Information Technology Reference
In-Depth Information
Weighted Matrix:
Binary Matrices:
j
R
=
{
8
,
12
}
R
=
{
2
,
3
,
6
,
6
,
3
}
R
=
{
4
,
12
,
1
.
5
,
2
.
75
}
R
=
{
2
,
12
,
6
,
0
}
i
w
j
1
2
1
4
1
8
1
A
1
=0
.
100
2
X
:
A
1
=0
.
001
2
X
:
X
:
A
1
=0
.
010
2
X
:
A
1
=0
.
001
2
A
2
=0
.
100
2
1
2
3
4
5
A
1
A
2
A
3
A
4
D
A
2
=0
.
011
2
A
2
=0
.
100
2
A
2
=0
.
110
2
16
8
4
6
3
2
3
Q
A
3
=0
.
000
2
A
3
=0
.
110
2
A
3
=0
.
001
2
A
3
=0
.
000
2
24
12
22
0
12
A
4
=0
.
000
2
A
4
=0
.
000
2
D
=0
.
000
2
A
4
=0
.
001
2
D
=0
.
000
2
A
4
=0
.
000
2
D
=0
.
001
2
D
=0
.
000
2
6
1
.
5
2
.
75
Invalid Solution
Valid Solution
Valid Solution
Valid Solution
W
11
5
.
5
d
=3
m
=
d
=1
m
=1
d
=3
m
=3
d
=3
m
=3
−
000
(b)
(c)
(d)
(e)
(a)
Fig. 4.
(a) Weighted matrix shown in Fig. 3(b). (b)-(e) Different binary matrices obtained by
finding
R
from
Q
for
C
A
=
2
32
.
Ta b l e 1 .
Results for Some Example Cases by
GDA
with Optimal
d
or Optimal
m
Optimal
d
Example
B
1
,
...
,
B
N
,0
Optimal
m
d
,m,I
>
1
,
M
lb
d
,m,I
>
1
,
M
lb
T d
[
d
1
,
...
,
d
N
+1
,
T
d
]
[
d
1
,
...
,
d
N
+1
,
T
d
]
Ex.1
7,14,15,0
12
4
[2,2,1,0,T5]
3,4,2,2
[2,0,2,0,T4]
3,3,2,1
Ex.2
1,4,61,0
5
6
[3,2,1,2,T8]
4,7,3,2
[4,0,1,0,T5]
4,4,3,1
Ex.3
3,6,9,12,15,0
8
4
[2,1,1,1,1,0,T6]
3,5,1,2
[2,1,0,1,0,0,T4]
3,3,1,1
Ex.4
4,6,8,9,12,15,0
14
4
[0,0,1,1,1,3,0,T6]
4,5,2,2
[0,1,1,0,0,3,0,T5]
4,4,2,1
Ex.5
1,3,52,0
33
6
[2,1,2,0,T5]
4,4,2,1
[1,1,2,1,T5]
3,4,1,2
Ta b l e 2 .
Results for Some Example Cases by
GDA
with Optimal
I
>
1
or Optimal
M
lb
Example
B
1
,
...
,
B
N
,0
Optimal
I
>
1
Optimal
M
lb
d
,m,I
>
1
,
M
lb
d
,m,I
>
1
,
M
lb
T d
[
d
1
,
...
,
d
N
+1
,
T
d
]
[
d
1
,
...
,
d
N
+1
,
T
d
]
Ex.1
7,14,15,0
12
4
[1,1,1,1,T4]
3,3,0,1
[2,2,0,1,T5]
4,4,2,1
Ex.2
1,4,61,0
5
6
[0,1,1,3,T5]
4,4,0,1
[3,3,1,0,T7]
6,6,4,1
Ex.3
3,6,9,12,15,0
8
4
[1,0,1,1,0,1,T4]
3,3,0,1
[3,1,0,1,0,0,T5]
4,4,2,1
Ex.4
4,6,8,9,12,15,0
14
4
[0,1,1,0,0,3,0,T5]
4,4,2,1
[0,1,1,0,0,3,0,T5]
4,4,2,1
Ex.5
1,3,52,0
33
6
[1,1,2,1,T5]
3,4,1,2
[2,1,2,0,T5]
4,4,2,1
be the total number of mix-split steps in the tree. The demand array [
d
1
,
...
,
d
N
+1
,
T
d
]
denotes that the required number of droplets of fluid
A
i
is
d
i
,and
T
d
denotes the total
number of input droplets required. Let
I
>
1
be the total number of extra (more than one)
droplets of
N
different fluids (except buffer
D
) required as inputs in the tree, and let
M
lb
be the number of mixers required for minimum time completion of the dilution
process. We present the results for several examples in Table 1 with the optimality cri-
terion of minimizing
d
or
m
. Table 2 provides the results with the optimality criterion
of minimizing
I
>
1
and
M
lb
of the dilution/mixing tree.
