IJPR (2015) A Distance-based Methodology for Increased Extraction Of Informatin in QFD

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International Journal of Production Research
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A distance-based methodology for increased
extraction of information from the roof matrices
in QFD studies
Zafar Iqbal, Nigel P. Grigg, K. Govindaraju & Nicola M. Campbell-Allen
To cite this article: Zafar Iqbal, Nigel P. Grigg, K. Govindaraju & Nicola M. Campbell-Allen
(2015): A distance-based methodology for increased extraction of information from
the roof matrices in QFD studies, International Journal of Production Research, DOI:
10.1080/00207543.2015.1094585
To link to this article: http://dx.doi.org/10.1080/00207543.2015.1094585
Published online: 20 Oct 2015.
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A distance-based methodology for increased extraction of information from the roof matrices
in QFD studies
Zafar Iqbala
, Nigel P. Grigga
*, K. Govindarajub
and Nicola M. Campbell-Allena
a
School of Engineering & Advanced Technology, Massey University, Palmerston North, New Zealand; b
Institute of Fundamental
Sciences, Massey University, Palmerston North, New Zealand
(Received 16 December 2014; accepted 29 August 2015)
Quality Function Deployment (QFD) is a process in which customer needs are operationalised into deliverable Technical
Characteristics (TCs) at the design stage. A system of matrices known as the House of Quality (HOQ) works collectively
to produce Final Weightings (FWs) for TCs, enabling prioritisation and focusing design activity. In prioritising TCs,
QFD practitioners often fail to fully integrate the diverse information within the HOQ. In this article, we address the
inclusion of ‘Roof Matrix Correlations’ (RMCs). We show that, while other heuristics have been developed to integrate
RMCs, they each have limitations and only result in changes to the FW values. We present a methodology based on the
Manhattan Distance Measure (MDM) that integrates RMC data into the FWs, but also measures the overall nature and
level of intercorrelation within the matrix. This facilitates a more efﬁcient selection of TCs because the MDM provides a
consistent informational basis for substituting negatively correlated TCs with better alternatives, and reducing duplication
of effort in cases of highly positively correlated TCs. Application of the method is illustrated through re-analysis of a
well-known, published QFD example. Our approach can help practitioners to avoiding duplicating effort or to address
contradictions between TCs in a timely fashion.
Keywords: Quality Function Deployment; Roof Correlation Matrix; Manhattan Distance Measure
1. Introduction
In the development of products, important objectives are to reduce product development time, whilst achieving the nec-
essary customer requirements and keeping production cost and effort as low as is practicable (Temponi, Yen, and Amos
Tiao 1999). Quality Function Deployment (QFD) helps engineers, product and process design teams to systematically
determine and prioritise design requirements when developing a product (or service) that maximises customer satisfac-
tion while meeting necessary technical requirements (Kim et al. 1998; Wang 1999). Some researchers view QFD in
wider terms as an effective strategic management system facilitating translation of strategic imperatives and policies into
measurable and achievable conceptual requirements (Killen, Walker, and Hunt 2005; Tsai, Lo, and Chang 2003);
enhancing the effectiveness of key activities (Yang, Yang, and Peng 2011); and helping to manage trade-offs between
client needs and organisational capacity (Chen, Yu, and Chang 2006).
The QFD process utilises a system of matrices collectively known as the ‘House of Quality (HOQ)’. This conceptual
map, resembling a house, brings together information relating to customer requirements, competitor performance and
engineering/technical characteristics (TCs) to help develop the product/service. The HOQ quantiﬁes the relationships
between: ‘Voice Of Customer’ requirements (VOCs); Customer Priority Ratings (CPRs); TCs required of the product or
service; and performance data relating to competitors (Hauser and Clausing 1988; Tan 2003). Figure 1 shows a standard
HOQ comprising its different elements, referred to – for perhaps obvious reasons – as ‘rooms’. Practitioners combine
the information in these room matrices to produce an overall importance rating, known as a Final Weight (FW), for each
of the TCs. The FWs determine the priority order in which TCs contribute to customer satisfaction.
In prioritising TCs, QFD practitioners often fail to fully integrate the diverse information within the HOQ. The pre-
sent paper derives from research systematically examining each room of the HOQ in order to provide mathematical and
statistical heuristics to improve the TC prioritisation and decision-making processes (Iqbal et al. 2014). The paper
focuses upon the triangular matrices of the HOQ referred to – again for obvious reasons – as the ‘roof’ matrices. These
are shown in Figure 1 as triangles marked A and B. Roof matrices contain pairwise intercorrelations between the VOCs
(triangle A) and the TCs (triangle B). We will focus our discussion on the TC matrix (B) but the discussion herein
*Corresponding author. Email: N.Grigg@massey.ac.nz
© 2015 Taylor & Francis
International Journal of Production Research, 2015
http://dx.doi.org/10.1080/00207543.2015.1094585
Downloadedby[MasseyUniversityLibrary]at18:1629October2015

applies equally to the VOC matrix (A). Situations where TCs have low (or zero) correlation values between them imply
that they are mutually independent: in this case, it is necessary to focus effort towards satisfying each individual TC in
order to meet product requirements. Alternatively, if TCs exhibit strong positive correlations with each other, then it fol-
lows that by satisfying one of these, the others will be simultaneously satisfied at least to some extent (Wasserman
1993). This is important for the practitioner to know as it can reduce some duplication of effort (Özgener 2003). If – in
the third instance – TCs exhibit strong negative correlations, then they are acting in mutual opposition. In this case,
achievement of one may reduce or negate achievement of the other.
In this paper, we first present a review of relevant literature relating to methods that have been developed to help
re-quantify FWs incorporating roof matrix correlations (RMCs), and discuss the potential benefits and shortcomings of
each. We then propose a method for utilising the correlations contained within the roof matrices to extract additional
information that will help design teams to eliminate or substitute certain TCs. This reduces the dimensionality of the
matrices and produces a more efficient set of TCs. Finally, we demonstrate the application of the new methods using a
published case that provides a good mix of positive, negative and zero RMCs, for effective illustration of the method.
2. Literature review
2.1 Relative importance of the ‘rooms’ in the HOQ matrix
The rooms of the HOQ shown in Figure 1 can be divided into two major groups:
(i) Compulsory: These are a compulsory part of the QFD process, necessary in computing the FWs of the TCs.
They are the minimum components required to compute FWs, and include VOCs, CPRs, TCs and the relation-
ship matrix shown in solid-shaded background in Figure 1.
(ii) Optional: These are not essential in the computation of FWs but their integration into the QFD process arguably
improves quantification of the FWs of TCs. These include competitor analysis, sales point data, correlations
between TCs, practical considerations, time to develop, etc.
Published case studies show that optional rooms of the HOQ are less frequently taken into account by QFD practi-
tioners in quantifying the FWs of TCs. These include: the roof matrices; competitors’ analysis (for VOCs and TCs);
‘sales point’; ‘manufacturing time’; and ‘cost’. Using Google Scholar, we searched published QFD examples published
between 1990 and 2014, containing the keyword combinations ‘QFD’, ‘Case study(ies)’, ‘correlation(s)’, ‘roof’, etc.
Using those that contained clearly explicated examples of the QFD process, we collected 75 published case studies.
Table 1 summarises the extent to which different authors have used the various rooms based on the 75 published
articles.
It is evident that some researchers and practitioners do, and others do not make use of the ‘optional’ rooms (i.e. the
full information) when calculating the FWs of TCs. Potential reasons for not using these will include cost, time, techni-
cal difficulties of measuring the attributes data and unavailability of methods to incorporate with FWs (Temponi, Yen,
Final Weights of TCs
Competitor(s)analysis
SalesPoints
Competitors’ analysis
Relationship Matrix
Voiceofcustomers
(VOCs)
Customer(s)PriorityRating
(CPRs)
Technical Characteristics
(TCs)
B
A
Time & Cost
Figure 1. A typical house of quality (HOQ, modified from Griffin and Hauser 1993).
2 Z. Iqbal et al.

and Amos Tiao 1999). For example, Chaudhuri and Bhattacharyya (2009) report difficulty in computing correlations for
all combinations, and so assume them to be constant. Tan, Xie, and Chia (1998) computed correlations between the
TCs but did not explicitly incorporate the correlations in prioritising the TCs. We contend that all rooms in the HOQ
are of potential relevance since they contain information that can impact FWs of TCs, and therefore should be integrated
into the ranking. Removing the effect of some rooms from the FWs score means that the TCs may end up ranked
inappropriately.
2.2 Correlation in roof matrices
The roof matrix (denoted B in Figure 1) is a pairwise matrix designed to show the impact of each TC on the others
(Esteban-Ferrer and Tricás 2012). Generally, the correlation (also referred to as interaction, association or impact)
between two variables measures the strength of relationship and is defined as the extent to which change in one will
determine the change in the second and vice versa. The ‘sign’ of correlation determines the direction: positive if the
direction of change in the first produces the same direction of change in the other; and negative if change in one brings
about an opposite change in the other (Franceschini and Rossetto 1998). The computed value of correlation magnitude
varies from –1 to 1, where 1 indicates perfect positive correlation and –1 indicates perfect negative correlation. Such
pairwise relationships can be: independent (no relationship); conflicting (negative correlation); or cooperative (positive
correlation). The weak or negative correlations are the generally traded off to find the best compromise, and the strong
positive correlations are considered together to avoid duplication of effort (Özgener 2003). Under the aegis of QFD
methodology, technical problems associated with TCs may emerge if the correlations between them are not reviewed
(Chien and Su 2003; Karsak, Sozer, and Alptekin 2003; Pakdil, Işın, and Genç 2012).
From the published case studies, we found that many researchers and practitioners have computed correlations
between the TCs but not subsequently incorporated these when subsequently prioritising the TCs (Tan, Xie, and Chia
1998; Wang, Xie, and Goh 1998). We also found that they generally provide only strengths of relationships, but not
underlying data-sets. If data-sets leading to correlation values are available then different multivariate techniques e.g.
explanatory factor analysis, principal component analysis, path analysis and interpretive structural modelling, etc. can
help to reduce or better analyse correlations (Sahney, Banwet, and Karunes 2006).
We also note that researchers and practitioners have employed a variety of symbols to represent the strength of rela-
tionship. Some researchers also quantify the relative strength using numeric scales with varying intervals. Table 2 shows
the use of diverse symbols, interval and numeric scales within a variety of published QFD articles.
2.3 Critique of previously developed methods for re-ranking roof matrices
In this section, we summarise some of the methods that have been developed and reported in literature by researchers
investigating re-ranking of the roof matrix elements, and identify possible shortcomings of these approaches under
certain circumstances.
2.3.1 Method I
Wasserman (1993) generated a new relationship matrix Rc
ij, where Rc
11 is created by normalising the sum of the products
of first relationship strength (in relationship matrix) of the first row by their corresponding correlations TC. Rc
12 is cre-
ated by normalising the sum of the products of second relationship strength of first row by their corresponding correla-
tions with second TC and so on. Equation (1) is used to generate the new relationship matrix. Ultimately, this new
relationship matrix helps to quantify new FWs. This method has been applied by Chang (2006) in his article on priori-
tising the TCs to enhance nursing home service quality.
Table 1. Summary statistics on the HOQ components used in 75 published case studies.
HOQ rooms used Number of studies Per cent (%) of 75 studies
Only compulsory rooms are used 26 35
+ Competitors of VOCs 18 24
+ Competitors of TCs 6 8
+ Correlations between TCs 36 48
+ Correlations between VOCs 2 3
International Journal of Production Research 3

Rc
ij ¼
Pn
k¼1 RikXCkj
Pn
j¼1
Pn
k¼1 RikXCkj
(1)
Wasserman assumed only positive correlations to generate a normalised relationship. This works for positive correlations
but there are two discrepancies to be considered. Firstly, if a TC has no correlation with others TCs, then row-wise
(VOCs) normalisation is not appropriate. In the illustrated example (Wasserman 1993, 60), the strength of relationships
of the ﬁrst VOC with TCs is 3 and 9, which following normalisation are transformed to .25 and .75, respectively. The
strength of relationship of the fourth VOC with same TCs is 1 and 3, which following normalisation are transformed to
.1 and .9. The point to be noted is that for ﬁrst VOC, strength of 9 is transformed to .75; and for fourth VOC a strength
of 9 converted to .9, which is contradictory. Secondly, Equation (1) may not work if the sum obtained from numerator
of Equation (1) turn out to be negative due to negative correlations.
2.3.2 Method II
Chan and Wu (2005) developed the following expression to re-quantify the VOCs.
Wc
j ¼
Xm
i¼1
WjXCij; i ¼ j ¼ 1; . . .; m (2)
where Wj are the existing FWs, Wc
j are the new FWs and Cij are correlations.
Equation (2) provides a linear combination of its correlation with other TCs weighted by the TCs initial FWs. It
overestimates FWs for positive correlations and underestimates them for negative correlations. Application of this
Table 2. Various linguistics–symbolic–numeric scale taken from published cases.
Authors
Linguistics–Symbolic–Numeric
scale Authors
Linguistics–Symbolic
scale
(Shin, Kim, and
Chandra 2002)
Weak Δ 1
(Wang, Xie, and Goh 1998)
Strong ●
Moderate O 3 Moderate ○
Strong ʘ 9 Weak Δ
(Bouchereau and
Rowlands 2000)
Strong +ve ʘ 9
(Hochman and O’Connell 1993)
Strong +ve ʘ
Weak +ve O 3 Positive ○
Weak –ve × –3 Negative *
Strong –ve * –9 Strong –ve **
(Pramod et al.
2006)
Weak ● 1
(Pun, Chin, and Lau 2000)
Strong +ve ●
Moderate ▲ 3 Moderate +ve ○
Strong ■ 9 Moderate –ve ×
(Deros et al. 2009) Strong Positive ʘ 9
(Crowe and Cheng 1996; Tsai, Lo,
and Chang 2003)
Strong +ve
Positive 3 Positive +
Negative –3 Negative –
Strong Negative Δ –9 Strong _ve
(Ramanathan and
Yunfeng 2009)
Weak 1
(Ramaswamy, Selladurai, and Gunasekaran
2002)
Very Strong
Medium 3
Strong
Strong 9
Moderate
Weak
(Sakao 2007) Positive + +9 (Thakkar, Deshmukh, and Shastree 2006) Positive ○
Negative – –9 Negative ×
(Liu 2011) Strong 0.8
(Stuart and Tax 1996)
Strong +ve ■
Moderate 0.6
Weak +ve □
Weak 0.2
Strong –ve ●
Weak –ve ○
4 Z. Iqbal et al.

approach together with difficulty of achieving a TC has been implemented by Ip and Jacobs (2006) to reprioritise the
TCs for the games industry. A deficiency of this approach is that the FWs change from most important to least impor-
tant if the weight of negative correlations is greater than the weight of positive correlations.
2.3.3 Method III
Sharma and Rawani (2008) computed new FWs by integrating current normalised FWs with correlations using the fol-
lowing expression
Wc
i ¼ NWi þ
Xm
j¼2
NWið1 þ NWjÞCij; i ¼ 1; . . .; n (3)
where Wc
j are the new FWs and NWj are the existing normalised FWs and Cij are correlations. The developed model
provides a linear combination of positive/negative correlations with FWs. The only problem is that if the TCs have
equal strength of correlation between them, then the FWs should remain the same. But Equation (3) does not provide
the same results of FWs under this condition.
2.3.4 Method IV
Pramod et al. (2006) employed QFD in a case study of a maintenance-intensive automobile service station. In order to
re-rank the estimated FWs of TCs using roof matrix, he added the normalised correlation weights of each TC to the
actual normalised FWs.
New Normalised FW of TC ¼ Percentage Normalised FW of TC
þ Percentage Normalised value of correlated weightage of each TC (4)
The above model uses the correlations of TCs to re-rank FWs. Practically this model adds the per cent proportion of
each TC correlations with others, which fails when one TC has more weight of negative than positive correlations. On
the other hand, if it has only positive correlations than just addition of normalised weight does make a linear change in
FWs.
As the above analysis shows, these methods focus only on re-ranking of FWs. Some make a linear change in FWs
while some do not. Most of the methods assume positive correlations to deal with (but hence ignore) the scenario of
negative correlation. In view of these shortcomings, we present an original distance-based approach. The developed
methodology is based on distances of given correlations from ideal correlations (whether for roof A or for roof B).
3. The Manhattan distance measure methodology
Various distance measurement methods are available in literature, with various applications. For example, Cook’s dis-
tance may help to determine outliers which can affect the accuracy of regression. Statistical distance is used to measure
the distance between two random variables, two probability distributions, or between a true and estimated value. In this
article, we employ the Manhattan Distance Measure (MDM) which measures the absolute distance between two points,
(i.e. d X; Yð Þ ¼
P
x À yj j).
In order to apply this distance measure, we first need to establish a consistent numeric scale. As we identified in
Table 2, researchers use various Linguistic–Symbolic–Numeric scales to represent the roof correlations. In other cases,
only qualitative scales and no numeric scale have been provided. As a first step in establishing the consistent scaling, if
a qualitative-numeric scale is given, we can transform the given scale to values between –1 and 1. If no scale is given,
on the other hand, then we can adopt appropriate correlations values to represent given strength of correlation.
Suppose that on one hand, we have a given correlation matrix, and on the other an ideal matrix consisting only of
value 1 (i.e. perfect correlations). Using the MDM methodology (i.e. distances of given TC correlations from ideal cor-
relations), we can measure both the strength of correlations within the matrices, and strength of correlations within each
TC. These distance measures allow us to re-rank the FW score of TCs, measuring the level of conflict between TCs
(due to negative correlations) and ultimately to reduce the number of TCs/VOCs to find most desirable set of TCs that
need to be addressed.
Before we proceed to develop methods to measure information from the correlation matrix, we have to see whether
or not all TCs are independent. i.e.

R ¼ ri;j
Â Ã
ðn;nÞ
¼
1; i ¼ j
0; otherwise
&
In this situation, FWs remain the same and reduction of TCs based on correlation is not possible.
3.1 Strength of correlation matrices (Distance of matrices)
Let R be a given matrix of pairwise correlations between n TCs.
R ¼ ri;j
Â Ã
ðn;nÞ
¼
1; i ¼ j
À1 ri;j 1; i 6¼ j
&
Let Z be an ideal matrix representing the ideal situation in which all the TCs are perfectly correlated (value 1).
Z ¼ zi;j
Â Ã
ðn;nÞ
; zi;j ¼ 1; 8i; j
Let E be a correlation matrix representing the most undesirable situation for any given matrix, namely of perfect nega-
tive correlation (value –1) between TCs.
E ¼ ei;j
Â Ã
ðn;nÞ
¼
1; i ¼ j
À1; i 6¼ j
&
Using Manhattan Distance, the distance of given correlation matrix R from ideal matrix Z is
d R; Zð Þ ¼
Xn
i¼1
Xn
j¼1
ri;j À zi;j

Using MDM, the distance of undesirable correlation matrix E from ideal matrix Z is.
d E; Zð Þ ¼
Xn
i¼1
Xn
j¼1
ei;j À zi;j

The normalised distance of given correlation matrix is given by
N R; Z; Eð Þ ¼
d R; Zð Þ
d E; Zð Þ
¼
Pn
i¼1
Pn
j¼1 ri;j À zi;j

Pn
i¼1
Pn
j¼1 ei;j À zi;j

; 0 NðR; Z; EÞ 1 (5)
The normalised distance 0 ≤ N(R, Z, E) ≤ 1 will be interpreted as follows: a closer distance to 0 implies that strong posi-
tive correlations exist in the system; a closer distance to 1 implies that strong negative correlations exist; and a distance
closer to 0.5 implies that there may be many independent relationships, and the negative and positive correlations have,
on balance, equal weight.
3.2 Level of conflict between TCs
In roof matrices, positive correlation between two TCs indicates that the TCs are mutually supportive, whereas negative
correlation between two TCs implies that one may adversely affect achievement of other (Han et al. 2001). The overall
level of conflict is indicated by the proportion of negative, positive and zero correlations in the RCM (comprised of
TCs). In our method, we first (step 1) quantify the MDM of positive (+), negative (–) and zero (0) correlations from
ideal correlations (for which the value is 1). Next (step 2), we obtain the overall proportion of +, – and 0 correlation
distances by comparing the total strength of correlation (as in Section 3.1).
To illustrate this approach: let R = ri,j be a given matrix of pairwise correlations, for which we will consider three
possible cases: namely ri,j are all positive; all negative; and all zero correlations between pairs of TCs.
6 Z. Iqbal et al.

R ¼ ri;j
Â Ã
ðn;nÞ
¼
ri;j0; 8i; j
or
ri;j ¼ 0; 8i; j
or
ri;j [ 0; 8i; j
8

:
Based on Manhattan Distance, we measure, separately, the respective distance of +, – and 0 correlations from ideal
correlation (=1)
d R; 1ð Þ ¼
Xn
i¼1
Xn
j¼1
ri;j À 1

Finally, the overall proportion (percentage) of +, – and 0 correlations is quantified using following equation
L ri;j
À Á
¼
d R; 1ð Þ
N R; Z; Eð Þ
Â 100 ¼
Pn
i¼1
Pn
j¼1 ri;j À 1

N R; Z; Eð Þ
Â 100;
ri;j0; 8i; j
or
ri;j ¼ 0; 8i; j
or
ri;j [ 0; 8i; j
8

:
(6)
In the situation, where there is a high percentage of negative correlations between TCs, then practitioners can use this
information as a basis for identifying any highly negatively correlated TC and substituting it with another. Once a possi-
ble substitute TC has been decided, practitioners can repeat the above process to measure the overall level of conflict
remaining between TCs in order to confirm whether the substitute has produced a better overall fit than the previous
TC.
3.3 Re-ranking of FWs
In order to re-rank the FWs based on correlation between TCs, first we find the normalised correlation strength of each
TC using the following steps:
Let R0
j : j ¼ 1. . .n be a column vector of correlations of any TC with other TCs.
R0
j ¼ r0
i;j
h i
ðn;1Þ
¼
1; i ¼ j
À1 ri;j 1; i 6¼ j

Let Z0
be an ideal column vector of correlations, representing the ideal situation in which one given TC is perfectly
correlated with others TCs.
Z0
¼ z0
i;1
h i
ðn;1Þ
; zi;1 ¼ 1; 8i
Let E0
j be a column vector of correlations representing the most undesirable situation in which any given TC is perfect
negative correlation (–1) with others TCs.
E0
j ¼ e0
i;j
h i
ðn;1Þ
¼
1; i ¼ j
À1; i 6¼ j

Using Manhattan Distance, the distance of each TC X0
j ; j ¼ 1; . . .; n correlations from ideal column vector Z0
is given by
d0
j R0
j; Z0

¼
Xn
i¼1
r0
i;j À z0
i;1

Using Manhattan Distance, the distance of extreme column vector E0
from ideal column vector Z0
, for j = 1, …, n is
given by

d0
j E0
j; Z0

¼
Xn
i¼1
e0
i;j À z0
i;1

Finally, the strength of each TC (normalised distance) for j = 1, …, n, can be obtained by
N0
j R0
:;j; Z0
; E0
j

¼
d0
j R0
j; Z0

d0
j E0
j; Z0
¼
Pn
i¼1 r0
i;j À z0
i;1

Pn
i¼1 e0
i;j À z0
i;1

; 0 N0
j R0
:;j; Z0
; E0
j

1 (7)
The normalised distance 0 N0
j R0
j; Z0
; E0
j

1 for each TC can be interpreted as follows: A Distance closer to 0 implies
that the TC has positive strong correlations with other TCs in the system; a distance closer to 1 implies that the TC has
strong negative correlations with other TCs, whereas a distance closer to 0.5 implies that some TCs may have indepen-
dent relationships with others, or that there may exist some negative and some positive correlations.
Using the normalised distance of each TC (Equation (7)), the following expression can be used to re-rank FWs.
Wc
j ¼ Wj Á 1 À N0
j

; j ¼ 1; . . .; n (8)
where Wc
j are new FWs, Wj are existing FWs and N0
j are normalised distance of each TC.
3.4 Reducing the TCs
We turn now to our central objective of reducing the number of TCs that need to be focused upon. Based on a given
correlation matrix, selection of which TC(s) to exclude can be considered in two ways.
3.4.1 Method 1
Using the distance measure method, we can ignore any number of the TCs and can select the best reduced group to
use. Let us suppose there are n TCs and k is the number of TCs to be ignored, so there M ¼ Cn
nÀk are the number of
different groups of TCs that can be selected. If k = 1 then M ¼ Cn
nÀ1 ¼ n, i.e. there are n different groups of size
(n – 1), (n – 1) that can be selected, and the normalised distance of each from ideal can be obtained. The matrix of TCs
with the highest normalised distance will be the best set of TCs to meet maximum customer satisfaction and vice versa.
If k = 2 then M ¼ Cn
nÀ2, i.e. there are nðnÀ1Þ
2 different groups of matrices with size (n – 2), (n – 2), again the reduced
matrix of TCs with the highest normalised distance will be the best set of TCs to meet maximum customer satisfaction,
and vice versa. In the similar way, we can exclude three or more TCs and can choose the best TCs to focus efforts
upon.
Generally, if M ¼ Cn
nÀk are the number of different matrices of TCs, from which the best correlated and least corre-
lated groups of TCs can be identiﬁed by the following expression.
Reduced Model ¼ N $
R$
; Z $
; Eð Þ ¼
d RnÀk; ZnÀkð Þ
d EnÀk; ZnÀkð Þ
¼
PnÀk
i¼1
PnÀk
j¼1 Ri;j À Zi;j

PnÀk
i¼1
PnÀk
j¼1 Ei;j À Zi;j

; where k ¼ 1; 2; . . .; n À 2 (9)
It intuitively makes sense that, from various selected groups of TCs, a group with higher normalised distance has lower
degree of intercorrelation, and is therefore ideal for inclusion into the list of TCs to focus efforts upon. But different
groups of TCs have different strengths of distance, so a selected group which has larger distance from an excluded
group are the most appealing for inclusion. This distance can be added as an adjusted distance (Adj distance) measure,
indicating large distance from ignored group, in order to select the most appropriate group for inclusion.
Adj Reduced Model ¼ Reduced Model þ Adj(distance) (10)
where Adj(distance) is the normalised distance of reduce matrix from ignored TC(s).
3.4.2 Method 2
Following the procedure in Section 3.3, we obtain the correlation strength of each TC using the distance measure
(Equation (7)). In order to eliminate one TC, the TC with lowest distance can be considered. To eliminate more TCs,
8 Z. Iqbal et al.

we can begin with the second lowest distance, then the third lowest and so on. Having presented our approach to
prioritising the TCs and reduce TC(s), we now test this method using a case taken from the extant literature. Using the
presented data from the case, we present a reanalysis based on our distance method.
4. Case-study application
The literature review reveals that since the development of QFD, practitioners have employed a variety of qualitative–
quantitative approaches (such as those listed in Table 2) to represent the correlations/interactions in roof matrix. In order
to best demonstrate the potential advantage of our MDM methods, we have selected a literature case which has a bal-
ance of positive, negative and zero correlations. The selected case comes from a well-known article (currently over 100
citations) focused upon improving manufacturing strategic planning written by Crowe and Cheng (1996).
The case organisation, North Press Metals, is a powdered metal product manufacturer company based in Pennsyl-
vania. The management team reported regular pressure of competition for their major clients. As a result, with the
help of top management, the authors firstly identified new market segments and then they developed a manufacturing
strategy plan using QFD. Due to its wider application, later Tsai, Lo, and Chang (2003) used the same case study, to
improve the FWs results by employing a fuzzy QFD model to convert the actual crisp FWs ranking to FWs fuzzy
ranking. We have selected this published case for two main reasons: firstly, the TC correlation matrix exhibited a bal-
ance of positive, negative and zero correlations; and secondly, the original authors did not integrate the roof matrix
into their selection of TCs. Although Crowe and Cheng suggested the selection of TCs which are positively corre-
lated, no significant attention was given to roof correlation matrix. Table 3 shows the pairwise correlations between
TCs for this case.
Table 4 shows normalised FWs and their ranking (without correlations effect), initially computed in the case study.
4.1 Strength of correlations matrix (Distance of matrix)
Using Equation (5), the strength of correlations between TCs is calculated to be = 0.47. This shows that the correlation
matrix exhibits neither strong positive nor strong negative correlation in an overall sense. In other words, the few strong
positive and negative correlations that are present in Table 3 are counterbalanced by the many low correlations that also
appear.
4.2 Level of conflict between TCs
The proportions (in percentage) of positive, negative and zero correlations were computed for the case study. Following
the procedure described in Section 3.2 and finally using Equation (6), we quantify the proportion of positive, negative
and no correlations which are 0.05, 0.11 and 0.31, respectively.
Table 3. Pairwise correlations between TCs (Crowe and Cheng 1996; Tsai, Lo, and Chang 2003).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 1 −0.9 0.5 0.5 0 0 0.9 0 0 0 0 0.5 0 0 0
2 −0.9 1 0 −0.5 0.5 0 0.9 −0.5 0 0.5 0 0 0 −0.5 0.5
3 0.5 0 1 −0.5 0 0 0 0 0 0.5 0 0 0 0.9 0
4 0.5 −0.5 −0.5 1 0.5 0 0 0.5 −0.5 0 0 0.5 0 0 0
5 0 0.5 0 0.5 1 0 0 0 0 0.5 −0.5 −0.5 0 0 0
6 0 0 0 0 0 1 0.5 −0.5 0 0 0.5 0 0.9 0 0.5
7 0.9 0.9 0 0 0 0.5 1 −0.9 0.5 0 0 0.5 0.5 0 0
8 0 −0.5 0 0.5 0 −0.5 −0.9 1 −0.5 0 0 −0.5 −0.5 0 0
9 0 0 0 −0.5 0 0 0.5 −0.5 1 −0.5 −0.9 0.9 0 0 0
10 0 0.5 0.5 0 0.5 0 0 0 −0.5 1 0 0 0.5 0 0
11 0 0 0 0 −0.5 0.5 0 0 −0.9 0 1 0 0 0 0.5
12 0.5 0 0 0.5 −0.5 0 0.5 −0.5 0.9 0 0 1 0 0 0
13 0 0 0 0 0 0.9 0.5 −0.5 0 0.5 0 0 1 0 0.5
14 0 −0.5 0.9 0 0 0 0 0 0 0 0 0 0 1 0
15 0 0.5 0 0 0 0.5 0 0 0 0 0.5 0 0.5 0 1

Table 4. Case study FWs and their ranking.
# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Normalised FWs 8.7 8.1 3.5 3.5 5.5 9.5 8.2 6.3 8.9 6.9 6.5 3.9 10.3 2.4 7.9
Ranking 4 6 14 13 11 2 5 10 3 8 9 12 1 15 7
Table 5. Proportion and percentage of correlations in the matrix.
Positive correlations Negative correlations No correlations
Proportion 0.05 0.11 0.31
Percentage 10.64% 24.01% 65.86%
Table 6. Re-ranking of FWs by using correlations.
# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
New Normalised FWs 9.04 7.65 3.64 3.44 5.40 10.23 9.30 4.76 7.74 7.17 6.02 3.69 11.09 2.31 8.51
New Ranking 4 7 13 14 10 2 3 11 6 8 9 12 1 15 5
Table 7. Reduced model and Adj reduced model for group of 12 TCs (for each TC elimination).
Eliminated TC Reduced model Adj (distance) Adj reduced model
1 0.476373626 0.215384615 0.6918
2 0.468131868 0.253846154 0.7220
3 0.468131868 0.212637363 0.6808
4 0.470879121 0.231868132 0.7027
5 0.470879121 0.20989011 0.6808
6 0.478571429 0.201648352 0.6802
7 0.484065934 0.242307692 0.7264
8 0.452197802 0.264285714 0.7165
9 0.462637363 0.264285714 0.7261
10 0.476373626 0.201648352 0.6780
11 0.465934066 0.212087912 0.6780
12 0.475824176 0.215384615 0.6912
13 0.478571429 0.215384615 0.6940
14 0.508791209 0.193406593 0.7022
15 0.479120879 0.231868132 0.7110
Note: TCs that can be eliminated are highlighted in bold type.
Table 8. Strength of each TC correlations (normalised distance).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.45 0.50 0.45 0.48 0.48 0.43 0.40 0.60 0.54 0.45 0.51 0.45 0.43 0.49 0.43
Table 9. FWs of TCs in ascending order.
7 15 6 13 1 10 3 12 4 5 14 2 11 9 8
0.40 0.43 0.43 0.43 0.45 0.45 0.45 0.45 0.48 0.48 0.49 0.50 0.51 0.54 0.60
10 Z. Iqbal et al.

4.3 Re-ranking of FWs score
Using the normalised FWs of TCs (shown in Table 4) and the strength of each TC correlations (these are presented later
in Table 8) through Equation (8), we obtained new normalised FWs and new ranking as shown in Table 6.
4.4 Reducing the TCs
We now apply the two methods presented in Section 3.4 to decide which TCs can be dropped from the analysis.
4.4.1 Method 1
Table 7 shows the results of Reduced Model analysis (Equation (9)), which shows the group of TCs obtained after
eliminating TC7 is optimal, and Adj Reduced Model (Equation (10)) also suggest the same group (the same procedure
using Equation (10) can be extended to eliminate two or more TCs).
4.4.2 Method 2
In order to reduce the number of TCs, we now determine the strength of each (Table 8) using Equation (7), and then
arrange them in ascending order (Table 9) to observe which TC has highest priority.
Method 2 results conclude that TC7 – having lowest distance from ideal correlations – is the best TC to eliminate
from the system, and the next best to eliminate is TC15. For the given case study, both methods suggest TC7 (‘Focus
on small orders’) has the largest correlation/interaction and may be eliminated.
4.5 Interpretation of results
The overall strength of correlation matrix was 0.47 (Section 4.1). The level of conflict measures (Table 5) further sup-
port the conclusion of low overall correlation, as there is only 10% proportion of positive correlations and 24% propor-
tion of negative correlations. This leads to the conclusion that most of TCs have no relation with each other. In
Section 4.3, re-ranking of FWs obtained demonstrates by comparing initially computed FWs that they been changed
substantially. It is observed by assuming various combinations of correlations between TCs that the method provides a
valid new set of FWs for the cases, where all TCs are independent or negatively correlated. In Section 4.4, both meth-
ods suggest elimination of TC7 will have the minimum effect on the system. In order to quantify results for the devel-
oped expressions, we used the statistical programming language ‘R’, in conjunction with MS Excel.
5. Discussion
In Section 2, we discussed four different methods to incorporate the RMCs into FWs of TCs, and the potential short-
comings of each under certain circumstances. Our method of re-ranking the FWs based on MDM, we believe, not only
overcomes these shortcomings, but also extracts other information from roof matrix as well which results in a consistent
methodology for selecting a parsimonious set of TCs on which to focus efforts, without a significant loss of customer
quality.
In relation to the case presented and analysed in Section 4, there is no evidence in the original article that the roof
correlations were integrated in any formalised or mathematical sense into the selection of TCs. The result is that all TCs
become incorporated into the final list of TCs requiring attention by the design team. This implies that resources will be
directed towards the achievement of each of the 15 resulting TCs. The method in this paper has identified TC7 as a TC
that could be potentially eliminated from the list without significant loss of overall quality, because of high intercorrela-
tion with other TCs. This means that resources can be directed towards the remaining 14 TCs. Applying the methodol-
ogy, it is therefore possible to eliminate further TCs from the list, identifying at each stage the most parsimonious, or
efficient, set of TCs to focus efforts upon. This can be valuable where resources are constrained within the organisa-
tions. In Appendix 1, we illustrate the application of the method with reference to two further and recently published
examples.
6. Conclusion
In this article, we explored the roof matrices of the HOQ, showing that the correlations between TCs have been
neglected in many case studies, and that ignoring any element from the HOQ that can potentially change the FWs of

TCs, and may result in inappropriate prioritisation of the TCs. It is revealed that researchers use a variety of symbolic–
linguistic–numeric scales in their case studies, and we also explored the different methods that have been developed to
integrate correlations with the FWs of TCs, and discussed limitations and weaknesses. Using the principle of MDM,
and the information contained within the ‘roof’ matrix showing correlation between TCs, we have proposed new meth-
ods which help not only to integrate those correlations into the FWs of TCs, but also help to measure level of conflict;
and have developed an associated methodology for reducing the dimensionality of the TCs. The practical result is that
less resource can be expended while still achieving a desirable QFD outcome from the point of view of customer and
technical requirements.
7. Limitations and further research
RMCs are an integral part of the recommended QFD process. To ignore or overlook, the data contained within these
matrices will therefore lead to TC rankings that do not reflect the full information contained within the HOQ matrices.
However, while we do advocate that RMCs should be integrated into FWs where possible, we are also aware that in
many cases pragmatic considerations may override mathematical or theoretical ones. Relevant practical considerations
can include: the effort or cost required to tackle or maximise a TC (relative to the benefits accrued); time constraints;
convenience and other inherent difficulties which will differ for every application of QFD (Wasserman 1993). In further
research based on RMC or other aspects of the HOQ, researchers might wish to integrate such practical, contextual
parameters in determining better models for prioritisation of FWs of TCs. Such research might involve empirical case
studies where mathematically derived approaches such as are presented herein are compared, on a cost-benefit basis,
with pragmatic approaches and solutions.
Disclosure statement
No potential conflict of interest was reported by the authors.
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Appendix 1
This appendix presents the application of the method developed to two more published cases. In both cases, using the distance-based
method proposed in the present paper, we are able to re-rank the TCs by extracting more information from the correlations contained
within the roof matrix, resulting in a new ranking. We are also able to suggest elimination of certain TCs which are highly correlated
with others. This high degree of intercorrelation implies that expending effort towards achieving these TCs may be unnecessary activ-
ity, since the achievement of other TCs will contribute towards these through the systemic association that exists between them.
Case 2
Consider the QFD context discussed in Chang (2012) who prioritised the TCs for the manufacturing ﬂexibility requirements of a food
company in Taiwan. In order to integrate the roof correlations with the FWs, the method developed by Chan and Wu (2005) was used
to integrate correlations with FWs. In the present article, we describe (Section 2.3) the shortcomings of this method.
Table A1 (from Chang 2012) shows that a large proportion of the TCs are positively correlated.
Table A2 shows the reported case study ranking of TCs, ignoring (ﬁrstly) the RMCs, while Table A3 shows the author’s
re-ranking of the FWs (following the Chan and Wu 2005) method, and taking into account the RMCs.
Table A4 shows the FWs and their rankings after applying the distance-based method proposed by the present authors. A clear
difference is observed in the obtained ranking of TCs.
Table A5 reveals that high percentage (87%) of the correlations are positive, with the balance being negligible.
Following our methods (Section 3.4, Method 1 and Method 2), we obtain Tables A6 and A7 which suggest the possibility of
eliminating the TC labelled as B.
Table A1. Pairwise correlations between TCs (case 2).
A B C D E F G H
A 1 0.3 0.1 0.3 0.9 0.3 0.9 0.3
B 0.3 1 0.9 0 0.9 0.9 0.1 0.3
C 0.1 0.9 1 0 0.3 0.3 0 0
D 0.3 0 0 1 0.3 0.3 0.3 0.9
E 0.9 0.9 0.3 0.3 1 0.3 0.3 0.3
F 0.3 0.9 0.3 0.3 0.3 1 0 0.3
G 0.9 0.1 0 0.3 0.3 0 1 0.1
H 0.3 0.3 0 0.9 0.3 0.3 0.1 1
Table A2. Ranking of FWs (Case 2).
# A B C D E F G H
Normalised FWs 9.06 10.95 8.45 18.54 14.79 12.72 10.74 14.76
Ranking 7 5 8 1 2 4 6 3
Table A3. Re-ranking of FWs using correlations by the author (Case 2).
# A B C D E F G H
Normalised FWs 11.85 14.36 3.23 19.57 20.65 12.93 2.33 15.08
Ranking 6 4 7 2 1 5 8 3
Table A4. Re-ranking of FWs using the proposed method (Case 2).
# A B C D E F G H
Normalised FWs 9.65 12.01 7.67 17.80 16.07 12.61 9.85 14.33
Ranking 7 5 8 1 2 4 6 3
14 Z. Iqbal et al.

Case 3
We next consider the case study reported in Ahmed and Amagoh (2010). In this case, the authors examined the measures which a
company should take to increase the demand of its products to become a leader in the glass manufacturing market in Central Asia.
Even although a detailed investigation of importance ratings of VoCs and relationship matrix was done, no apparent integration of
roof matrix data was carried out in this case, in the FW and ranking of TCs.
Table A8 gives the pairwise correlations for case 3. This shows a relatively high proportion of zero correlations, together with
some weak and some strong positive correlations.
Table A9 shows the original ranking of TCs, ignoring the RMCs.
After applying our method, the re-ranked FWs are shown in Table A10. The ranking of TC4, TC5 and TC6 changed after adjusting
for the correlations affect. Table A11 shows that there are no negative correlations, but a large number of TCs have zero correlations
with each other.
After applying the proposed method, we obtain Tables A12 and A13, which suggest the possibility of eliminating TC3.
Table A5. Percentage of correlations strength in the matrix (Case 2).
Percentage 87% 0% 23%
Table A6. Reduced model and Adj reduced model after each TC elimination (Case 2).
A 0.215116 0.330233 0.545349
B 0.220833 0.345349 0.566182
C 0.218605 0.295349 0.513953
D 0.222093 0.306977 0.52907
E 0.189535 0.334884 0.524419
F 0.164583 0.28125 0.445833
G 0.203125 0.297674 0.500799
H 0.206977 0.309302 0.516279
Table A7. Strength of each TC correlations (normalised distance, Case 2).
A B C D E F G H
0.28 0.26 0.39 0.35 0.26 0.33 0.38 0.34

Table A10. Re-Ranking of FWs using correlations (Case 3).
# 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Normalised-FWs 15.9 4.23 13.69 3.27 6.06 6.52 6.49 6.55 5.57 6.05 4.69 2.80 5.56 12.62
Re-Ranking 1 12 2 13 7 5 6 4 9 8 11 14 10 3
Table A9. Ranking of FWs (Case 3).
# 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Normalised FWs 15.89 4.21 13.55 3.27 6.07 6.54 6.54 6.54 5.61 6.07 4.67 2.80 5.61 12.62
Ranking 1 12 2 13 7 4 5 6 9 8 11 14 10 3
Table A11. Percentage of correlations strength in the matrix (Case 3).
Percentage 17% 0% 83%
Table A8. Case 2: pairwise correlations between TCs (Case 3).
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 1 0.7 0 0 0.1 0 0 0 0 0 0.5 0 0 0
2 0.7 1 0.7 0 0 0 0 0 0 0 0 0 0 0.7
3 0 0.7 1 0 0.7 0 0 0.7 0 0.7 0 0 0 0
4 0 0 0 1 0 0.1 0 0 0 0 0.5 0.7 0 0
5 0.1 0 0.7 0 1 0 0 0 0 0 0 0 0 0
6 0 0 0 0.1 0 1 0 0 0 0 0.7 0 0 0
7 0 0 0 0 0 0 1 0 0 0 0 0 0 0
8 0 0 0.7 0 0 0 0 1 0.1 0 0 0 0 0.5
9 0 0 0 0 0 0 0 0.1 1 0 0 0 0 0
10 0 0 0.7 0 0 0 0 0 0 1 0 0 0 0
11 0.5 0 0 0.5 0 0.7 0 0 0 0 1 0 0 0
12 0 0 0 0.7 0 0 0 0 0 0 0 1 0 0
13 0 0 0 0 0 0 0 0 0 0 0 0 1 0
14 0 0.7 0 0 0 0 0 0.5 0 0 0 0 0 1
16 Z. Iqbal et al.

Table A13. Strength of each TC correlations (normalised distance) (Case 3).
1 2 3 4 5 6 7 8 9 10 11 12 13 14
0.0700 0.0652 0.0610 0.0700 0.0730 0.0730 0.0778 0.0700 0.0772 0.0736 0.0676 0.0736 0.0778 0.706
Table A12. Reduced model and Adj reduced model after each TC elimination (Case 3).
1 0.460897 0.110577 0.571474
2 0.466026 0.129167 0.595192
3 0.470513 0.145192 0.615705
4 0.460897 0.119551 0.580449
5 0.457692 0.097436 0.555128
6 0.457692 0.104487 0.562179
7 0.452564 0.089103 0.541667
8 0.460897 0.098397 0.559295
9 0.453205 0.090705 0.54391
10 0.457051 0.086218 0.543269
11 0.463462 0.121795 0.585256
12 0.457051 0.105449 0.5625
13 0.452564 0.089103 0.541667
14 0.460256 0.110897 0.571154

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