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an
d
th
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s
ec
o
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d
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is
f
u
zz
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[
1
]
.
Fu
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p
tim
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tec
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n
i
q
u
es
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e
u
s
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f
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b
u
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First
liter
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e
is
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b
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B
ellm
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Z
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ak
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th
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h
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p
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f
f
u
zz
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m
b
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in
[
2
]
.
B
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[
3
]
s
o
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lem
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in
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Z
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r
m
an
n
’
s
m
eth
o
d
.
Das
et
a
l
.
[
4
]
in
tr
o
d
u
ce
d
th
e
n
ew
alg
o
r
it
h
m
with
a
g
r
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m
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p
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b
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o
r
r
ea
l
-
life
p
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b
lem
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.
Nah
ar
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a
l
.
[
5
]
ar
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co
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v
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tin
g
m
u
lti
-
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b
jectiv
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m
o
d
al
in
to
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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I
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2088
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weig
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Sen
[
6
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in
tr
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ce
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a
n
in
ter
m
ed
iate
m
eth
o
d
f
o
r
p
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th
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d
e
v
elo
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m
e
n
t
o
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r
u
r
al
ar
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with
m
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Ack
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liter
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[
7
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p
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a
m
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ely
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u
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ev
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ev
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.
T
h
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in
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u
in
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Sp
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u
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ea
r
p
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a
m
m
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o
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b
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Ah
m
ad
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Ad
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am
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[
8
]
.
I
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ay
b
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d
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id
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in
to
th
r
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ca
teg
o
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ies
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d
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f
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p
ly
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W
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Ar
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A
s
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Ab
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[
9
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o
p
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p
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f
s
tu
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f
o
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E
x
p
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d
in
g
o
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th
is
wo
r
k
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Ash
r
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l
.
[
1
0
]
i
n
tr
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d
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ce
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s
p
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f
u
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en
h
an
ce
th
e
co
n
ce
p
t.
Ash
r
af
et
a
l
.
[
1
1
]
d
ev
elo
p
e
d
s
ev
er
al
ag
g
r
e
g
atio
n
o
p
er
ato
r
s
f
o
r
s
p
h
er
ical
f
u
zz
y
d
o
m
b
i
(
SD)
av
er
ag
in
g
,
an
d
o
r
d
er
ed
a
v
er
ag
i
n
g
.
h
y
b
r
id
av
er
a
g
in
g
,
g
eo
m
etr
ic,
an
d
h
y
b
r
id
g
e
o
m
etr
ic
av
e
r
ag
i
n
g
to
ef
f
icien
tly
tack
le
d
ec
is
io
n
-
m
ak
in
g
p
r
o
b
lem
s
.
T
h
ese
o
p
er
at
o
r
s
in
th
e
c
o
n
tex
t
o
f
f
u
zz
y
s
ets
ar
e
r
e
v
o
lu
tio
n
ar
y
.
As
p
ar
t
o
f
an
au
to
m
ated
s
to
r
ag
e
a
n
d
r
etr
iev
al
s
y
s
tem
s
tech
n
o
lo
g
y
s
elec
tio
n
ch
allen
g
e.
Gar
g
et
a
l
.
[
1
2
]
d
ev
elo
p
ed
an
d
en
h
an
ce
d
im
m
er
s
iv
e
ag
g
r
e
g
atio
n
p
r
o
ce
d
u
r
es
f
o
r
T
-
s
p
h
e
r
ical
f
u
zz
y
s
ets
in
m
u
lti
-
attr
ib
u
te
d
ec
is
i
o
n
-
m
ak
i
n
g
.
Giu
ler
ia
an
d
B
ajaj
[
1
3
]
in
tr
o
d
u
c
ed
th
e
in
n
o
v
ativ
e
co
n
ce
p
t
o
f
T
-
s
p
h
er
ical
f
u
zz
y
g
r
ap
h
s
,
in
clu
d
in
g
t
h
eir
ar
ith
m
etic
o
p
er
atio
n
s
,
an
d
a
p
p
lied
th
em
to
b
u
s
in
ess
lo
g
is
tics
m
an
ag
em
en
t
d
ec
is
io
n
-
m
ak
i
n
g
an
d
s
er
v
ice
r
esto
ass
ess
m
en
t
p
r
o
b
lem
s
Fu
r
th
er
in
g
th
eir
ef
f
o
r
ts
.
Ku
tlu
Gu
n
d
o
g
d
u
an
d
Kah
r
am
an
[
1
4
]
p
io
n
ee
r
e
d
th
e
MU
L
T
I
MO
OR
A
m
eth
o
d
o
lo
g
y
to
s
o
lv
e
p
er
s
o
n
n
el
s
elec
tio
n
p
r
o
b
lem
s
.
T
o
m
ak
e
d
ec
is
io
n
-
m
ak
in
g
ev
en
m
o
r
e
ac
ce
s
s
ib
le.
T
h
is
ex
ten
s
io
n
allo
wed
f
o
r
m
o
r
e
p
r
ec
is
e
an
d
n
u
an
ce
d
d
ec
is
io
n
-
m
a
k
in
g
.
G
ü
n
d
o
ğ
d
u
an
d
Kah
r
am
a
n
[
1
5
]
p
u
s
h
e
d
th
e
lim
its
o
f
th
e
VI
KOR
m
eth
o
d
b
y
in
tr
o
d
u
cin
g
th
e
s
p
h
er
ical
f
u
z
zy
VI
KOR
(
S
F
-
VI
KOR)
ap
p
r
o
ac
h
an
d
s
u
cc
ess
f
u
lly
ap
p
ly
in
g
it
to
s
elec
t
a
war
eh
o
u
s
e
p
lace
m
e
n
t,
d
em
o
n
s
tr
atin
g
its
s
u
p
er
io
r
p
e
r
f
o
r
m
an
c
e
J
in
et
a
l.
[
1
6
]
,
[
1
7
]
in
t
r
o
d
u
ce
d
s
p
h
er
ical
f
u
zz
y
en
tr
o
p
y
to
id
en
tif
y
u
n
k
n
o
wn
c
r
iter
io
n
weig
h
t
in
f
o
r
m
atio
n
an
d
p
r
o
p
o
s
ed
n
e
w
lo
g
ar
ith
m
ic
o
p
er
atio
n
s
o
n
s
p
h
er
ical
f
u
zz
y
s
ets.
L
iu
et
a
l
.
[
1
8
]
,
[
1
9
]
in
tr
o
d
u
ce
d
th
e
L
t
-
SF
Ns
o
p
er
ato
r
,
wh
ich
e
v
alu
ates
lan
g
u
a
g
e
v
alu
e
u
n
d
er
s
tan
d
in
g
am
o
n
g
th
e
p
u
b
lic.
T
h
ey
th
en
d
e
v
elo
p
ed
th
e
L
t
-
SF
weig
h
ted
a
v
er
ag
in
g
o
p
e
r
ato
r
,
in
teg
r
atin
g
lan
g
u
ag
e
ev
alu
atio
n
k
n
o
wled
g
e.
B
u
ild
in
g
o
n
th
ese
co
n
ce
p
ts
,
th
e
au
th
o
r
s
en
h
a
n
c
ed
th
e
T
ODI
M
ap
p
r
o
ac
h
an
d
estab
lis
h
ed
an
MA
B
AC
m
et
h
o
d
o
lo
g
y
b
ased
o
n
L1
-
SF
Na,
a
g
en
er
aliza
tio
n
o
f
p
ictu
r
e
f
u
zz
y
s
ets.
Ullah
et
a
l
.
[
2
0
]
,
[
2
1
]
p
r
o
p
o
s
ed
n
o
v
el
s
im
ilar
ity
m
etr
ics
,
s
u
ch
as
co
s
in
e
s
im
ilar
ity
m
e
asu
r
em
en
ts
,
g
r
e
y
s
im
ilar
ity
m
ea
s
u
r
es,
an
d
s
et
-
th
eo
r
etica
l
s
im
ilar
ity
m
ea
s
u
r
es
ap
p
lied
to
a
co
n
s
tr
u
ctio
n
m
ate
r
ial
id
en
tific
atio
n
p
r
o
b
lem
in
t
h
e
co
n
tex
t o
f
s
p
h
er
ical
f
u
zz
y
s
ets an
d
T
-
s
p
h
er
ical
f
u
zz
y
s
ets.
Z
en
g
et
a
l
.
[
2
2
]
d
e
v
is
ed
a
n
o
v
el
ap
p
r
o
ac
h
f
o
r
h
y
b
r
id
s
p
h
e
r
ical
f
u
zz
y
s
ets
u
s
in
g
r
o
u
g
h
s
et
co
n
ce
p
ts
b
y
im
p
lem
en
tin
g
a
co
v
e
r
in
g
-
b
ased
s
p
h
er
ical
f
u
zz
y
r
o
u
g
h
s
et
(
C
SF
R
S)
m
o
d
el
with
in
th
e
T
OPSIS
f
r
am
ewo
r
k
.
Z
h
eu
g
et
a
l
.
[
2
3
]
p
r
o
p
o
s
ed
an
an
aly
s
is
f
o
r
o
p
tim
izin
g
th
e
c
er
am
ic
f
ib
er
s
u
s
in
g
t
h
e
d
if
f
er
en
tial
m
eth
o
d
.
T
h
e
m
u
lti
Go
al
Fu
zz
y
p
r
o
b
lem
s
wer
e
d
is
cu
s
s
ed
u
s
in
g
elem
en
tar
y
T
r
a
n
s
f
o
r
m
atio
n
b
y
Sh
r
i
v
astav
a
[
2
4
]
.
Pro
f
it
m
ax
im
izatio
n
in
th
e
s
m
all
m
ec
h
an
ical
in
d
u
s
tr
y
d
u
e
to
th
e
ap
p
licatio
n
o
f
L
in
ea
r
Pro
g
r
am
m
in
g
was
ex
p
lo
r
e
d
b
y
J
ain
et
al
.
[
2
5
]
.
T
h
e
p
r
im
ar
y
o
b
jectiv
e
o
f
th
e
r
esear
ch
is
to
ad
d
r
ess
th
e
ch
allen
g
es
in
o
p
tim
izin
g
m
u
lti
-
o
b
jectiv
e
lin
ea
r
p
r
o
b
lem
s
wh
e
n
th
e
d
ata
is
n
o
t
d
eter
m
in
is
tic,
a
c
o
m
m
o
n
s
ce
n
ar
io
in
i
n
d
u
s
tr
i
al,
ec
o
n
o
m
ic,
a
n
d
en
g
in
ee
r
in
g
ap
p
licatio
n
s
.
E
x
i
s
tin
g
ap
p
r
o
ac
h
es
eith
er
lack
ef
f
ec
tiv
e
d
ef
u
zz
if
icatio
n
o
r
o
v
er
s
im
p
lify
f
u
zz
y
p
ar
am
eter
s
,
lead
in
g
to
in
ac
cu
r
ate
o
r
s
u
b
o
p
tim
al
s
o
lu
tio
n
s
.
T
h
e
f
o
llo
win
g
g
ap
was
f
o
u
n
d
an
d
it
h
as
b
ee
n
ad
d
r
ess
ed
in
th
is
wo
r
k
:
i
)
L
ac
k
o
f
r
o
b
u
s
t
d
e
f
u
zz
if
icatio
n
m
e
th
o
d
s
th
at
c
ap
tu
r
e
th
e
n
u
an
ce
o
f
tr
a
p
ez
o
id
al
f
u
zz
y
p
ar
am
eter
s
;
ii
)
I
n
ad
eq
u
ate
in
teg
r
atio
n
o
f
s
tatis
tical
to
o
ls
in
th
e
ch
an
g
e
m
u
ltip
le
o
b
jectiv
es
in
to
th
e
s
in
g
le
o
b
jectiv
e
in
o
p
tim
izatio
n
p
r
o
ce
s
s
;
an
d
iii
)
Ab
s
en
ce
o
f
a
u
n
if
ied
f
r
a
m
ewo
r
k
th
at
c
o
m
b
in
es
u
n
ce
r
tain
ty
m
o
d
elin
g
,
d
ef
u
zz
i
f
icatio
n
,
an
d
m
u
lti
-
o
b
jectiv
e
o
p
tim
izatio
n
.
T
h
is
s
tu
d
y
c
o
n
s
id
er
s
th
e
s
tatis
tical
m
ea
n
ap
p
r
o
ac
h
es
to
h
an
d
le
th
e
f
u
zz
y
m
u
lti
-
o
b
jecti
v
e
lin
ea
r
p
r
o
g
r
a
m
m
in
g
p
r
o
b
lem
.
T
h
e
co
r
e
m
eth
o
d
o
lo
g
y
in
v
o
lv
es a
s
f
o
llo
ws:
a.
Fu
zz
y
r
ep
r
esen
tatio
n
:
Ob
jectiv
e
an
d
co
n
s
tr
ain
ts
o
f
th
e
lin
ea
r
p
r
o
g
r
a
m
m
in
g
p
r
o
b
lem
ar
e
p
r
ep
ar
ed
b
y
u
s
in
g
T
r
ap
ez
o
id
al
f
u
zz
y
n
u
m
b
er
s
b
ec
au
s
e
o
f
its
r
ec
o
g
n
ized
f
o
r
m
at
th
at
m
o
r
e
s
u
cc
ess
f
u
lly
r
ep
r
esen
ts
th
e
v
ag
u
en
ess
an
d
i
m
p
r
ec
is
io
n
p
r
esen
t in
th
e
r
ea
l
-
wo
r
l
d
d
ata
co
m
p
ar
ed
to
cr
is
p
o
r
tr
ian
g
u
la
r
f
u
zz
y
n
u
m
b
er
s
.
b.
Statis
t
ical
m
ea
n
ap
p
r
o
ac
h
:
T
h
e
n
o
v
elty
o
f
th
is
ap
p
r
o
ac
h
lies
in
u
s
in
g
s
tatis
tical
m
ea
n
to
ch
an
g
e
th
e
m
u
lti
-
o
b
jectiv
e
f
u
n
ctio
n
s
in
to
th
e
s
in
g
le
o
b
jectiv
e
f
u
n
ctio
n
in
p
lace
o
f
weig
h
ted
s
u
m
m
eth
o
d
.
c.
Op
tim
izatio
n
tech
n
iq
u
e:
T
h
e
c
lass
ic
al
B
ig
-
M
m
eth
o
d
is
em
p
lo
y
ed
with
th
e
h
elp
o
f
T
o
r
a
S
o
f
twar
e
to
s
o
lv
e
th
is
tr
an
s
f
o
r
m
ed
p
r
o
b
le
m
an
d
o
b
tain
ed
a
n
o
p
tim
al
s
o
lu
tio
n
.
2.
M
E
T
H
O
D
T
h
is
r
esear
ch
d
is
tin
g
u
is
h
es
its
elf
th
r
o
u
g
h
s
ev
er
al
in
n
o
v
ativ
e
co
n
tr
ib
u
tio
n
s
,
th
e
f
ir
s
t
in
teg
r
at
io
n
o
f
th
e
s
tatis
t
ical
m
ea
n
ap
p
r
o
ac
h
with
Yag
er
'
s
r
an
k
in
g
f
u
n
ctio
n
in
th
e
co
n
tex
t
o
f
FMOL
PP
s
in
v
o
lv
in
g
t
r
ap
ez
o
id
a
l
f
u
zz
y
n
u
m
b
er
s
.
T
h
eo
r
etica
l
r
ef
in
em
en
t
o
f
h
o
w
f
u
zz
y
d
ata
is
tr
an
s
lated
in
to
cr
is
p
v
alu
es
-
m
ain
tain
in
g
th
e
in
f
o
r
m
atio
n
al
in
teg
r
ity
o
f
f
u
zz
y
p
ar
am
eter
s
t
h
r
o
u
g
h
o
u
t
th
e
t
r
an
s
f
o
r
m
atio
n
.
A
u
n
if
ied
co
m
p
u
tatio
n
al
f
r
am
ewo
r
k
th
at
s
y
s
tem
atica
lly
h
an
d
les u
n
ce
r
tain
ty
,
f
u
zz
if
ica
tio
n
,
an
d
m
u
lti
-
o
b
jectiv
e
o
p
ti
m
izatio
n
,
wh
ich
ca
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
6
,
Decem
b
e
r
20
25
:
5
7
0
8
-
5
7
1
6
5710
b
e
g
en
er
alize
d
ac
r
o
s
s
v
ar
io
u
s
d
o
m
ain
s
.
T
h
e
m
eth
o
d
o
l
o
g
y
al
lo
ws
f
o
r
m
o
r
e
r
ea
lis
tic
m
o
d
eli
n
g
o
f
u
n
ce
r
tain
ties
th
at
ar
is
e
in
ec
o
n
o
m
ic
p
lan
n
i
n
g
,
p
r
o
d
u
ctio
n
s
ch
ed
u
lin
g
,
an
d
lo
g
is
tics
,
wh
er
e
v
a
g
u
e
h
u
m
an
ju
d
g
m
en
ts
o
f
ten
s
h
ap
e
d
ec
is
io
n
v
ar
ia
b
les.
2
.
1
.
F
uzzy
s
et
L
e
t
X
b
e
a
n
o
n
-
e
m
p
t
y
s
e
t
.
A
f
u
z
z
y
s
e
t
̃
i
n
X
i
s
c
h
a
r
a
c
t
e
r
i
z
e
d
b
y
i
t
s
m
e
m
b
e
r
s
h
i
p
f
u
n
c
t
i
o
n
̃
:
→
[
0
,
1
]
a
n
d
̃
(
)
i
s
i
n
t
e
r
p
r
e
t
e
d
as
t
h
e
d
e
g
r
e
e
o
f
m
e
m
b
e
r
s
h
i
p
o
f
e
l
e
m
e
n
t
x
i
n
f
u
z
z
y
s
e
t
A
f
o
r
e
ac
h
∈
.
2
.
2
.
M
ulti
-
o
bje
ct
iv
e
f
uzzy
lin
ea
r
pro
g
ra
mm
ing
pro
blem
I
f
al
l
t
h
e
p
ar
am
et
er
s
o
f
a
li
n
ea
r
p
r
o
g
r
a
m
m
i
n
g
p
r
o
b
l
em
(
L
PP
)
ar
e
p
r
ese
n
t
e
d
i
n
t
er
m
s
o
f
v
a
g
u
e
n
ess
i.
e
.
f
u
zz
y
n
u
m
b
e
r
s
t
h
en
o
u
r
L
P
p
r
o
b
le
m
is
id
en
ti
f
ie
d
as
a
f
u
zz
y
lin
ea
r
p
r
o
g
r
a
m
m
i
n
g
p
r
o
b
l
em
(
FLPP)
a
n
d
if
F
L
P
p
r
o
b
l
em
c
o
n
s
is
ts
o
f
m
o
r
e
t
h
an
o
n
e
o
b
j
ec
t
iv
e
f
o
r
a
p
a
r
ti
cu
la
r
m
o
d
al
t
h
e
n
it
i
s
m
o
d
al
n
a
m
el
y
k
n
o
w
as
a
m
u
lti
-
o
b
je
cti
v
e
f
u
zz
y
li
n
ea
r
p
r
o
g
r
a
m
m
i
n
g
p
r
o
b
le
m
.
I
n
o
u
r
s
tu
d
y
,
f
u
z
zy
n
u
m
b
e
r
s
a
r
e
ass
i
g
n
ed
as
̃
,
̃
.
He
r
e
we
co
n
s
i
d
e
r
MO
F
L
PP
as
=
∑
̃
=
1
∀
∈
Su
b
je
ct
t
o
∑
̃
≤
̃
=
1
1
≤
≤
∃
>
0
2
.
3
.
F
uzzy
t
ra
pezo
ida
l num
ber
L
et
,
,
,
ar
e
r
ea
l
n
u
m
b
e
r
s
th
e
n
i
f
t
h
ese
n
u
m
b
er
s
c
a
n
b
e
a
r
r
a
n
g
e
d
i
n
th
e
f
o
ll
o
w
in
g
m
an
n
er
̃
=
(
,
,
,
)
T
h
en
̃
=
(
,
,
,
)
t
h
is
f
u
z
zy
n
u
m
b
e
r
is
k
n
o
wn
as
f
u
zz
y
t
r
a
p
e
zo
id
al
n
u
m
b
er
s
i
f
its
m
em
b
e
r
s
h
i
p
f
u
n
ct
io
n
ca
n
b
e
r
e
p
r
ese
n
t
ed
b
y
t
h
e
f
u
n
c
tio
n
̃
(
)
=
{
−
(
−
)
−
≤
≤
1
≤
≤
(
+
)
−
≤
≤
+
0
No
w
h
er
e
we
p
r
esen
t
th
e
a
r
ith
m
etic
o
p
er
atio
n
f
o
r
f
u
zz
y
tr
a
p
ez
o
id
al
n
u
m
b
er
s
as
f
o
llo
ws
.
L
e
t
̃
(
1
,
1
,
1
,
1
)
a
n
d
̃
=
(
2
,
2
,
2
,
2
)
a
r
e
r
e
p
r
e
s
e
n
t
i
n
g
tw
o
f
u
z
z
y
t
r
a
p
e
z
o
i
d
a
l
n
u
m
b
e
r
s
d
e
f
i
n
e
>
0
,
∈
:
̃
=
(
1
,
1
,
1
,
1
)
<
0
,
∈
:
̃
=
(
1
,
1
,
−
1
,
−
1
)
̃
+
̃
=
(
1
+
2
,
1
+
2
,
1
+
2
,
1
+
2
)
2
.
4
.
Ra
n
k
ing
f
un
ct
io
n
L
et
ℛ
is
a
f
u
n
ct
io
n
wh
ic
h
m
ath
ev
er
y
f
u
zz
y
n
u
m
b
er
(
ℛ
)
in
t
h
e
r
ea
l
lin
e
i.e
.
ℛ
:
(
ℛ
)
→
ℛ
.
No
w
h
er
e
we
p
r
esen
tin
g
th
e
o
r
d
er
o
n
(
ℛ
)
as f
o
llo
ws
̃
≤
̃
⇔
ℛ
(
̃
)
≤
ℛ
(
̃
)
,
̃
≥
̃
⇔
ℛ
(
̃
)
>
ℛ
(
̃
)
̃
≃
̃
⇔
ℛ
(
̃
)
=
ℛ
(
̃
)
,
̃
≤
̃
⇔
̃
≥
̃
W
h
er
e
̃
an
d
̃
b
elo
n
g
in
(
ℛ
)
.
Her
e
we
s
p
ec
ially
f
o
cu
s
ed
o
n
o
n
ly
ab
o
u
t
lin
ea
r
r
a
n
k
in
g
f
u
n
c
tio
n
i.e
.
a
r
an
k
in
g
f
u
n
ctio
n
ℛ
d
ef
in
e
s
u
ch
th
at
ℛ
(
̃
+
̃
)
=
ℛ
(
̃
)
+
ℛ
(
̃
)
∀
̃
,
̃
∈
(
ℛ
)
No
w
tak
in
g
th
e
(
)
as lin
ea
r
r
an
k
in
g
f
u
n
ctio
n
as f
o
llo
ws:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
C
o
mp
u
ta
tio
n
a
l m
o
d
ellin
g
u
n
d
er u
n
ce
r
ta
in
ty:
s
ta
tis
tica
l m
ea
n
a
p
p
r
o
a
ch
to
…
(
A
r
ti S
h
r
iva
s
ta
va
)
5711
ℛ
(
̃
)
=
1
2
∫
(
̃
+
s
up
̃
)
1
0
W
h
ich
r
ed
u
ce
to
ℛ
(
̃
)
=
1
2
(
+
)
+
1
4
(
−
)
T
h
en
f
o
r
f
u
zz
y
tr
ap
ez
o
id
al
n
u
m
b
er
̃
=
(
1
,
1
,
1
,
1
)
an
d
̃
=
(
2
,
2
,
2
,
2
)
W
e
h
av
e
̃
≥
̃
⇔
1
2
(
1
+
1
)
+
1
4
(
1
−
1
)
≥
1
2
(
2
+
2
)
+
1
4
(
2
−
2
)
2
.
5
.
Arit
hm
e
t
ic
o
pera
t
io
n
I
n
th
is
s
u
b
-
s
ec
tio
n
,
we
ar
e
g
o
in
g
to
p
r
esen
t
th
e
o
p
er
atio
n
p
r
o
ce
d
u
r
es
f
o
r
th
e
ad
d
itio
n
an
d
m
u
ltip
licatio
n
o
f
two
f
u
zz
y
tr
a
p
ez
o
id
al
n
u
m
b
er
s
.
L
et
̃
=
(
1
,
1
,
1
,
1
)
an
d
̃
=
(
2
,
2
,
2
,
2
)
b
e
two
tr
ap
ez
o
id
al
f
u
zz
y
n
u
m
b
er
s
th
en
̃
+
̃
=
(
1
,
1
,
1
,
1
)
+
(
2
,
2
,
2
,
2
)
=
(
1
+
2
,
1
+
2
,
1
+
2
,
1
+
2
)
̃
=
−
(
1
,
1
,
1
,
1
)
=
(
−
1
,
−
1
,
1
,
1
)
I
f
̃
≥
0
a
n
d
̃
≥
0
t
h
e
n
,
̃
×
̃
=
(
1
,
1
,
1
,
1
)
+
(
2
,
2
,
2
,
2
)
=
(
1
2
,
1
2
,
1
2
,
1
2
)
2
.
6
.
Cha
nd
ra
Sen’s m
et
ho
d
T
h
e
s
tep
s
in
v
o
lv
ed
in
th
e
alg
o
r
ith
m
[
6
]
ar
e:
−
Ap
p
ly
th
e
B
ig
M
Me
th
o
d
an
d
d
eter
m
in
e
th
e
o
p
tim
u
m
s
o
lu
tio
n
f
o
r
ev
er
y
o
b
jectiv
e
f
u
n
ctio
n
.
−
L
et
Ma
x
=
χ
k
,
wh
er
e
k
=
1,
2,
3
,
…
,
g
,
=
χ
k
,
wh
er
e
=
+
1
,
+
2
…
ℎ
.
−
C
a
l
c
u
l
a
t
e
B
1
a
n
d
B
2
,
w
h
e
r
e
B
1
=
m
a
x
(
|
|
,
w
h
e
r
e
k
=
1,
2,
3
,
…
,
g
,
B
2
=
m
i
n
|
|
,
w
h
e
r
e
K
=
g
+
1
,
g
+
2
,
…
,
h.
−
C
alcu
late
th
e
v
alu
e
o
f
th
e
m
ea
n
b
y
d
if
f
er
e
n
t m
ea
n
m
et
h
o
d
s
.
a.
A
r
ith
m
etic
m
ea
n
m
eth
o
d
Ar
ith
m
etic
m
ea
n
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
,
Ar
ith
m
etic
m
ea
n
b
y
av
er
ag
e
.
.
=
1
+
2
2
b.
Q
u
ad
r
atic
m
ea
n
m
et
h
o
d
Qu
ad
r
atic
m
ea
n
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
,
Qu
ad
r
atic
m
ea
n
b
y
a
v
er
ag
e
=
√
(
1
2
+
2
2
)
2
c.
G
eo
m
etr
ic
m
ea
n
m
et
h
o
d
Geo
m
etr
ic
m
ea
n
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
,
Geo
m
etr
ic
m
ea
n
b
y
av
er
ag
e
=
√
1
×
2
d.
H
ar
m
o
n
ic
m
ea
n
m
eth
o
d
Har
m
o
n
ic
m
ea
n
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
,
Har
m
o
n
ic
m
ea
n
b
y
a
v
er
ag
e
=
2
1
1
+
1
2
e.
H
er
o
n
ian
m
ea
n
m
eth
o
d
Her
o
n
ian
m
ea
n
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
,
Her
o
n
ian
m
ea
n
b
y
av
er
a
g
e
.
=
1
3
(
1
+
√
1
×
2
+
2
)
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
I
n
th
is
s
tu
d
y
,
we
co
n
s
id
er
a
m
u
lti
-
o
b
jectiv
e
f
u
zz
y
lin
ea
r
p
r
o
g
r
am
m
in
g
p
r
o
b
lem
with
f
o
u
r
o
b
jectiv
es
an
d
s
ix
co
n
s
tr
ain
ts
with
tr
ap
ez
o
id
al
f
u
zz
y
n
u
m
b
er
s
.
Ou
r
p
r
o
b
lem
s
ar
e
as f
o
llo
ws
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
6
,
Decem
b
e
r
20
25
:
5
7
0
8
-
5
7
1
6
5712
Mu
ltip
le
o
b
jectiv
e
1
=
(
3
,
8
,
11
,
13
)
1
+
(
3
,
6
,
8
,
10
)
2
+
(
3
,
8
,
11
,
13
)
3
+
(
3
,
4
,
5
,
7
)
4
2
=
(
8
,
9
,
12
,
14
)
1
+
(
7
,
8
,
10
,
12
)
2
+
(
3
,
8
,
11
,
13
)
3
+
(
3
,
4
,
5
,
7
)
4
3
=
(
4
,
8
,
12
,
16
)
1
+
(
9
,
13
,
17
,
21
)
2
+
(
7
,
8
,
10
,
12
)
3
+
(
8
,
9
,
12
,
14
)
4
4
=
(
9
,
11
,
12
,
28
)
1
+
(
13
,
15
,
16
,
32
)
2
+
(
10
,
12
,
13
,
27
)
3
+
(
7
,
9
,
10
,
26
)
4
Su
b
ject
to
co
n
s
tr
ain
t
(
3
,
5
,
6
,
22
)
1
+
(
5
,
7
,
8
,
24
)
2
+
(
4
,
6
,
7
,
23
)
3
+
(
6
,
8
,
9
,
25
)
4
≤
271
.
75
(
5
,
7
,
8
,
24
)
1
+
(
6
,
8
,
9
,
25
)
2
+
(
7
,
9
,
10
,
26
)
3
+
(
10
,
12
,
13
,
27
)
4
≤
411
.
75
(
8
,
10
,
11
,
27
)
1
+
(
8
,
10
,
11
,
27
)
2
+
(
8
,
10
,
11
,
27
)
3
+
(
12
,
14
,
15
,
31
)
4
≤
573
.
75
(
6
,
8
,
9
,
25
)
1
+
(
9
,
11
,
12
,
28
)
2
+
(
9
,
11
,
12
,
28
)
3
+
(
8
,
10
,
11
,
27
)
4
≤
385
.
5
(
9
,
11
,
12
,
28
)
1
+
(
13
,
15
,
16
,
32
)
2
+
(
12
,
14
,
15
,
31
)
3
+
(
9
,
11
,
12
,
28
)
4
≤
539
.
5
(
11
,
13
,
14
,
30
)
1
+
(
14
,
16
,
17
,
33
)
2
+
(
15
,
17
,
18
,
34
)
3
+
(
13
,
15
,
16
,
32
)
4
≤
759
.
5
1
,
2
,
3
,
4
≥
0
T
o
s
o
l
v
e
t
h
e
o
b
j
ec
t
iv
e
f
u
n
ct
io
n
th
e
r
a
n
k
i
n
g
f
u
n
cti
o
n
o
f
th
e
tr
ap
ez
o
i
d
al
n
o
.
L
et
̃
=
(
m
1
,
n
1
,
α
1
,
β
1
)
a
n
d
̃
=
(
m
2
,
n
2
,
α
2
,
β
2
)
.
No
w
t
h
e
r
a
n
k
in
g
o
f
t
h
e
T
r
a
p
ez
o
i
d
al
n
o
.
is
(
̃
)
=
1
2
(
+
)
+
1
4
(
−
)
,
T
h
en
(
8
,
9
,
12
,
14
)
=
1
2
(
8
+
9
)
+
1
4
(
14
−
12
)
=
6
.
No
w
f
u
l
ly
f
u
zz
y
li
n
e
ar
p
r
o
g
r
a
m
m
i
n
g
p
r
o
b
le
m
r
e
d
u
c
es
in
t
h
i
s
f
o
r
m
Ob
jectiv
e
f
u
n
ctio
n
s
:
1
=
6
1
+
5
2
+
6
3
+
4
4
2
=
9
1
+
8
2
+
6
3
+
4
4
3
=
7
1
+
12
2
+
8
3
+
9
4
4
=
14
1
+
17
2
+
15
3
+
12
4
Su
b
ject
to
:
8
1
+
10
2
+
9
3
+
11
4
≤
271
.
75
10
1
+
11
2
+
12
3
+
15
4
≤
411
.
75
13
1
+
13
2
+
13
3
+
17
4
≤
573
.
75
11
1
+
14
2
+
14
3
+
13
4
=
385
.
5
14
1
+
18
2
+
17
3
+
14
4
=
539
.
5
16
1
+
19
2
+
20
3
+
18
4
=
759
.
5
1
,
2
,
3
,
4
≥
0
a.
First f
u
zz
y
o
b
jectiv
e
f
u
n
ctio
n
1
=
6
1
+
5
2
+
6
3
+
4
4
Su
b
je
ct
t
o
:
8
1
+
10
2
+
9
3
+
11
4
≤
271
.
75
10
1
+
11
2
+
12
3
+
15
4
≤
411
.
75
13
1
+
13
2
+
13
3
+
17
4
≤
573
.
75
11
1
+
14
2
+
14
3
+
13
4
=
385
.
5
14
1
+
18
2
+
17
3
+
14
4
=
539
.
5
16
1
+
19
2
+
20
3
+
18
4
=
759
.
5
1
,
2
,
3
,
4
≥
0
T
h
e
o
p
tim
ized
v
alu
e
is
1
=
−
25426
.
7
.
b
.
Seco
n
d
f
u
zz
y
o
b
jectiv
e
f
u
n
ctio
n
2
=
9
1
+
8
2
+
6
3
+
4
4
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
C
o
mp
u
ta
tio
n
a
l m
o
d
ellin
g
u
n
d
er u
n
ce
r
ta
in
ty:
s
ta
tis
tica
l m
ea
n
a
p
p
r
o
a
ch
to
…
(
A
r
ti S
h
r
iva
s
ta
va
)
5713
Su
b
je
ct
t
o
:
8
1
+
10
2
+
9
3
+
11
4
≤
271
.
75
10
1
+
11
2
+
12
3
+
15
4
≤
411
.
75
13
1
+
13
2
+
13
3
+
17
4
≤
573
.
75
11
1
+
14
2
+
14
3
+
13
4
=
385
.
5
14
1
+
18
2
+
17
3
+
14
4
=
539
.
5
16
1
+
19
2
+
20
3
+
18
4
=
759
.
5
1
,
2
,
3
,
4
≥
0
T
h
e
o
p
tim
ized
v
alu
e
is
2
=
−
25349
.
4
c.
T
h
ir
d
f
u
zz
y
o
b
jectiv
e
f
u
n
ctio
n
3
=
7
1
+
12
2
+
8
3
+
9
4
Su
b
ject
to
:
8
1
+
10
2
+
9
3
+
11
4
≤
271
.
75
10
1
+
11
2
+
12
3
+
15
4
≤
411
.
75
13
1
+
13
2
+
13
3
+
17
4
≤
573
.
75
11
1
+
14
2
+
14
3
+
13
4
=
385
.
5
14
1
+
18
2
+
17
3
+
14
4
=
539
.
5
16
1
+
19
2
+
20
3
+
18
4
=
759
.
5
1
,
2
,
3
,
4
≥
0
T
h
e
o
p
tim
ized
v
alu
e
is
3
=
25863
.
7
d
.
Fo
u
r
th
f
u
zz
y
o
b
jectiv
e
f
u
n
ctio
n
4
=
14
1
+
17
2
+
15
3
+
12
4
Su
b
je
ct
t
o
:
8
1
+
10
2
+
9
3
+
11
4
≤
271
.
75
10
1
+
11
2
+
12
3
+
15
4
≤
411
.
75
13
1
+
13
2
+
13
3
+
17
4
≤
573
.
75
11
1
+
14
2
+
14
3
+
13
4
=
385
.
5
14
1
+
18
2
+
17
3
+
14
4
=
539
.
5
16
1
+
19
2
+
20
3
+
18
4
=
759
.
5
1
,
2
,
3
,
4
≥
0
T
h
e
o
p
tim
ized
v
alu
e
is
4
=
26095
.
1
T
ab
le
1
s
h
o
ws th
e
i
n
itial tab
le
:
T
ab
le
1
.
I
n
itial
tab
le
O
b
j
e
c
t
i
v
e
s
χ
k
|
|
V
a
l
u
e
o
f
B
1
a
n
d
B
2
1
-
2
5
4
2
6
.
7
2
5
4
2
6
.
7
B
1
=
2
5
4
2
6
.
7
2
-
2
5
3
4
9
.
4
2
5
3
4
9
.
4
3
2
5
8
6
3
.
7
2
5
8
6
3
.
7
B
2
=
2
5
8
6
3
.
7
4
2
6
0
9
5
.
1
2
6
0
9
5
.
1
−
Ar
ith
m
etic
Me
an
=
1
+
2
2
=
25426
.
7
+
25863
.
7
2
=
25645
.
2
−
Q
u
ad
r
atic
Me
an
=
√
(
1
2
+
2
2
)
2
=
√
(
25426
.
7
2
+
25863
.
7
2
)
2
=
25646
.
1
−
Geo
m
etr
ic
Me
an
=
√
1
×
2
=
√
25426
.
7
×
25863
.
7
=
25644
.
3
−
Har
m
o
n
ic
Me
an
=
2
1
1
+
1
2
=
2
1
2
5
4
2
6
.
7
+
1
25863
.
7
=
25643
.
4
−
Her
o
n
ian
Me
an
=
1
3
(
1
+
√
1
×
2
+
2
)
=
25644
.
9
−
Me
an
Dev
iatio
n
=
(
1
+
2
)
−
(
3
+
4
)
=
{
(
6
1
+
5
2
+
6
3
+
4
4
)
+
(
9
1
+
8
2
+
6
3
+
4
4
)
}
−
{
(
7
1
+
12
2
+
8
3
+
9
4
)
+
(
14
1
+
17
2
+
15
3
+
12
4
)
}
No
w
th
e
o
b
jectiv
e
f
u
n
ctio
n
is
co
n
v
er
tin
g
in
th
is
f
o
r
m
.
Ma
x
by
ℎ
=
ℎ
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
6
,
Decem
b
e
r
20
25
:
5
7
0
8
-
5
7
1
6
5714
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
=
(
−
6
1
−
16
2
−
11
3
−
13
4
)
/
25645
.
2
=
(
−
0
.
000234
1
−
0
.
000624
2
−
0
.
000429
3
−
0
.
000507
4
)
Ob
jectiv
e
f
u
n
ctio
n
:
=
(
−
0
.
000234
1
−
0
.
000624
2
−
0
.
000429
3
−
0
.
000507
4
)
Su
b
ject
to
:
8
1
+
10
2
+
9
3
+
11
4
≤
271
.
75
10
1
+
11
2
+
12
3
+
15
4
≤
411
.
75
13
1
+
13
2
+
13
3
+
17
4
≤
573
.
75
11
1
+
14
2
+
14
3
+
13
4
=
385
.
5
14
1
+
18
2
+
17
3
+
14
4
=
539
.
5
16
1
+
19
2
+
20
3
+
18
4
=
759
.
5
1
,
2
,
3
,
4
≥
0
Ma
x
P b
y
ℎ
=
−
25625
.
01
.
Similar
ly
,
Q
u
ad
r
atic
M
ean
:
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
−
Max
P b
y
=
(
−
6
1
−
16
2
−
11
3
−
13
4
)
/
25646
.
1
.
.
=
(
−
0
.
000234
1
−
0
.
000624
2
−
0
.
000429
3
−
0
.
000507
4
)
Geo
m
etr
ic
Me
an
:
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
−
Ma
x
P b
y
=
(
−
6
1
−
16
2
−
11
3
−
13
4
)
/
25644
.
3
.
=
(
−
0
.
000234
1
−
0
.
000624
2
−
0
.
000429
3
−
0
.
000507
4
)
Har
m
o
n
ic
Me
an
:
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
−
Ma
x
P b
y
=
−
6
1
−
16
2
−
11
3
−
13
4
25643
.
.
=
(
−
0
.
000234
1
−
0
.
000624
2
−
0
.
000429
3
−
0
.
000507
4
)
Her
o
n
ian
Me
an
:
=
(
∑
=
1
−
∑
ℎ
=
+
1
)
.
.
−
Ma
x
P b
y
=
(
−
6
1
−
16
2
−
11
3
−
13
4
)
/
25644
.
9
.
=
(
−
0
.
000234
1
−
0
.
000624
2
−
0
.
000429
3
−
0
.
000507
4
)
W
ith
s
am
e
co
n
s
tr
ain
ts
.
Op
tim
ized
v
alu
e
o
f
FF
MO
L
PP
b
y
m
ea
n
a
p
p
r
o
ac
h
es is
−
Ma
x
P b
y
ℎ
=
−
25625
.
01
,
Ma
x
P b
y
=
−
25625
.
01
−
Ma
x
P b
y
=
−
25625
.
01
,
Ma
x
P b
y
=
−
25625
.
01
−
Ma
x
P b
y
=
−
25625
.
01
4.
CO
NCLU
SI
O
N
Fu
zz
y
m
u
lti
-
o
b
jectiv
e
lin
ea
r
p
r
o
g
r
a
m
m
in
g
p
r
o
b
lem
s
with
tr
ap
ez
o
id
al
n
u
m
b
e
r
s
p
r
esen
t
r
em
ar
k
ab
le
im
p
r
o
v
em
e
n
t
in
o
p
tim
izin
g
r
e
s
u
lts
wh
en
we
a
r
e
u
s
in
g
t
h
e
p
r
o
p
o
s
ed
s
tatis
tical
m
ea
n
ap
p
r
o
ac
h
m
eth
o
d
s
.
Her
e
we
f
o
u
n
d
th
at
f
r
o
m
o
b
tain
ed
r
esu
lts
ar
e
th
e
s
am
e
f
o
r
m
all
th
e
ap
p
licab
le
co
n
d
itio
n
s
,
with
th
is
o
b
s
er
v
atio
n
ab
ilit
y
o
f
d
ec
is
io
n
-
m
ak
in
g
h
as
b
ee
n
en
h
an
ce
d
,
esp
ec
ially
in
am
b
ig
u
o
u
s
co
n
d
itio
n
s
.
T
h
e
p
r
o
p
o
s
ed
f
r
am
ewo
r
k
d
em
o
n
s
tr
ates
co
n
s
is
ten
t
o
p
tim
al
r
esu
lts
ac
r
o
s
s
d
if
f
er
en
t
s
tati
s
tica
l
m
ea
s
u
r
es,
in
d
icatin
g
its
r
o
b
u
s
tn
ess
an
d
r
eliab
ilit
y
in
h
an
d
l
in
g
v
ag
u
e
n
ess
an
d
im
p
r
ec
is
io
n
in
h
er
en
t
in
r
ea
l
-
wo
r
ld
d
ec
is
io
n
-
m
ak
in
g
s
ce
n
ar
io
s
.
T
h
e
u
s
e
o
f
T
OR
A
s
o
f
twar
e
f
o
r
im
p
lem
en
tatio
n
ad
d
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0
,
re
sp
e
c
ti
v
e
ly
,
a
n
d
o
b
tain
e
d
h
e
r
M
.
Tec
h
in
c
o
m
p
u
ter
sc
ien
c
e
fr
o
m
BUIT,
B
h
o
p
a
l
i
n
2
0
1
2
.
S
h
e
a
lso
re
c
e
iv
e
d
a
P
h
.
D
.
i
n
m
a
th
e
m
a
ti
c
s
fro
m
Ra
b
in
d
ra
n
a
th
Tag
o
re
Un
i
v
e
rsity
,
Bh
o
p
a
l
in
2
0
1
9
.
S
h
e
h
a
s
b
e
e
n
h
o
n
o
re
d
wit
h
t
h
e
S
rij
a
n
Aw
a
r
d
fo
r
b
e
st
tea
c
h
e
r
in
d
isc
re
te
stru
c
tu
re
fro
m
M
P
CS
T
wit
h
2
3
y
e
a
rs
o
f
tea
c
h
in
g
e
x
p
e
rien
c
e
.
S
h
e
i
s
c
u
rre
n
tl
y
a
n
a
ss
o
c
iate
p
r
o
fe
ss
o
r
in
t
h
e
De
p
a
rtme
n
t
o
f
M
a
t
h
e
m
a
ti
c
s
a
t
Ra
b
in
d
ra
n
a
th
Tag
o
r
e
Un
iv
e
rsity
(RNTU)
in
B
h
o
p
a
l.
He
r
re
se
a
rc
h
in
tere
sts
in
c
lu
d
e
m
a
th
e
m
a
ti
c
a
l
m
o
d
e
ll
in
g
,
o
p
e
ra
ti
o
n
re
se
a
rc
h
,
g
ra
p
h
t
h
e
o
ry
,
c
o
m
p
u
tatio
n
a
l
m
a
th
e
m
a
ti
c
s,
a
n
d
d
isc
re
te
m
a
th
e
m
a
ti
c
s,
a
n
d
sh
e
h
a
s
c
o
-
a
u
th
o
re
d
n
u
m
e
ro
u
s
n
a
ti
o
n
a
l
a
n
d
in
ter
n
a
ti
o
n
a
l
jo
u
rn
a
ls
a
n
d
c
o
n
fe
re
n
c
e
p
a
p
e
rs.
S
h
e
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
b
h
a
rti
s
a
x
e
n
a
0
6
0
@g
m
a
il
.
c
o
m
.
Ra
m
a
k
a
n
t
B
h
a
r
d
w
a
j
is
p
re
se
n
tl
y
wo
r
k
i
n
g
a
s
p
r
o
fe
ss
o
r,
De
p
a
rtme
n
t
o
f
M
a
th
e
m
a
ti
c
s,
Am
it
y
U
n
iv
e
rsit
y
Ko
l
k
a
ta
sin
c
e
2
8
Au
g
u
st
2
0
1
9
.
He
is
D.S
c
.
fro
m
APS
Un
iv
e
rsity
[
S
tate
G
o
v
e
rn
m
e
n
t
Un
iv
e
rsit
y
M
P
],
P
h
.
D.
fro
m
Ba
rk
a
tu
ll
a
h
U
n
iv
e
rsit
y
Bh
o
p
a
l
i
n
m
a
th
e
m
a
ti
c
s.
He
h
a
s
wo
r
k
e
d
a
s
d
e
p
u
t
y
d
irec
to
r
(re
se
a
rc
h
a
n
d
d
e
v
e
lo
p
m
e
n
t),
Tec
h
n
o
c
ra
ts
G
ro
u
p
o
f
I
n
stit
u
tes
Bh
o
p
a
l,
S
e
p
2
0
1
4
to
2
7
Au
g
u
st
2
0
1
9
.
He
a
lso
wo
rk
e
d
i
n
TRUBA
G
ro
u
p
Bh
o
p
a
l
fro
m
2
0
0
6
t
o
Au
g
u
st
2
0
1
4
.
He
v
isit
e
d
L
o
n
d
o
n
,
Taiwa
n
,
Th
a
il
a
n
d
,
a
n
d
Vie
tn
a
m
to
d
e
li
v
e
r
lec
tu
re
s
in
in
tern
a
ti
o
n
a
l
c
o
n
fe
re
n
c
e
s.
Twe
lv
e
stu
d
e
n
ts
h
a
v
e
b
e
e
n
a
wa
rd
e
d
P
h
.
D
.
a
n
d
fiv
e
a
re
c
u
rre
n
tl
y
u
n
d
e
r
h
is
g
u
id
a
n
c
e
.
He
h
a
s
p
u
b
l
ish
e
d
2
0
0
re
se
a
rc
h
p
a
p
e
rs
in
i
n
tern
a
ti
o
n
a
l
a
n
d
n
a
ti
o
n
a
l
j
o
u
r
n
a
ls
i
n
c
lu
d
in
g
S
CI,
S
c
o
p
u
s,
a
n
d
o
t
h
e
rs.
F
i
v
e
p
a
ten
ts
a
re
a
lso
p
u
b
li
s
h
e
d
b
y
h
im.
Dr
Bh
a
rd
wa
j
h
a
s
p
u
b
li
sh
e
d
se
v
e
n
b
o
o
k
s
o
n
m
a
th
e
m
a
ti
c
s
(0
1
M
.
Tec
h
.
,
0
3
B
.
E.
,
0
2
p
h
a
rm
a
c
y
,
0
1
re
se
a
rc
h
o
rien
ted
).
He
h
a
s
re
se
a
rc
h
c
o
ll
a
b
o
ra
ti
o
n
with
a
c
a
d
e
m
icia
n
s
fro
m
th
e
USA,
Ca
n
a
d
a
,
T
h
a
il
a
n
d
,
Om
a
n
,
Eg
y
p
t,
Ba
h
ra
i
n
,
a
n
d
Vie
tn
a
m
.
He
h
a
s
c
o
m
p
lete
d
tw
o
re
se
a
rc
h
p
ro
jec
ts
fro
m
M
P
CS
T
Bh
o
p
a
l.
He
is
a
lso
h
a
n
d
li
n
g
th
re
e
c
o
n
su
l
tan
c
ies
fro
m
NG
Os
.
Dr
Bh
a
rd
wa
j
o
rg
a
n
ize
d
1
2
i
n
tern
a
ti
o
n
a
l/
n
a
ti
o
n
a
l
c
o
n
fe
re
n
c
e
s/se
m
in
a
rs
a
s
o
rg
a
n
izi
n
g
se
c
re
tary
.
H
e
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
rk
b
h
a
r
d
wa
j1
0
0
@g
m
a
il
.
c
o
m
.
Adi
ty
a
G
h
o
sh
is
p
re
se
n
t
l
y
wo
r
k
in
g
a
s
a
ss
o
c
iate
p
ro
fe
ss
o
r,
De
p
a
rtme
n
t
o
f
M
a
th
e
m
a
ti
c
s,
Am
it
y
Un
i
v
e
rsity
Ko
lk
a
ta
sin
c
e
1
2
Au
g
u
st
2
0
2
4
.
He
is
P
h
.
D.
fr
o
m
G
a
u
h
a
ti
Un
iv
e
rsity
i
n
m
a
th
e
m
a
ti
c
s.
He
h
a
s
wo
rk
e
d
a
s
a
ss
istan
t
d
irec
to
r
(q
u
a
li
ty
a
ss
u
ra
n
c
e
a
n
d
a
c
c
re
d
it
a
ti
o
n
)
in
Ad
a
m
a
s
Un
iv
e
rsity
,
Ko
l
k
a
ta
a
lo
n
g
wit
h
a
ss
o
c
iate
p
ro
fe
ss
o
r,
m
a
th
e
m
a
ti
c
s.
P
rio
r
to
th
a
t
,
h
e
wa
s
wo
r
k
in
g
a
s
a
n
a
ss
istan
t
p
r
o
fe
ss
o
r
o
f
m
a
th
e
m
a
ti
c
s
in
Ro
y
a
l
G
ro
u
p
o
f
In
stit
u
t
io
n
s,
G
u
wa
h
a
ti
(p
re
se
n
tl
y
k
n
o
w
n
a
s
Ro
y
a
l
G
lo
b
a
l
U
n
iv
e
rsit
y
)
fro
m
1
2
Ju
l
y
2
0
1
0
t
o
2
5
M
a
y
2
0
1
7
.
Un
d
e
r
h
is
g
u
i
d
e
sh
i
p
,
o
n
e
c
a
n
d
i
d
a
te
h
a
s
b
e
e
n
a
wa
rd
e
d
P
h
.
D.
He
h
a
s
1
4
re
se
a
rc
h
p
a
p
e
rs
i
n
d
iffere
n
t
n
a
ti
o
n
a
l
a
n
d
i
n
tern
a
ti
o
n
a
l
jo
u
rn
a
ls
a
n
d
b
o
o
k
c
h
a
p
ters
.
He
c
a
n
b
e
c
o
n
tac
te
d
a
t
e
m
a
il
:
a
a
d
it
y
a
.
g
h
o
sh
0
9
@
g
m
a
il
.
c
o
m
.
S
a
ty
e
n
d
r
a
Na
r
a
y
a
n
is a p
ro
fe
ss
o
r
o
f
a
p
p
li
e
d
c
o
m
p
u
ti
n
g
i
n
th
e
F
a
c
u
lt
y
o
f
Ap
p
li
e
d
S
c
ien
c
e
a
n
d
Tec
h
n
o
l
o
g
y
a
t
S
h
e
ri
d
a
n
I
n
stit
u
te o
f
Tec
h
n
o
lo
g
y
i
n
O
n
tario
,
Ca
n
a
d
a
f
o
r
th
e
p
a
st 2
5
y
e
a
rs (rec
e
n
tl
y
c
ro
ss
-
a
p
p
o
i
n
ted
w
it
h
Alg
o
m
a
Un
iv
e
rsity
,
Bra
m
p
to
n
Ca
m
p
u
s).
He
h
o
ld
s a
P
h
.
D.
fro
m
th
e
Un
iv
e
rsity
o
f
Wate
rlo
o
a
n
d
a
m
a
ste
r’s
d
e
g
re
e
fro
m
t
h
e
Un
iv
e
rsity
o
f
Ca
li
f
o
rn
ia
with
sp
e
c
ializa
ti
o
n
s
in
a
p
p
li
e
d
c
o
m
p
u
ti
n
g
.
S
a
t
y
e
n
d
ra
h
a
s
3
0
+
y
e
a
rs
o
f
tea
c
h
in
g
a
n
d
re
se
a
rc
h
e
x
p
e
rien
c
e
in
t
h
e
field
o
f
a
p
p
li
e
d
c
o
m
p
u
ti
n
g
,
g
e
o
sc
ien
c
e
s,
a
n
d
g
e
o
-
e
n
g
in
e
e
rin
g
,
a
n
d
h
a
s
p
u
b
li
sh
e
d
se
v
e
ra
l
p
a
p
e
rs,
a
rti
c
les
,
a
n
d
re
p
o
rts.
Be
fo
re
jo
i
n
i
n
g
S
h
e
rid
a
n
,
S
a
ty
e
n
d
ra
wo
rk
e
d
i
n
th
e
sc
ien
ti
fic
c
o
m
p
u
ti
n
g
i
n
d
u
str
y
a
n
d
wa
s
in
v
o
lv
e
d
i
n
c
o
ll
a
b
o
ra
ti
v
e
re
se
a
rc
h
in
th
e
a
re
a
o
f
e
n
v
iro
n
m
e
n
tal
g
e
o
p
h
y
sic
s
wit
h
t
h
e
Lam
o
n
t
Earth
Ob
se
rv
a
t
o
ry
,
Un
iv
e
rsity
o
f
Co
l
u
m
b
ia.
He
h
a
s
a
lso
c
o
n
tri
b
u
ted
si
g
n
ifi
c
a
n
tl
y
to
,
a
n
d
e
sta
b
li
sh
e
d
stro
n
g
re
se
a
rc
h
p
a
rtn
e
rsh
i
p
s
with
,
th
e
In
d
ian
I
n
stit
u
te
o
f
S
c
ien
c
e
s,
Ba
n
g
a
lo
re
,
In
d
ia;
Re
g
io
n
a
l
Re
se
a
rc
h
Lab
o
ra
to
r
y
(RRL),
Bh
o
p
a
l,
In
d
ia;
In
stit
u
t
Tek
n
o
l
o
g
i
Ba
n
d
u
n
g
,
Ba
n
d
u
n
g
,
I
n
d
o
n
e
sia
;
M
in
istr
y
o
f
Re
se
a
rc
h
a
n
d
Tec
h
n
o
l
o
g
y
,
In
d
o
n
e
sia
;
P
T
Ep
h
in
d
o
,
Ja
k
a
rta,
In
d
o
n
e
sia
;
a
n
d
th
e
Un
i
v
e
rsity
o
f
Wate
rlo
o
,
On
tari
o
,
Ca
n
a
d
a
.
S
a
ty
e
n
d
ra
is
a
re
c
ip
ien
t
o
f
S
h
e
rid
a
n
’s
b
e
st
tea
c
h
in
g
e
x
c
e
ll
e
n
c
e
a
wa
rd
o
f
m
e
rit
a
n
d
th
e
Lea
d
e
rsh
ip
i
n
F
a
c
u
lt
y
Tea
c
h
i
n
g
Aw
a
rd
in
re
c
o
g
n
it
i
o
n
o
f
h
is
e
x
e
m
p
lary
e
ffo
rts
t
o
i
n
flu
e
n
c
e
,
m
o
ti
v
a
te,
a
n
d
in
s
p
ire
stu
d
e
n
ts.
S
a
ty
e
n
d
ra
h
a
s
b
e
e
n
a
n
a
c
ti
v
e
m
e
m
b
e
r
o
f
v
a
rio
u
s
c
o
m
m
it
tee
s
a
t
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h
e
rid
a
n
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.
g
.
,
stu
d
e
n
t
a
d
v
ise
m
e
n
t
a
n
d
su
c
c
e
ss
a
d
v
is
o
ry
c
o
m
m
it
tee
s,
S
h
e
rid
a
n
c
re
a
ti
v
e
in
d
u
stries
c
o
m
m
it
tee
)
a
n
d
h
a
s
b
e
e
n
a
c
o
o
r
d
in
a
t
o
r
a
n
d
a
d
v
ise
r
fo
r
se
v
e
ra
l
p
ro
g
ra
m
s.
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c
a
n
b
e
c
o
n
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