As observed previously in Section 9.1, a large size HoQ inevitably tends to attenuate the visibility and facilities of analysis of the information contained there. To avoid a consequent debasement of advantages deriving from QFD, two further methods are proposed, in such a way as:
• To make easier the compilation of the correlation matrix among technical
characteristics or among customer requirements
• To automatically verify the presence of technical characteristics or customer needs not related to other requirements or technical characteristics in the relationship matrix, respectively
• To identify the minimum set covering of technical characteristics able to cover all customer requirements
9.2.1 METHOD TO SUPPORTTHE COMPILATIONOF
THE CORRELATION MATRIX
The compilation of the correlation matrix can entail a big waste of time if the
project manages many technical characteristics. To make this activity easier, one can think of automating the procedure, empowering the designer to establish only the sign of correlation.
To better understand the proposed method, it is important to define the meaning of correlationand sign of correlation.
In QFD, two technical characteristics are defined to be correlated if variations on the first one determine variations on the second and vice versa. The sign defines the direction of such a correlation (positive, if positive variations of the first are connected with positive variations of the other; negative, if the opposite). A signif- icant example about the meaning of correlation and sign of correlation can be found in Hauser and Clausing (1988).
At present, designers who use QFD establish correlations among characteristics
on the basis of merely qualitative reasonings [Cohen, 1995; Wasserman, 1993]. They
do not take into account how technical characteristics influence customer require-
ments by means of the “content” of relationship matrix R∈ℜm,n (m is the number
of customer requirements and n is the number of technical characteristics).
By observing a generic relationship matrix, it may be noted that in many
cases correlated characteristics influence the same customer requirements. This
remark can be used as a starting point to build a partially automatic tool to define indirectly correlations among technical characteristics (to be used together with qualitative analysis).
How to Improve the Use of Quality Function Deployment 119 As a matter of fact, if the i-th characteristic influences u-th, v-th, etc. requirements, it is likely that the j-th characteristic correlated to it influences the same requirements. Moreover, if the dependence among characteristics induced by the action of the same requirements may imply the presence of a correlation, the opposite it is not necessarily true. In fact, it can be demonstrated that a correlation between two characteristics
may exist, without induced links on requirements in the relationship matrix.
Consequently, the method proposed here, which investigates the induced depend-
ence, can highlight only a fraction of the total correlations. The presence of an
induced dependence on the requirements is, therefore, a necessary but not sufficient condition to state that two characteristics are correlated. It is the designer, playing a new role of “validator,” who must confirm the possible sufficiency.
To formulate the existence of dependencies induced by requirements an n-
dimensional space constituted by a set of column vectors bi∈ ℜn is considered
(each one associated to a well-defined technical characteristic in the relationship
matrix). Supposing that the relationship matrix R is filled adopting the symbol
to individuate strong relationships, the symbol for medium relationships, and the
symbol for weakrelationships, the coefficients of vectors bi (∀i = 1, …, n) are determined as in the following:
∀i,j if rij = or or
then
bij = 1
Thus, by starting from the symbolic matrix R a new binary matrix B∈ℜm,n is
created.* Matrix B columns are then normalized producing another matrix N∈ℜm,n,
with the columns named vi (∀i = 1, …, n). The example in Figure 9.1 can give a
better idea of the building process of matrix N.
To represent the effects of the interdependence between i-th and j-th character- istics, the coefficient qij (scalar product of vectors vi) is introduced:
By calculating qij for all pairs of vectors in the N matrix, it is possible to
determine the characteristics dependence matrixQ:
Q∈ℜn,n is symmetrical, with q
ii = 1; ∀i = 1, …, n.
* Alternatively to the proposed procedure, matrix B could be defined as making a distinction between strong, medium, and weak relationships.
qij T i j n
j i j
=vi ⋅ =v cos
( )
v v, ∀, = …1 , ,Q=N NT
120 Advanced Quality Function Deployment
Matrix Q expresses the degree of induced dependence among technical charac-
teristics with reference to their capacity of influencing the same customer requirements. It may be observed that, if one works with large matrices, the determination of
matrix Q reveals the existence of columns or rows without relation with other
columns or rows of the matrix R, respectively. This fact is highlighted by appearance
of some zeros in the main diagonal of Q.
To allow the filling of the roof of the HoQ, information contained in Q is
compared with a prefixed threshold k (with 0 ≤k≤ 1); ∀i, j if qij > k then a potential
correlation between characteristics i-th and j-th is admitted, else this correlation is supposed nonexistent. So starting from Q, a new matrix “roof” is built.
At this point, the designer, on the basis of coefficients establishes, in an inter- active manner, the real existence of correlations (sufficiency condition) and its sign.
9.2.2 MINIMUM SET COVERING OF TECHNICAL CHARACTERISTICS
In some situations it is important to define the minimum set of characteristics able to interact with all customer requirements.
The definition of priorities as carried out according to the traditional scoring method [Akao, 1990] presents some counterindication because it does not take into account dependencies among characteristics themselves. In fact, in many applica- tions it happens that in the first ranking positions only strongly dependent charac- teristics appear, thus influencing only a limited part of customer requirements.
In some situations, however, it may be important for the designer to establish the minimum set of characteristics able to focus the design attention toward the main characteristics of a project development. Obviously, this does not mean that some characteristics can be neglected during the process development of a new product, but that the project can be organized in such a way as to give more importance to those characteristics that have more impact with customer requirements.
FIGURE 9.1 Example of the building process of matrix N starting from matrix R.
ˆ Q ˆ
Q
How to Improve the Use of Quality Function Deployment 121 The search for the minimum set of technical characteristics covering all customer requirements is a classic combinatorial optimization problem known as the set covering problem [Nemhauser and Wolsey, 1988; Parker and Rardin, 1988].
In more detail, if M = {1, …, m} is a finite set and {Mj}, for j ∈ N = {1, …, n},
a given collection of subsets of M, we say that F ⊆ N covers M if .
The sets Mj are known as covering sets. If pj is the cost associated to each Mj, the
minimum set covering problem becomes that of minimum-cost set covering. The search for the minimum number of columns (technical characteristics) able
to cover all rows (customer requirements) is a set covering problem with pj = 1,
∀j ∈ N. The set covering problem belongs to the set of NP-complete problems, which have a nonpolynomial computational complexity [Parker and Rardin, 1988]. In our specific case, since the aim is to give an agile supporting tool to the designer in the management of large relationship matrices R, a heuristic algorithm [Nemhauser and Wolsey, 1988] has been utilized (see Appendix). The algorithm has a polynomial computational complexity, and it is particularly suitable to give quick responses in a short time.