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function,
∂
f u
/ ' s
. If the amplitudes of un-
certainty, i.e. the dispersions of DVs and DPs
do not change, the dispersion of the performance
function
∆
f
will solely depend on the gradients
∂
f u
/ ' s
. It is obvious that a change in the
dispersion
∆
f
i
(due to the variation of the ith DV
or DP) will be more than
∆
f
j
(due to the varia-
tion of the jth DV or DP) if the associated im-
portance measure
I
i
>
or vice versa. Hence,
it seems to be more logical to use the importance
measure of the associated gradients as well in
defining the measure of robustness. A new mea-
sure of robustness is thus proposed in the present
RDO study by redefining the performance dis-
persion as following:
Integrating the information presented so far, the
proposed RDO scheme can be finally represented
as an equivalent DDO problem as following:
f
f
∆
∆
f
f
minimize:
(
1
−
α
)
+
α
w
*
*
w
such that:
g
+
k
∆
g
≤
0
,
j
=
1 2
,
, ...
...,
m
(29)
j
j
jw
In the above,
k
j
is as defined in Eq. (5). The
above nonlinear optimization problem can be
solved by available optimization techniques. Once
the optimum design point is obtained by the pro-
posed importance factor based RDO approach,
the dispersion of the performance function can
be evaluated at the optimum design point using
Eq. (2). It may be noted that the importance fac-
tors are directly incorporated in the optimization
formulation and such factors are evaluated at
updated design point during each iteration cycle
of the optimization process. Doing so, more sen-
sitive DVs and DPs are automatically got re-
amplified yielding a better robust solution in
lesser computational time. Basically, the incor-
poration of the importance factors changes the
feasible domain of the original optimization
problem and captures the more flat zone of the
performance function. In this regard, it is worth
mentioning that many researchers adopt a separate
sub-problem to find a search direction for a quick
convergence towards the robust optima (Lee, &
Park, 2001; Wang et al., 2009). The search direc-
tion should emphasize on the more sensitive
variables for an efficient and quick convergence
to the robust optima. As the importance factor
based dispersion index includes this aspect of
assigning more importance to more sensitive DVs
and DPs, the proposed RDO procedure con-
verges to robust optima efficiently without requir-
ing any such separate sub-problem for direction
finding.
2
2
N
N
∂
∂
f
u
I
∂
∂
f
u
∂
∂
f
u
∑
1
∑
∆
f
=
∆
u
,
where,
I
=
N
w
fi
i
fi
i
=
i
i
k
=
1
k
(27)
In the above,
∆
f
w
is the new dispersion index
and
I
fi
is the importance factor for the ith DV or
DP. It is to be noted here that the importance fac-
tor as defined above is multiplied by the total
number (N) of the DVs and DPs. However, the
summation of all such factors will be always equal
to N, whatever is the individual value of the im-
portance factor. This will keep the consistency of
the definition of this index to measure the robust-
ness with the usual dispersion index used in the
conventional RDO approach.
The basic idea to improve the robustness of the
performance function using the new index utilizing
the importance factors proportional to the impor-
tance of the gradients of the performance function
can be readily extended to the constraints as well.
A new dispersion index to achieve the robustness
of jth constraint feasibility is defined as:
2
2
∂
∂
g
u
I
∂
∂
g
u
∂
g
N
N
∑
1
∑
u
k
k
j
j
j
∆
g
=
∆
u
,
where
,
I
=
N
jw
g
i
g
∂
ji
ji
i
=
=
1
i
i
(28)
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