs in the new method is same as in the traditional method. Therefore, the top four are relatively stable, namely: Cooling mode (TA₇), power consumption (TA₂), input power (TA₉) and insulation layer thickness (TA₄).

Finally, taking computing time of this task into account in Matlab, we get that the computing time is about 0.7789s in the new method and about 1.1991s in the traditional method.

5 Conclusions

In general, in order to determine the importance of TAs in fuzzy QFD, the paper proposed a new method, which mainly includes three parts: (1) Date acquisition and review. The important level matrix of CRs and the relation matrix between CRs and TAs are determined in this part; (2) Determining the fuzzy importance of TAs. Here are two innovative points: (a) For solving constrained nonlinear programming problems, the exact penalty function is used. What’s more, in the traditional exact penalty function methods, the constraint penalty parameter need to be continuously increased, which makes it not efficient in practical computing of Matlab. So the objective penalty function method with objective penalty parameter is proposed; (b) The new method adopts the fuzzy method approximate fuzzy number method, which only takes h = 0 and h = 1 by solving the proposed two constrained non-linear programming problems to obtain the fuzzy importance of each TA; (3) Rating the fuzzy importance of TAs. In this part, the TOPSIS method based on weighted Hamming distance is proposed to get the raking of TAs. This method takes fully into account the uncertain preferences of decision makers.

Finally, we can draw the following conclusions from the Examples 1 and 2: (1) In the part of calculating the fuzzy importance of TAs, the value of $(Y_{j})_{h = 0}^{L}$ ,(Y_j) _h=1 and $(Y_{j})_{h = 1}^{U}$ are obtained by the new method in numerical examples. This paper obtains the results which close to and even slightly better than the traditional one in [34]; (2) When taking the different value of β, the different rating of TAs is obtained by the new method. Therefore, according to the same problem, the most suitable plan will be provide for the decision makers; (3) The numerical experiments of this paper are completed with the help of Matlab. As can be seen from the Examples 1 and 2, the new method takes about 20 percent less time than the traditional method in [34]. The new method of integrating objective penalty function and fuzzy TOPSIS method based on the weighted Hamming distance, which fully considers the principle of people-oriented in fuzzy QFD, can make different decisions according to different types of customers. What’s more, it’s easy to calculate and it can saves a lot of time when the data is huge.

In short, a new method to rating TAs in fuzzy QFD is proposed, which can provide more suitable options for decision makers. Furthermore, the new method also may be proved to apply in some cases that the level important matrix and the relation matrix of fuzzy QFD are defined by trapezoidal fuzzy numbers or intuitionistic fuzzy numbers in future studies.

Footnotes

Acknowledgment

The authors acknowledge the support from the General Program of National Natural Science Foundation of China under Grants 11671250.

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