Information Technology Reference
In-Depth Information
Related Inputs:
CF
Level
3
CF
20
.
25
32
Level
CF
C
A
=
2
32
A
1
=
16
A
32
A
3
=
12
A
32
19
.
75
32
A
2
=
24
A
32
A
4
=
22
A
32
D
=
0
32
CF
Level
17
.
5
32
23
32
20
32
16
.
5
32
Weighted Matrix:
A
4
A
2
CF
Level
1
2
1
4
1
8
18
32
20
32
17
32
1
1
A
3
A
4
A
1
A
A
1
16
8
4
6
3
2
3
A
A
2
24
12
A
2
A
4
A
2
A
3
A
1
A
2
A
3
A
4
A
A
3
12
6
1
.
5
2
.
75
Output Parameters:
Output Parameters:
Output Parameters:
Output Parameters:
A
A
4
22
0
11
5
.
5
d
m
W
I
(
A
2
,A
3
,A
4
)
d
m
W
I
(
A
2
,A
3
,A
4
)
d
m
W
I
(
A
1
,A
2
)
d
m
W
I
(
A
1
,A
2
,A
3
,A
4
)
A
D
000
(a)
(b)
(c)
(d)
(e)
Fig. 3.
(a) Weighted matrix for input
CF
s. (b)-(e) Four different dilution trees for
C
A
=
2
32
.
3.1
Examples of Generalized Dilution from Related Inputs
An example of producing four related inputs of fluid
A
(
A
1
,A
2
,A
3
and
A
4
)asthe
27
by-products of dilution process for
C
t
=
32
from pure sample and buffer fluids by
twoWayMix
[6] is shown in Fig. 2(b). Consider that we need to produce a target droplet
with
C
A
=
2
32
from the supply of these by-products (related inputs) and buffer solution
D
. Four different dilution trees for target
CF C
A
=
2
32
are shown in Fig. 3(b)-(e).
The best solution (dilution tree) among all these four solutions is shown in Fig. 3(d),
as it takes minimum number of mix-split steps to produce the target droplet. However,
if only three related inputs
A
2
,A
3
and
A
4
are available after using
A
1
is some other
dilution process, then the dilution tree shown in Fig. 3(c) is the best solution, as it has
less
m
and consumes less
I
than the dilution tree shown in Fig. 3(b).
A solution (dilution tree) can be envisaged with the help of some underlying data of
the
weighted matrix
(
) that is computed from the
CF
s of related inputs as shown in
Fig. 3(a). For
A
i
, the portion of
A
contributed to the final concentration follows a geo-
metric progression with a common ratio of
W
1
2
, depending on the level of tree at which
has 24, 12, 6, 3 for
A
2
. In the dilution tree of
Fig. 3(c), the contributions of
A
2
and
A
3
to the target
CF
are
it is used. For example, in Fig. 3(a)
W
6
32
3
32
, respectively,
as they are at depth 2 of the tree, whereas that of
A
4
is
1
32
, as it is at depth 1 of the tree.
This is easy to visualize from the basic principles of dilution. The selection of contribu-
tions from
and
for the target
CF
provides the binary fractions of the constituent fluids
corresponding to a dilution/mixing tree. For example, the binary fractions of
A
2
,
A
3
and
A
4
for the dilution tree shown in Fig. 3(c) can be written as 0
.
01
2
, 0
.
01
2
and 0
.
10
2
,
respectively. A selection of numbers from
W
is
valid
, if a dilution/mixing tree can be
constructed using
Min-Mix
from the binary fractions corresponding to the selection.
W
3.2
Problem Formulation and Proposed Algorithm
First, we represent
b
i
as
B
i
T
2
d
,where
B
i
sand
T
are positive real
numbers. For each input
A
i
,(
d
+1) geometric progression (G.P.) terms with the first
term
B
i
and common ratio
2
, i.e.,
B
i
,
B
2
,
B
4
,
2
d
(for all
i
)and
C
A
as
,
B
i
···
2
d
, are stored in an one-dimensional
array
W
[
i
]. A two-dimensional array
W
of size (
N
+1)
×
(
d
+1)stores the G.P. terms
1
2
j
for all
N
inputs and
D
. Each element
W
[
i,j
] is associated with a weight
w
j
=
