Determination of Target Values of Engineering Characteristics in QFD Using Uncertain Programming
 Yunwen Miao^{1},
 Yuanyuan Liu^{1}Email author and
 Yizeng Chen^{2}
DOI: 10.1186/s404670150040x
© Miao et al. 2015
Received: 19 September 2015
Accepted: 10 November 2015
Published: 8 December 2015
Abstract
Quality function deployment (QFD) is a new product development tool remarked with interpreting customer requirements into engineering characteristics of the design process. On account of the inherent imprecise and uncertain elements in the weights of customer requirements, the relationships between customer requirements and engineering characteristics, and the correlations among engineering characteristics, uncertain variables are preferred to be applied in this paper. By taking advantage of expected value modelling to determine the target values of engineering characteristics in handling different practical design scenarios, two uncertain programming models are proposed for optimizing the QFD process in an uncertain environment. Subsequently, the proposed uncertain models are implemented in a motor car design for quality development.
Keywords
Quality function deployment Engineering characteristic Uncertain variable Expected value modellingIntroduction
Nowadays, global economy has raised fierce international market competition and rapid technological change, especially in traditional tangible products made by manufacturing enterprises. For the sake of occupying sales and profits in larger market and longterm development, more and more economical products with high quality are designed and generated to cater diverse customer perceptions and expectations. As far as the continuous development of an existing product is concerned, the use of quality function development (QFD) has gained extensive global support. Originated in Japan in the late 1960s [1], QFD was known as a customerdriven product design methodology, which is meant to promote the quality of products. It is a systematic method contributing to translating various customer requirements (CRs) into several engineering characteristics (ECs) of the product for achieving higher customer satisfaction. So far, QFD has been applied in many other fields, including supply chain management [2, 3], investment [4], product selection and assignment [5], etc.
The product development process based on QFD is to determine a set of x _{1},x _{2},⋯,x _{ n } for ECs of the new/improved product to match or exceed the degree of overall customer satisfaction of all competitors in the target market with limited organizational resources. It is a complex decision process with multiple variables, requiring tradeoff and optimizing all kinds of conflicts contained in HoQ. As an important branch of QFD research, more and more systematic and rational programmings with different considerations to determine the target values of ECs have achieved flourishing advances in the last few decades. Generally, a traditional and classic programming in QFD planning is designed to achieve maximum overall customer satisfaction in the constraints of two functional relationships including levels of attainment between CRs and ECs and that among ECs, and other constraints of resources including cost, resource, technology, etc.
In order to formulate the programming, it is critical to first determine the relative importance of CRs and the functional relationships. According to the previous literature, they were either confirmed by subjective assessments and judgments expressed as crisp, random or fuzzy variables [7–10], or by the frequent application of the fuzzy linear and nonlinear regressions methods [11–14]. Even though the latter one seems more objective, practically speaking, it is much less feasible due to sparse data collection.
It can be seen that, among them, fuzzy modelling approaches were popular since they have applied fuzziness based on the fuzzy set theory [15] to define the imprecise elements in HoQ. Therefore, the objective function and constraints of different programmings can be incorporated with fuzzy parameters to get better results close to reality. On this basis, numerous studies have been conducted on how to obtain a set of target values of ECs. Chen et al. proposed a fuzzy expected value modelling approach for target setting, which simultaneously took maximizing the overall customer satisfaction and minimizing the design cost into consideration [16]. Sener and Karsak developed some fuzzy mathematical programming models combining the functional relationships obtained from a fuzzy regression based on nonlinear programming and an integrated fuzzy linear regression and fuzzy multiple objective programming approach to determining target values of ECs [17, 18]. Zhong et al. proposed a fuzzy chanceconstrained programming model with the objective of minimizing the fuzzy expected cost and the chance constraint of overall customer satisfaction [19].
Until now, most of the variables or parameters applied in QFD process were either crisp values or fuzzy ones. However, it is usually not appropriate enough because both the probability theory and the fuzzy set theory may sometimes lead to counterintuitive results [20]. In this paper, we put forward a new method based on the uncertainty theory proposed by Liu [21] and redefined in Liu [22]. Similarly to fuzzy optimization models, the uncertain variables involved like the relative importance of CRs, the uncertain relationship between CRs and ECs, the uncertain correlations among ECs and the variable cost to fulfill one unit of ECs, will be predefined by experts in a vague way using uncertain variables rather than crisp values. So as to effectively determine the target values of ECs in handling practical design scenarios, two uncertain programming models using expected value modelling (EVM) are generated under the objectives of maximizing the overall customer satisfaction and minimizing the total design cost, respectively.
The rest of the article is organized as follows. In Section Preliminaries, some preliminaries of uncertain variable, uncertainty distribution, and uncertain programming are described. In Section Uncertainty theory, two uncertain programming models using EVM for QFD planning in an uncertain environment are proposed to determine the target values of ECs. Finally, Section Uncertain Expected Value Modelling for QFD Planning illustrates a numerical example of a motor car design, which is presented to demonstrate the performance of the proposed approach.
Preliminaries
Uncertainty theory is an efficient mathematical system to deal with indeterminacy, which plays an crucial role to measure expert statistics and subjective estimations. In this section, some basic knowledge of uncertainty theory is introduced for describing the approaches of EVM applied in the uncertain programming method. The reader may refer to Liu [20–22] for more details.
Uncertainty theory
Definition 1.

(i) \(\mathcal {M} \{\Gamma \}=1\) for the universal set Γ;

(ii) \(\mathcal {M}\{\Lambda \}+\mathcal {M}\{\Lambda ^{c}\}=1\) for any event Λ;

(iii) For every countable sequence of events Λ _{1},Λ _{2},⋯, we have$$\begin{array}{@{}rcl@{}} \mathcal{M}\left\{\bigcup_{i=1}^{\infty} \Lambda_{i}\right\}\leq \sum\limits_{i=1}^{\infty} \mathcal{M}\{\Lambda_{i}\}. \end{array} $$
where Λ _{ k } are arbitrarily chosen events from \(\mathcal {L}_{k}\) for k=1,2,⋯, respectively.
Based on the four axioms of uncertain measure given above, a formal definition of uncertain variable is presented as follows.
Definition 2.
(Liu [21]) An uncertain variable is a measurable function ξ from an uncertainty space \((\Gamma, \mathcal {L}, \mathcal {M})\) to the set of real numbers such that {ξ∈B} is an event for any Borel set B.
To better describe uncertain variables, the concept of uncertainty distribution is adopted. In many cases, it is more sufficient to know the uncertainty distribution than the uncertain variable itself.
Definition 3.
for any real number x.
Definition 4.
Suppose that a regular uncertainty distribution Φ(x) has an inverse function on the range of x with 0<Φ(x)<1, and the inverse function Φ ^{−1}(α) exists on the open interval (0,1) and is unique for each α∈(0,1), then the inverse function Φ ^{−1}(α) is called the inverse uncertainty distribution of an uncertain variable ξ, which is vital in operations of independent uncertain variables with regular uncertainty distributions.
In order to ensure that we can separately define uncertain variables on different uncertainty spaces, in 2009, Liu defined the independence of uncertain variables in the following mathematical forms.
Definition 5.
for any Borel sets B _{1},B _{2},…,B _{ n } of real numbers.
Therefore, regarding strictly monotone functions of independent uncertain variables with regular uncertainty distributions, the operational law was given by Liu [22].
Theorem 1.
With respect to uncertain measure, the expected value of an uncertain variable is the average value, which is able to be represented by the inverse uncertainty distribution as follows.
Definition 6.
provided that at least one of the two integrals is finite.
Theorem 2.
Thus, referring to the operational law of strictly monotone function of independent uncertain variables, the expected value can be calculated as follows.
Theorem 3.
Uncertain Expected Value Modelling for QFD Planning
QFD is a planning and problemsolving tool for product development, the core of which is House of Quality (HoQ) embedded with four matrices, i.e., relative importance matrix of CRs, relationship matrix between CRs and ECs, correlations matrix among ECs, and target value matrix of ECs.
According to information provided in HoQ, the purpose of product planning process is usually to determine target values of ECs to maximize the overall customer satisfaction with limited organizational resources and technologies, or to minimize the design cost under a preferred acceptable overall customer satisfaction. In reality, we frequently lack observed data, and the estimated probability distribution may be far from the cumulative frequency [20, 25]. In order to get over this difficulty, based on uncertainty theory, the uncertain programming method using EVM is proposed in this section to deal with QFD planning problem in an uncertain environment.
Problem Notations and Explanations

–CR _{ i }: the ith customer requirement, i=1,2,⋯,m;

–EC _{ j }: the jth engineering characteristic, j=1,2,⋯,n;

–Comp _{ q }: the qth competitor, q=1,2,⋯,p;

–R: the original uncertain relationship matrix between CRs and ECs, the element r _{ ij } of which denotes the uncertain relation measure between CR _{ i } and EC _{ j };

–P: the uncertain correlation matrix among ECs, the element p _{ kj } of which denotes the uncertain correlation measure between EC _{ j } and EC _{ k };

–R’: the modified uncertain relationship matrix between CRs and ECs by accommodating the uncertain correlation matrix P, in which the element r ij′ denotes the modified uncertain relationship measure between CR _{ i } and EC _{ j };

–Y: the uncertain vector of customer perception of CRs, Y=(y _{1},y _{2},⋯, y _{ m })^{ T }, and y _{ i } is the customer perception of the satisfaction degree of CR _{ i }, i=1,2,⋯,m;

–W: the uncertain relative importance vector of CRs, W=(w _{1},w _{2},⋯, w _{ m })^{ T }, among which w _{ i } is the uncertain relative importance of CR _{ i }, i=1,2,⋯,m;

–X: the vector of level of attainment of ECs, X=(x _{1},x _{2},⋯,x _{ n })^{ T }, and x _{ j } is the level of attainment of EC _{ j }, 0≤x _{ j }≤1, j=1,2,⋯,n;

–V: the importance vector of ECs, V=(v _{1},v _{2},⋯,v _{ n })^{ T }, in which v _{ j } is the uncertain importance of EC _{ j }, j=1,2,⋯,n;

– S _{ q }: the overall customer satisfaction of the qth competitor, q=1,2,⋯,p;

– l _{ j }: the target value of EC _{ j }, j=1,2,⋯,n;

–C: the total product design cost;

– C _{ F }: the fixed part of design cost;

– C _{ V }: the variable part of design cost;

– C _{ j }: the uncertain cost required for achieving x _{ j }, j=1,2,⋯,n;

– c _{ j }: the uncertain cost required for improving each one unit of EC _{ j }, j=1,2,⋯,n;

–B: the budget of product development.
The relationships between CRs and ECs are generated in the relationship matrix R, the body of HoQ, in which, each vector can be denoted as r _{ ij }. Generally, the relationships summit to a pile of predefined uncertain variables measured by experts. The forms of uncertain variables defined in Formula (8) can also be adopted to illustrate the strength of relationship, i.e., concave uncertainty distribution can explain the “strong” relationship between a certain CR and EC, while convex and linear ones represent “weak” and “medium”, respectively.
Similarly, correlations among ECs can be represented as above uncertain variables defined by experts as well, which are illustrated in the correlation matrix P, the roof of HoQ, and P _{ kj } denotes the correlation between EC _{ k } and EC _{ j }. Concave and convex uncertainty distributions can reflect dependence like “positive” and “negative” among ECs, respectively. It is certain that, the EC is defined as the strongest dependence on itself in the construction of the correlation matrix [26], i.e., p _{ jj } is defined as the maximum degree in the correlation matrix.
Normalizing the Target Values of ECs
where 0≤x _{ qj }≤1.
Calculation of Overall Customer Satisfaction
The overall customer satisfaction can be calculated through the integration of four matrices W, R, P, and X, which denote the relative importance of CRs, the relationships between CRs and ECs, the correlations among ECs and the target values of ECs, respectively.
Formulation of Development Resources
where the cost coefficient c _{ j } denotes the cost needed when EC _{ j } is fully improved, i.e., a cost c _{ j } will be required if one unit of attainment of the EC _{ j } is fulfilled. Since the price of one unit material usually vibrates in an interval in the market, we adopt linear uncertain variables listed below to define c _{ j }, which will be applied in the calculation later.
denoted by \(\mathcal {L}(a,b)\), where a and b are real numbers with a<b. In this case, a is the price of lower limit of one unit material in the market, and b is the upper limit.
Uncertain Programming Using EVM
In the above objective functions and constraints, w _{ i }, r _{ ik }, p _{ kj }, and c _{ j } are predefined uncertain variables mentioned in Section 3, the calculation of which will be described in more detail later. Notably, it seems quite appropriate and rational to utilize these uncertain variables in practical product design scenarios. However, it is hard for mathematical calculation since uncertain variables are not as straightforward as crisp ones.
For this reason, in order to build unambiguous uncertain programming model for QFD planning, two uncertain programming models using EVM are proposed in this section, in which the underlying philosophy is based on selecting the decision with the maximum expected returns.
where \(\bar {E}(S)\) is the normalized expected value of the overall customer satisfaction.
in which the constraint guarantees that the expected value of the total cost required for the new/improved product will not exceed the expected value of the budget.
where S ^{′} represents the preferred acceptable overall customer satisfaction. The definition of S ^{′} depends on the decisionmakers’ preference and subjectivity on customer satisfaction of products.
Numerical Example
To demonstrate the feasibility and effectiveness of the proposed uncertain programming models, the development of a new type of motor car is introduced as an example in this section. Applying QFD into the process aims to investigate the influence of target values of ECs on the overall customer satisfaction and the total design cost, which will provide a dynamic routine to guide the design team to determine a new set of target values for ECs.
A corporation is improving a new model of motor car to enhance competitiveness and occupy larger market, thus a survey regarding an initial market among users was done. With respect to the survey data in the market and feedbacks from users, five major CRs are identified to be the most significant concerns of the customers. i.e., “reducing the noise of car” (CR_{1}), “enhancing the acceleration” (CR_{2}), “saving fuel” (CR_{3}), “improving security” (CR_{4}), and “seat comfort” (CR_{5}), respectively.
The house of quality of a motor car
−  +  −  +  +  

Engineering characteristics  EC_{1}  EC_{2}  EC_{3}  EC_{4}  EC_{5}  
x _{1}  x _{2}  x _{3}  x _{4}  x _{5}  
EC_{1}  x ^{ 6 } (α ^{ 1 / 6 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  
EC_{2}  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 6 } (α ^{ 1 / 6 })  x ^{ 2 } (α ^{ 1 / 2 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  
Uncertain correlation matrix  EC_{3}  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 2 } (α ^{ 1 / 2 })  x ^{ 6 } (α ^{ 1 / 6 })  x ^{ 4 } (α ^{ 1 / 4 })  x ^{ 1 / 9 } (α ^{ 9 }) 
EC_{4}  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 4 } (α ^{ 1 / 4 })  x ^{ 6 } (α ^{ 1 / 6 })  x ^{ 1 / 9 } (α ^{ 9 })  
EC_{5}  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 6 } (α ^{ 1 / 6 })  
Customer  Uncertain weighs of  Uncertain relationship matrix between customer requirements and engineering characteristics  
requirements  customer requirements  
CR_{1} \(\mathcal {Y}_{1}\)  x ^{ 4 } (α ^{ 1 / 4 })  x ^{ 6 } (α ^{ 1 / 6 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 2 } (α ^{ 1 / 2 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 }) 
CR_{2} \(\mathcal {Y}_{2}\)  x ^{ 2 } (α ^{ 1 / 2 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 4 } (α ^{ 1 / 4 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 }) 
CR_{3} \(\mathcal {Y}_{3}\)  x ^{ 2 } (α ^{ 1 / 2 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 6 } (α ^{ 1 / 6 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 }) 
CR_{4} \(\mathcal {Y}_{4}\)  x ^{ 6 } (α ^{ 1 / 6 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 6 } (α ^{ 1 / 6 })  x ^{ 1 / 9 } (α ^{ 9 }) 
CR_{5} \(\mathcal {Y}_{5}\)  x ^{ 1 / 4 } (α ^{ 4 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 1 / 9 } (α ^{ 9 })  x ^{ 4 } (α ^{ 1 / 4 }) 
Units  dB  Horsepower  Gallon  Kg  M^{3}  
Comp_{1}  80  75  0.042  23  0.18  
Comp_{2}  65  70  0.034  24  0.20  
Comp_{3}  65  80  0.028  23  0.18  
Technical measures  Comp_{4}  75  60  0.032  15  0.14 
Comp_{5}  95  80  0.030  20  0.19  
Min  60  60  0.027  15  0.14  
Max  95  90  0.042  25  0.21  
Cost coefficients  \(\mathcal {L}(8,12)\)  \(\mathcal {L}(9,12)\)  \(\mathcal {L}(24,26)\)  \(\mathcal {L}(14,16)\)  \(\mathcal {L}(7,10)\) 
Different types of uncertain variables with different meanings for grading
Type  Uncertain variables  W  R  P 

I  Φ(x)=x ^{1/9}  extremely unimportant  extremely weak  strong negative 
II  Φ(x)=x ^{1/6}  very unimportant  very weak  very negative 
III  Φ(x)=x ^{1/4}  quite unimportant  quite weak  quite negative 
IV  Φ(x)=x ^{1/2}  some important  weak  weak negative 
V  Φ(x)=x  moderately important  medium  medium 
VI  Φ(x)=x ^{2}  important  strong  weak positive 
VII  Φ(x)=x ^{4}  quite important  quite strong  quite positive 
VIII  Φ(x)=x ^{6}  very important  very strong  very positive 
IX  Φ(x)=x ^{9}  extremely important  extremely strong  strong positive 
Relative importance for five ECs through normalized expected values
EC_{1}  EC_{2}  EC_{3}  EC_{4}  EC_{5}  

E(v _{ j })  2.18  2.91  3.67  3.17  1.78 
\(\bar {E}(v_{j})\)  0.16  0.21  0.27  0.23  0.13 
Ranking  4  3  1  2  5 
Rankings for five companies through normalized expected values
Comp_{1}  Comp_{2}  Comp_{3}  Comp_{4}  Comp_{5}  

\(\bar {E}(S_{q})\)  0.43  0.67  0.79  0.27  0.60 
Ranking  4  2  1  5  3 
In order to improve the existing design process, the resources are needed to be allocated more properly under the limit of a budget which will cover equipments and materials. Through investigation, the budget is determined to be 100 units in terms if all the resources are taken into consideration. As illustrated in Section 3.4, C _{ F } is the fixed cost in the development of design process which will set to be 50 units in this numerical example while C _{ j } is the variable part required to improve one unit of EC _{ j }, and each c _{ j } of individual EC _{ j } is uncertain to determine. Thus, as shown in Table 1, five linear uncertain variables are applied to c _{ j } to express such circumstance.
Solutions of UP1
EC_{1}  EC_{2}  EC_{3}  EC_{4}  EC_{5}  

x _{ j }  1  1  0.24  1  1 
l _{ j }  60  90  0.0384  25  0.21 
\(\bar {E}(v_{j})/E(c_{j})\)  0.0160  0.0200  0.0108  0.0153  0.0152 
Ranking  2  1  5  3  4 
Solutions of UP2 with different values of S ^{′}
S ^{′}  x _{1}  x _{2}  x _{3}  x _{4}  x _{5}  E(C _{ V }) 

0.1  0  0.4762  0  0  0  5.0001 
0.2  0  0.9524  0  0  0  10.0002 
0.3  0.5625  1  0  0  0  16.1250 
0.4  1  1  0  0.1304  0  22.4575 
0.5  1  1  0  0.5652  0  28.9795 
0.6  1  1  0  1  0  35.5000 
0.7  1  1  0  1  0.7693  42.0390 
0.8  1  1  0.2593  1  1  50.4825 
0.9  1  1  0.6296  1  1  59.7425 
1  1  1  1  1  1  69.0000 
The above five pairs of dynamic roadmaps would assist the design team to determine the target values of the five ECs to improve the design of motor car by taking competition requirements, the technical feasibility and financial factors into account. For example, if our company (Comp_{1}) wants to rank foremost among the competitors, the preferred acceptable overall customer satisfaction should at least match or exceed that of Comp_{3} (0.79), which is the current leader among five competitors. It will become more convenient for a design team to determine the target values for ECs of the improved car based on above five pairs of plots by uncertain programming, e.g., the calculated target values of ECs, l _{1} is 60 dB, l _{2} is 90 horsepower, l _{3} is smaller than 0.0384 gallon, l _{4} is 25 kg, and l _{5} is 0.21 m^{3}, respectively, along with the overall customer satisfaction of 0.79. Accordingly, the variable costs for improving them are (8, 12), (9, 12), (5.76, 6.24), (14, 16), and (7, 10). If the fixed cost is 50 units, the variable cost will be greater than (43.76, 56.24), which implies that the expected value of the total design cost will be at 100 units.
In resource constraints, the fixed cost and budget are crisp values predefined by experts; actually, it can also be defined as uncertain variables or other different crisp numbers in optimizing the models, which will lead to more comparative results.
To some extent, incorporating uncertainty theory into QFD would assist the company to better define the vagueness and ambiguity in the design process and achieve more rational results in determining the target values of ECs and obtain higher overall customer satisfaction.
Conclusions
In this paper, the basic idea of uncertain programming which includes expected value modeling has been applied to model the QFD planning process in an uncertain environment. On the basis of uncertainty theory, novel modeling approaches have been put up to determine the target values of ECs in QFD.
On account of the imprecise and uncertain elements in the development process, uncertain variables of regular uncertainty distributions have been adopted to define the relative importance of each CR, the uncertain relationship between CRs and ECs and the correlation among the ECs, while linear uncertain variables are applied to describe the variable cost of improving one unit of individual EC. The illustrated example of quality improved problem of a motor car showed that the proposed approach can model the process effectively in an uncertain environment by taking competition requirements, the technical feasibility and financial factors into consideration.
A new method based on uncertainty theory, namely, uncertain programming using EVM has been introduced in QFD to determine the target values of engineering characteristics in an uncertain environment of different real life scenarios. The work can be extended to many angles with respect to diverse uncertain features in HoQ. Much more can be done in this area, which may lead to more fruitful achievements.
Declarations
Acknowledgements
This work was supported in part by a grant from the National Natural Science Foundation of China (No. 71272177).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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