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Axiomatic Design 269

h({PV}) ln|[B] [A]|

h({PV}) ln|[C]|
There are two components of complexity design: (1) that due to
variability ( h(DP)) and (2) that due to coupling vulnerability
( ln|A|). The term coupling vulnerability of Theorem 8.1 has a broad-
er meaning than the numerical values of the sensitivity coefficients,
the argument of the design matrix determinant. There are three ingre-
dients in the coupling complexity component: mapping, sensitivity, and
dimension. The mapping ingredient refers to the binary variable Z ij
denoting the mapping process between the functional domain and the
physical domain and is defined as

Z ij 1 if FR i → DP j }
{ 0 elsewhere

In other words, the mapping variable represents the position of the
nonzero sensitivity coefficients in the design matrix A. The sensitivity
ingredient refers to the magnitude and sign of nonzero A ij ∂FR i /∂DP j
coefficients. The dimension ingredient refers to the size of the design
problem: m, the number of FRs and p, the number of DPs in the
squared design matrix. We view our interpretation of the complexity
component due to vulnerability as the mathematical form of the Simon
(1981) complexity definition.
The theme of Theorem 8.1 is that the designer experiences two
complexity components in attaining an FR (in the physical map-
ping) or a DP (in the process mapping) if she or he does not know
how its mapped-to variables vary (the variability component) and
at what scale they vary (the vulnerability component). For an
uncoupled design, the value of |[A]| is the product of the diagonal
p
elements, |[A]| Π i 1 A ii , and the complexity component due to
p
sensitivity is
i 1 ln |A ii |. The total uncoupled design complexity
p
(assuming that all DPs are normal information sources) is
i 1 ln
(2 e i A ii ) nats.
8.7 Case Study: Axiomatic Design
of the Water Faucet*
The water faucet case study, introduced in Fig. 8.4, has been selected
as an illustrative example because first, it is very familiar design to
everyone and second, it illustrates nicely several axiomatic design

*El-Haik (2005).