Fotoenergetikada nanostrukturali yarimo‘tkazgich materiallar II xalqaro ilmiy anjumani

2.3 Employment of the “Intersection” Multi
–
Objective Optimization in the 
Material Selection of RF 
–
 MEMS Shunt Capacitive Switches
The values of the optimal material indices for various materials are given in 
Table 1 [1]. 
Table 1 Optimal material index for various materials [1] 
Mat. 
Young’s 
modulus 
Electrical 
resistive 
c
o
efficient 
Thermal 
conductivity 
Fracture 
strength 
(

f
/
E
) 

10
3
 
E
(GPa) 

e
(Ω m) 
10
−8

(W/m∙K) 

f
(MPa) 
Ni 
193 
6.99 
90 
345 
1.7876 
Au 
70 
2.44 
315 
220 
3.1429 
Al 
70 
2.82 
204 
47 
0.6714 
Ag 
83 
1.59 
407 
110 
1.3253 
Pt 
168 
10.5 
73 
125 
0.7440 
Cu 
117 
1.68 
386 
314 
2.6838 
Cr 
279 
12.9 
90 
370 
1.3262 
W 
411 
5.28 
163 
1725 
4.1971 
Co 
209 
6.24 
69 
675 
3.2297 
Fe 
211 
9.61 
73 
540 
2.5592 
In the new the new multi – object optimization method [4], the favorable 
probability of a material performance index of the beneficial indicator in the 
material selection process is positively correlative to this material index linearly, 
i.e. 
,
1, 2, ..., 
=1, 2, ..., .
,
ij
ij
ij
j
ij
P
X
i
n; j
m
P
X




(1) 
In Eq. (1), 
X
ij
 
is the
 j
th
 
material property index of the
 i
th
 
candidate
 
material;
 P
ij
 
represents the
 
partial favorable probability of the beneficial material property 
indicator 
X
ij
;
n
is the total number of candidate materials in the material group 
involved; 
m
is the total number of material property indicators of each candidate 
material in the group; 

j
is the normalized factor of the 
j
th
material performance 
indicator. 

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Fotoenergetikada nanostrukturali yarimo‘tkazgich materiallar

II xalqaro ilmiy anjumani 
 
19-20 noyabr 2021 yil 
103 
Furthermore, the summation of each 
P
ij
for the index 
i
in 
j
th
material factor is 
normalized equaling to 1 according to the general principle of probability theory 
[4], i.e., 
1
1
n
ij
i
P



, which naturally results in 
.
(2) 
j
X
is the arithmetic mean value of the material performance indicator in the 
material group involved.
Equivalently, the partial favorable probability of the unbeneficial (cost) material 
property indicator 
X
ij
is negatively correlative to its material property indicator 
linearly, i.e. 
max
min
max
min
(
),
1, 2, ..., 
=1, 2, ..., .
(
),
ij
j
j
ij
ij
j
j
j
ij
P
X
X
X
i
n; j
m
P
X
X
X








(3) 
In Eq. (3), 
X
j
max
and 
X
j
min
represent the maximum and minimum values of the 
material property indicator 
X
j
in the material group, respectively; Furthermore, the 
normalized factor of the 
j
th
material performance indicator

j
is
max
min
1
(
)
j
j
j
j
n X
X
n X








. 
(4) 
Moreover, according to basic probability theory [4], the total / comprehensive 
favorable probability of the
i
th
candidate material is the product of its partial 
favorable probability of each material property indicator 
P
ij 
in the overall selection, 
i.e.,
1
2
1
m
i
i
i
im
ij
j
P
P
P
P
P






. 
(5) 
The total favourable probability of a candidate is the unique decisive index in 
the overall selection process competitively. The main characteristic of the new
 
"intersection" method for MOO is that the treatment for both beneficial property 
index and unbeneficial property index is equivalent and conformable without any 
artificial factors or subjective scaling factor. 
The partial favorable probabilities of the indices of 
E
0.5
,
 

and 

e
and 

f
 
/
E
and 
the total favorable probabilities are assessed according the Equations (1) through 
(5), respectively, which are shown in Table 2. In addition, the ranking here by 
using the new multi – object optimization method is given in Table 2 together with 
those of Vikor and Topsis from for comparison [1]. 
It can be seen from Table 2 that the appropriate material from the new multi – 
object optimization method is Cu, which is different from those of Vikor and 
Topsis from [1], this is because of the inherent defects of impersonal and 
subjective factors in Vikor and Topsis [4]. 

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