Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
19,
No.
1,
July
2020,
pp.
85
90
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v19i1.pp85-90
r
85
A
systematic
appr
oach
f
or
solving
mixed
constraint
fuzzy
balanced
and
unbalanced
transportation
pr
oblem
Priyanka
A.
P
athade
1
,
Ahmed
A.
Hamoud
2
,
Kirtiwant
P
.
Ghadle
3
1,2,3
Department
of
Mathematics,
Dr
.
Babasaheb
Ambedkar
Marathw
ada
Uni
v
ersity
,
India
2
Department
of
Mathematics,
T
aiz
Uni
v
ersity
,
Y
emen
Article
Inf
o
Article
history:
Recei
v
ed
Jul
16,
2019
Re
vised
Dec
19,
2019
Accepted
Jan
14,
2020
K
eyw
ords:
Best
candidate
method
Fuzzy
ranking
technique
Fuzzy
transportation
problem
Mix
ed
trapezoidal
fuzzy
numbers
T
ri
vial
fuzzy
numbers
ABSTRA
CT
In
present
article
a
mix
ed
type
transpor
tation
problem
is
considered.
Most
of
the
transportation
problems
in
real
life
situation
ha
v
e
mix
ed
type
transportation
problem
this
type
of
transportation
problem
cannot
be
solv
ed
by
usual
methods.
Here
we
attempt
a
ne
w
concept
of
Best
Candidate
Method
(BCM)
to
obtain
the
optimal
solution.
T
o
determine
the
compromise
solution
of
balanced
mix
ed
fuzzy
transportation
problem
and
unbalanced
mix
ed
fuzzy
transportation
problem
of
trapezoidal
and
tri
vial
fuzzy
numbers
with
ne
w
BCM
solution
procedure
has
been
applied.
The
method
is
illustrated
by
numerical
e
xamples.
Copyright
©
2020
Insitute
of
Advanced
Engineeering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Priyanka
A.
P
athade,
Department
of
Mathematics,
Dr
.
Babasaheb
Ambedkar
Marathw
ada
Uni
v
ersity
,
Aurang
abad-431004
(M.S.)
India.
Email:
priyankapathade88@gmail.com
1.
INTR
ODUCTION
The
transportation
problem
w
as
initially
de
v
eloped
by
Hitchcock
[1]
in
1941.
The
transport
ation
problem
w
as
assumed
that
all
the
parameters
is
sure
about
the
e
xact
v
alues.
Precise
nature
of
the
goods
is
v
ery
important
in
the
transport
rout.
But
in
real
life
we
can’
t
represent
e
xactness
or
preciseness.
So
we
ha
v
e
f
ace
imprecise
conditions.
Also
the
situations
are
dif
ferent
in
dif
ferent
conditions
since
v
arious
parameters
not
be
kno
wn
precisely
.
Due
to
uncontrollable
f
actors
we
ag
ain
f
ace
to
mix
ed
type
constaint
problems.
This
type
of
problems
ha
v
e
less
information
in
literature
also.
There
are
no
ef
ficient
methods
to
solv
e
mix
ed
type
problems.
Therefore,
fuzzy
numbers
introduced
by
zadeh
[2].
Zimmerman
[3]
introduced
a
huge
information
about
fuzzy
in
his
fuzzy
set
theory
and
applications.
He
discussed
crispness,
v
agueness,
uncertainty
,
fuzziness
in
his
w
ork
and
suitable
e
xamples
also.
Gani
[4]
solv
ed
a
mix
ed
constraint
transportation
problem
under
fuzzy
en
vironment.
Also
the
y
highlighted
the
steps
by
usi
ng
a
simple
flo
w
chart.
Mandal
and
Hussain
[5]
also
studied
mix
ed
constraint
the
y
described
the
algorithm
to
find
an
optimal
more-for
-less
solution.
Gupta
and
Bari
[6]
determined
the
multi
objecti
v
e
capacited
transportation
problem
(MOCTP)
with
mix
ed
constraint.
The
y
pro
vide
a
solution
procedure
also.
K
umar
and
Hussain
[7]
proposed
a
method
which
is
easy
to
understand
and
apply
for
finding
intuistionistic
fuzzy
optimal
solution.
The
y
used
mix
ed
intuitionistic
fuzzy
transportation
problem.
Ghadle
and
P
athade
[8]
compare
bal
anced
and
unbalanced
fuzzy
transportation
problem
by
using
he
xagonal
fuzzy
numbers
and
rob
ust
ranking
technique.
Ahmed
and
Khan
[9]
de
v
eloped
a
ne
w
algorithm
for
finding
an
initi
al
basic
feasible
solution
of
a
transportation
problem
with
fuzzy
approach.
Prabha
and
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
86
r
ISSN:
2502-4752
V
imala
[10]
used
a
strate
gy
to
solv
e
mix
ed
intuitionistic
fuzzy
transportation
problem
by
Best
Candidate
Method.
Nidhi
et
al.
[11]
used
mix
ed
constraint
in
fuzzy
en
vironment.
Ghadle
and
P
athade
[12]
studied
generalized
he
xagonal
and
octagonal
fuzzy
numbers
by
ranking
method.
Moreo
v
er
,
properties
of
the
fuzzy
transportation
ha
v
e
been
studied
by
se
v
eral
authors
[13–22].
2.
PRELIMIN
AR
Y
In
this
section,
we
collect
some
basic
definitions
that
will
be
important
to
us
in
the
sequel
[23–27].
Definition
2..1
A
fuzzy
set
is
c
har
acterized
by
a
member
ship
function
mapping
element
of
a
domain,
space
,
or
the
univer
se
of
discour
se
X
to
the
unit
interval
[0
;
1]
i:e
A
=
f
(
A
(
x
);
x
2
X
)
g
:
Her
e
A
(
x
)
:
X
!
[0
;
1]
is
a
mapping
called
the
de
gr
ee
of
member
ship
value
of
x
2
X
in
the
fuzzy
set
A
.
These
member
ship
gr
ades
ar
e
often
r
epr
esented
by
r
eal
number
s
r
anking
fr
om
[0
;
1]
:
Definition
2..2
A
fuzzy
set
~
A
is
defined
on
univer
sal
set
of
r
eal
number
s
is
said
to
be
g
ener
alized
fuzzy
number
if
its
member
ship
function
~
A
has
the
following
attrib
utes:
•
~
A
(
x
)
:
R
!
[0
;
1]
is
continuous;
•
~
A
(
x
)
=
0
for
all
x
2
(
1
;
a
]
S
[
d;
1
);
•
~
A
(
x
)
is
strictly
incr
easing
on
[
a;
b
]
and
strictly
decr
easing
on
[
c;
d
]
;
•
~
A
(
x
)
=
w
for
all
x
2
[
b;
c
]
,
wher
e
0
<
w
1
:
Definition
2..3
A
fuzzy
number
~
A
=
(
a;
b;
c;
d
)
is
said
to
be
a
tr
apezoidal
fuzzy
number
if
its
member
ship
function
is
given
by
the
following
e
xpr
ession:
~
A
(
x
)
=
8
>
>
<
>
>
:
x
a
b
a
;
a
x
b
1
;
b
x
c
d
x
d
c
;
c
x
d
0
;
other
w
ise
(1)
Definition
2..4
A
fuzzy
number
~
A
=
(
a;
b;
c;
d
)
is
said
to
be
a
trivial
tr
apezoidal
fuzzy
number
if
and
only
if
•
a=b=c=d
•
it’
s
member
ship
function
is
given
by
,
~
A
(
x
)
=
1
;
x
=
a
0
;
other
w
ise
(2)
Definition
2..5
Properties
of
T
rapezoidal
Fuzzy
Number
1.
A
tr
apezoidal
fuzzy
number
~
A
=
(
a;
b;
c;
d
)
is
said
to
be
non-ne
gative
(non-positive)
tr
apezoidal
fuzzy
number
i.e
.
~
A
0(
~
A
0)
if
and
only
if
a
0
(
c
0)
.
A
tr
apezoidal
fuzzy
number
is
said
to
be
positive
(ne
gative
)
tr
apezoidal
fuzzy
number
i.e
.
~
A
>
0
(
~
A
<
0)
if
and
only
if
a
>
0
(
c
<
0)
.
2.
T
wo
tr
apezoidal
fuzzy
number
~
A
1
=
(
a;
b;
c;
d
)
and
~
A
2
=
(
a;
b;
c;
d
)
ar
e
said
to
be
equal.
3.
A
zer
o
tr
apezoidal
fuzzy
number
is
denoted
by
~
O
=
(0
;
0
;
0
;
0)
.
4.
A
tr
apezoidal
fuzzy
number
~
A
=
(
a;
b;
c;
d
)
is
a
tr
aingular
fuzzy
number
if
b
=
c:
Definition
2..6
Let
~
A
=
(
a;
b;
c;
d
)
be
a
tr
apezoidal
fuzzy
number
.
Then
its
-cut
is
given
by
(
L
;
U
)
=
(
b
a
)
+
a;
d
(
d
c
)
,
wher
e
2
[0
;
1]
:
Definition
2..7
A
tr
apezoidal
fuzzy
number
~
A
=
(
a;
b;
c;
d
)
is
said
to
be
a
fuzzy
zer
o
if
R
(
~
A
)
=
0
:
Definition
2..8
Fuzzy
number
s
have
no
natur
al
or
der
b
ut
in
the
decision
making
pr
ocess
fuzzy
number
s
must
be
systematic.
W
ith
the
help
of
systematic
or
der
of
number
s
we
mak
e
good
decision
in
fuzzy
en
vir
onment.
Her
e
we
use
Rob
ust
r
anking
function.
If
~
A
=
(
a;
b;
c;
d
)
is
a
fuzzy
number
then
the
Rob
ust
r
anking
is
defined
by
,
R
(
~
A
)
=
R
1
0
(0
:
5)(
L
;
U
)
d
,
wher
e
(
L
;
U
)
is
the
-cut
of
the
tr
apezoidal
fuzzy
number
~
A:
R
(
~
A
)
=
a
+
b
+
c
+
d
4
F
or
any
two
fuzzy
number
s,
~
A
1
;
~
A
2
we
have
the
following
comparision,
(a)
~
A
1
<
~
A
2
if
and
only
if
R
(
~
A
1
)
<
R
(
~
A
2
)
,
(b)
~
A
1
>
~
A
2
if
and
only
if
R
(
~
A
1
)
>
R
(
~
A
2
)
,
(c)
~
A
1
=
~
A
2
if
and
only
if
R
(
~
A
1
)
=
R
(
~
A
2
)
.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
19,
No.
1,
July
2020
:
85
–
90
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
87
3.
NEW
BEST
CANDID
A
TE
METHOD
Step
1:
Must
check
the
matrix
is
balanced,
if
the
total
supply
equal
to
the
total
demand
then
the
matrix
is
balanced
and
if
the
total
supply
is
not
equal
to
the
total
demand.
So
the
transportat
ion
cost
to
this
ro
w
or
column
assigned
to
zero.
Step
2:
The
b
e
st
candidates
are
selected
by
choosing
minimum
cost
for
minimization
problem
and
maximum
cost
for
maximization
problems.
Step
3:
Assign
as
much
as
possible
to
the
cell
with
the
smallest
unit
cost
(or
highest)
in
the
entire
tableau.
If
there
is
a
tie
then
choose
arbitrarily
.
Step
4:
Check
each
ro
w
(and
column
)
has
atleast
one
best
candidate.
Step
5:
Identify
the
ro
w
with
the
smallest
cost
candidate
from
the
chosen
combination.
Then
allocate
the
demand
and
the
supply
as
much
as
possible
to
the
v
ariable
with
the
least
unit
cost
in
the
selected
ro
w
or
column.
Step
6:
Also,
we
should
adjust
the
supply
and
demand
by
crossing
out
the
ro
w/column
to
be
then
assigned
to
zero.
If
the
ro
w
or
column
not
assigned
to
zero,
then
we
check
the
sel
ected
ro
w
if
it
has
an
element
in
the
chosen
combination,
then
we
elect
it.
Step
7:
Elect
the
ne
xt
least
cost
from
the
chosen
combination
and
repeat
Step
5
until
all
columns
and
ro
ws
is
e
xhausted.
4.
NUMERICAL
EXAMPLES
4.1.
Example
1.
Consider
the
follo
wing
mix
ed
fuzzy
transportation
problem.
Here
the
a
v
ailability
of
the
problem:
Solution:
The
gi
v
en
mix
ed
fuzzy
transportation
problem
is
a
balanced
fuzzy
transportation
problem.
Here
the
cost
matrix
contains
v
e
real
numbers
and
rest
are
trapezoidal
fuzzy
numbers.
Here
the
a
v
ailability
of
the
product
a
v
ailable
at
the
three
origins
and
the
demand
of
the
product
a
v
ailable
at
the
four
destinations,
unit
cost
of
the
product
from
each
origin
to
each
destination
is
represented
by
trapezoidal
fuzzy
numbers
sho
wn
in
T
able
1.
T
able
1.
Mix
ed
constraint
balanced
fuzzy
transportation
d
1
d
2
d
3
d
4
Supply
s
1
2
:
5
(1,
3,
4,
6)
(9,
11,
12,
14)
(5,
7,
8,
11)
(1,
6,
7,
12)
s
2
1.75
0.5
(5,6,7,8)
1.5
(0,
1,
2,
3)
s
3
(3,5,6,8)
8.5
(12,15,16,19)
(7,9,10,12)
(7,10,12,17)
Demand
(5,
7,
8,
10)
(1,
5,
6,
10)
(1,
3,
4,
7)
(1,2,3,5)
W
e
used
the
definition
of
tri
vial
trapezoidal
fuzzy
numbers
t
o
con
v
ert
the
real
v
alues
as
trapez
oidal
fuzzy
numbers
sho
wn
in
T
able
2.
Ra
n
ki
ng
of
trapezoidal
fuzzy
numbers
is
important
task
to
solv
e
the
problem.
So
we
used
the
rob
ust
ranking
method
to
rank
t
he
trapezoidal
and
tri
vial
trapezoidal
fuzzy
numbers.
W
e
applied
the
rob
ust
ranking
method
and
reduced
table
is
sho
wn
in
T
able
3.
T
able
2.
T
ri
vial
trapezoidal
transportation
d
1
d
2
d
3
d
4
Supply
s
1
(2.5,2.5,2.5,2.5)
(1,3,4,6)
(9,11,12,14)
(5,7,8,11)
(1,6,7,12)
s
2
(1.75,1.75,1.75,1.75)
(0.5,0.5,0.5,0.5)
(5,6,7,8)
(1.5,1.5,1.5,1.5)
(0,1,2,3)
s
3
(3,5,6,8)
(8.5,8.5,8.5,8.5)
(12,15,16,19)
(7,9,10,12)
(7,10,12,17)
Demand
(5,7,8,10)
(1,5,6,10)
(1,3,4,7)
(1,2,3,5)
T
able
3.
Ranking
of
mix
ed
trapezoidal
fuzzy
transportation
d
1
d
2
d
3
d
4
Supply
s
1
2.5
3.5
11.5
7.75
6.5
s
2
5.5
0.5
6.5
1.5
1.5
s
3
5.5
8.5
15.5
9.5
11.5
Demand
7.5
5.5
3.75
2.75
A
systematic
appr
oac
h
for
solving
mixed...
(Priyanka
A.
P
athade)
Evaluation Warning : The document was created with Spire.PDF for Python.
88
r
ISSN:
2502-4752
After
ranking
the
numbers
we
shall
determine
the
whole
tableau
and
able
to
used
the
ne
w
best
candidate
method
sho
wn
in
T
able
4.
T
able
4.
Selection
of
Best
candidates
in
transportation
d
1
d
2
d
3
d
4
Supply
s
1
2.5
3.5
11.5
7.75
6.5
s
2
5.5
0.5
6.5
1.5
1.5
s
3
5.5
8.5
15.5
9.5
11.5
Demand
7.5
5.5
3.75
2.75
It
is
ob
vious
from
the
T
able
5.
that
the
optimal
solution
obtained
by
ne
w
best
candidate
method.
W
e
calculate
the
cost
and
get
the
result
as:
T
able
5.
Apply
ne
w
best
candidate
method
d
1
d
2
d
3
d
4
Supply
s
1
2.5
2
3
:
5
3
:
75
11
:
5
2
:
75
7
:
75
6.5
s
2
5.5
1
:
5
0
:
5
6.5
1.5
1.5
s
3
7
:
5
5
:
5
4
8
:
5
15.5
9.5
11.5
Demand
7.5
5.5
3.75
2.75
=
(11
:
5)(3
:
75)
+
(7
:
75)(2
:
75)
+
(0
:
5)(1
:
5)
+
(5
:
5)(7
:
5)
+
(8
:
5)(4)
=
140
:
4375
4.2.
Example
2.
Consider
the
follo
wing
mix
ed
fuzzy
transportat
ion
problem.
Here
the
a
v
ailability
of
the
product
a
v
ailable
at
the
three
origins
and
the
demand
of
the
product
at
three
destinations.
Unit
cost
of
the
product
from
each
origin
to
each
destination
is
represented
by
mix
ed
trapezoidal
fuzzy
numbers
sho
wn
in
T
able
6.
T
able
6.
Mix
ed
constraint
unbalanced
fuzzy
transportation
d
1
d
2
d
3
Supply
s
1
4
(2,4,5,6)
(1,5,6,7)
(1,2,3,4)
s
2
4.75
2.75
(0,1,2,3)
(1,4,5,5)
s
3
(5,6,8,9)
(1,5,6,7)
5.5
(1,3,5,7)
Demand
(1,2,4,6)
(0,1,1,2)
(3,5,7,8)
Solution:
The
gi
v
en
mix
ed
fuzzy
transportation
problem
is
a
unbalanced
fuzzy
transportation
problem.
Here
the
cost
matrix
contains
four
real
numbers
and
rest
are
trapezoidal
fuzzy
numbers.
Here
the
a
v
ailability
of
the
product
a
v
ailable
at
the
three
origins
and
the
demand
of
the
product
a
v
ailable
at
the
three
destinations,
unit
cost
of
the
product
from
each
origin
to
each
destination
is
represented
by
trapezoidal
fuzzy
numbers
sho
wn
in
T
able
6.
The
abo
v
e
problem
is
unbalanced
so
we
mak
e
it
balanced
sho
wn
in
T
able
7.
T
able
7.
T
ri
vial
trapezoidal
transportation
d
1
d
2
d
3
Supply
s
1
(4,4,4,4)
(2,4,5,6)
(1,5,6,7)
(1,2,3,4)
s
2
(4.75,4.75,4.75,4.75)
(2.75,2.75,2.75)
(0,1,2,3)
(1,4,5,5)
s
3
(5,6,8,9)
(1,5,6,7)
(5.5,5.5,5.5,5.5)
(1,3,5,7)
s
4
0
0
0
0.5
Demand
(1,2,4,6)
(0,1,1,2)
(3,5,7,8)
W
e
used
the
definition
of
tri
vial
trapezoidal
fuzzy
numbers
to
con
v
ert
the
real
v
alues
as
trapezoidal
fuzzy
numbers
sho
wn
in
abo
v
e
T
able.
W
e
used
the
rob
ust
ranking
method
to
rank
the
trapezoidal
and
tri
vial
trapezoidal
fuzzy
numbers
we
us
ed
the
rob
ust
ranking
method
which
is
also
applied
in
tri
vial
trapezoidal
fuzzy
numbers
also
and
the
reduced
table
is
sho
wn
in
T
able
8.
After
ranking
the
numbers
has
been
deter
mined
by
the
whole
tableau
and
able
to
used
the
ne
w
best
candidate
method
sho
wn
in
T
able
9,
then
apply
ne
w
best
candidate
method
T
able
10.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
19,
No.
1,
July
2020
:
85
–
90
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
89
T
able
8.
Ranking
of
mix
ed
trapezoidal
fuzzy
transportation
d
1
d
2
d
3
Supply
s
1
4
4.25
4.75
2.5
s
2
4.75
2.75
1.5
3
s
3
7
4.75
5.5
4
s
4
0
0
0
0.5
Demand
3.25
1
5.75
T
able
9.
Selection
of
best
candidates
in
transportation
d
1
d
2
d
3
Supply
s
1
4
4.25
4.75
2.5
s
2
4.75
2.75
1.5
3
s
3
7
4.75
5.5
4
s
4
0
0
0
0.5
Demand
3.25
1
5.75
T
able
10.
Apply
ne
w
best
candidate
method
d
1
d
2
d
3
Supply
s
1
2
:
5
4
4.25
4.75
2.5
s
2
4.75
1
2
:
75
2
1
:
5
3
s
3
7
0
:
75
4.75
5.5
3
:
25
4
s
4
0
0
0
0
:
5
0.5
Demand
3.25
1
5.75
W
e
calculate
the
cost
and
get
the
result
as,
=
(4)(2
:
5)
+
(2
:
75)(1)
+
(1
:
5)(2)
+
(7)(0
:
75)
+
(5
:
5)(3
:
25)
+
(0)(0
:
5)
=
38
:
875
5.
CONCLUSION
In
this
paper
ne
w
method
is
proposed
for
finding
t
he
optimal
solution
of
mix
ed
trapezoidal
fuzzy
transportation
problem.
The
balanced
and
unbalanced
mix
ed
trapezoidal
fuzzy
transportation
problem
are
discussed
and
a
numerical
e
xample
is
solv
ed
to
illustrate
the
proposed
method.
The
proposed
method
is
easy
to
apply
for
solving
the
fuzzy
transportation
problem
of
mix
ed
type.
In
real
life
situations
this
type
of
ne
w
ideas
are
necessary
to
f
ace
the
problems
because
we
can
apply
the
discussed
methods.
REFERENCES
[1]
F
.
Hitchcock,
”The
distrib
ution
of
a
product
from
se
v
eral
sources
to
numerous
localities,
”
J
ournal
of
Maths.
Phys.
(1941),
20,
pp.224-230.
[2]
L.
Zadeh,
”Fuzzy
Sets”,
Information
and
Contr
ol
,
(1965),
8,
pp.338-353.
[3]
H.
Zimmerman,
”Fuzzy
Set
Theory
and
its
Applications,
”
F
ourth
Edition,
Springer
Science
Business
Media,
LLC
Ne
w
Y
ork,
2001.
[4]
A.
Gani,
”Mix
ed
constraint
fuzzy
transshipment
problem,
”
Applied
mathematical
siences
,
(2012),
48(6),
pp.2385-2394.
[5]
R.
Mandal
,
M.
Hussain,
”Solving
the
transportation
problem
with
mix
ed
constraints,
”
International
jour
-
nal
of
Mana
g
ement
and
Business
Studies
,
(2012),
2,
pp.95-99.
[6]
N.
Gupta,
A.
Bari,
”Fuzzy
multi-objecti
v
e
capacitated
transportation
problem
with
mix
ed
constrant,
”
J
ournal
of
Statistics
Applications
and
Pr
obability
,
(2014),
3,
pp.201-209.
[7]
P
.
K
umar
,
R.
Hussain,
”A
s
ystematic
approach
for
solving
mix
ed
intuitionistic
fuzzy
transportation
prob-
lem,
”
International
J
ournal
of
Pur
e
and
Applied
Mathematics
,
(2014),
92(2),
pp.181-190.
[8]
K.
Ghadle,
P
.
P
athade,
”Optimal
solution
of
balanced
and
unbalanced
fuzzy
transportation
problem
using
he
xagonal
fuzzy
numbers,
”
International
J
ournal
of
Mathematical
Resear
c
h
,
(2016),
5(2),
pp.131-137.
[9]
N.
Ahemed,
A.
Khan,
”Solution
of
mix
ed
type
transporta
tion
problem:
A
fuzzy
Approach,
”
A
utomatica
Di
Calculator
e
,
(2015),
11,
pp.20-31.
[10]
S.
Prabha,
S.
V
imala,
”A
strate
gy
to
solv
e
mix
ed
intuitionistic
fuzzy
transportation
problem
by
BCM,
”
Middle
East
J
ournal
of
Scientific
Reser
ac
h
,
(2017),
25,
pp.374-379.
[11]
N.
Joshi,
S.
Chauhan,
”A
Ne
w
Approach
to
solv
e
mix
ed
constrant
transportation
problem
under
fuzzy
en
viornment,
”
International
J
ournal
of
Computer
s
and
T
ec
hnolo
gy
,
(2017),
16,
pp.6895-6902.
[12]
K.
Ghadle,
P
.
P
athade,
”Solving
transportation
problem
with
generalized
he
xagonal
and
generalized
oc-
tagonal
fuzzy
numbers
by
ranking
method,
”
Global
J
ournal
of
Pur
e
an
d
Applied
Mathematics
,
(2017),
13,
pp.6367-6376.
A
systematic
appr
oac
h
for
solving
mixed...
(Priyanka
A.
P
athade)
Evaluation Warning : The document was created with Spire.PDF for Python.
90
r
ISSN:
2502-4752
[13]
P
.
P
athade,
K.
Ghadle,
A.
Hamoud,
”Optimal
solution
solv
ed
by
t
riangular
intuitionistic
fuzzy
transporta-
tion
problem,
”
Adv
ances
in
Intelligent
Systems
and
Computing
,
(2010),
1025,
pp.379-385.
[14]
K.
Hussain,
A.
Hamoud,
N.
Mohammed,
”Some
ne
w
uniqueness
results
for
fractional
inte
gro-dif
ferential
equations,
”
Nonlinear
Functional
Analysis
and
Applications
,
(2019),
24(4),
pp.827-836.
[15]
A.
Hamoud,
N.
Mohammed,
K.
Ghadle,
”A
study
of
some
ef
fecti
v
e
techniques
for
solving
V
olterra-
Fredholm
inte
gral
equations,
”
Dynamics
of
Continuous,
Di
screte
and
Impulsi
v
e
Systems
Series
A:
Math-
ematical
Analysis
,
(2019),
26,
pp.389-406.
[16]
A.
Hamoud,
K.
Ghadle,
P
.
P
athade,
”An
e
xistence
and
con
v
er
gence
results
for
Caputo
fractional
V
olterra
inte
gro-dif
ferential
equations,
”
Jordan
Journal
of
Mathematics
and
Statistics
,
(2019),
12(3),
pp.307-327.
[17]
A.
Hamoud,
K.
Ghadle,
”Some
ne
w
e
xistence,
uniqueness
and
con
v
er
gence
results
for
fractional
V
olterra-
Fredholm
inte
gro-dif
ferential
equations,
”
J.
Appl.
Comput.
Mech.
,
(2019),
5(1),
pp.58-69.
[18]
A.
Hamoud,
K.
Ghadle,
”Existence
and
uniqueness
of
the
solution
for
V
olterra-Fredholm
inte
gro-
dif
ferential
equations,
”
Journal
of
Siberian
Federal
Uni
v
ersity
.
Mathematics
&
Ph
ysics
,
(2018),
11(6),
pp.692-701.
[19]
A.
Hamoud,
K.
Ghadle,
S.
Atshan,
”The
approximate
solut
ions
of
fractional
inte
gro-dif
ferential
equations
by
using
modified
Adomian
decomposition
method,
”
Khayyam
J.
Math.
(2019),
5(1),
pp.21-39.
[20]
N.
Mohammed,
L.
Sultan,
S.
Lomte,
”Pri
v
ac
y
preserving
outsourcing
algorithm
for
tw
o-point
linear
boundary
v
alue
problems,
”
Indonesian
Journal
of
Electrical
Engineering
and
Computer
Science
,
(2019).
[21]
N.
Mohammed,
S.
Lomte,
”Secure
and
ef
ficient
outsourcing
of
lar
ge
scale
linear
fractional
programming,
”
Adv
ances
in
Intelligent
Systems
and
Computing,
Springer
Nature
Sing
apore
,
1025
(2020).
[22]
N.
Mohammed,
L.
Sultan,
A.
Hamoud,
S.
Lomte,
”V
erifiable
secure
computation
of
linear
fractional
programming
using
certificate
v
alidation,
”
International
Journal
of
Po
wer
Electronics
and
Dri
v
e
Systems
,
(2020),
11(1),
pp.284-290.
[23]
A.
Hamoud,
A.
Azeez,
K.
Ghadle,
”A
study
of
some
iterati
v
e
methods
for
solving
fuzzy
V
olterra-
Fredholm
inte
gral
equations,
”
Indonesian
J.
Elec.
Eng.
&
Comp.
Sci.
,
(2018),
11(3),
pp.1228-1235.
[24]
A.
Hamoud,
K.
Ghadle,
”Homotop
y
analysis
method
for
the
first
order
fuzzy
V
olterra-Fredholm
inte
gro-
dif
ferential
equations,
”
Indonesian
J.
Elec.
Eng.
&
Comp.
Sci.
,
(2018),
11(3),
pp.857-867.
[25]
A.
Ha
moud,
K.
Ghadle,
”Modified
Adomian
decomposition
method
for
solving
f
uzzy
V
olterra-Fredholm
inte
gral
equations,
”
J.
Indian
Math.
Soc.
,
(2018),
85(1-2),
pp.52-69.
[26]
A.
Hamoud,
K.
Ghadle,
”Usage
of
the
homotop
y
analysis
method
for
solving
fractional
V
olterra-Fredholm
inte
gro-dif
ferential
equation
of
the
second
kind,
”
T
amkang
J.
Math.
(2018),
49(4),
pp.301-315.
[27]
K.
Ghadle,
P
.
P
athade,
A.
Hamoud,
”An
impro
v
ement
to
one’
s
BCM
for
the
balanced
and
unbalanced
transshipment
problems
by
using
fuzzy
numbers,
”
T
rends
in
Mathematics
,
(2018),
pp.271-279.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
19,
No.
1,
July
2020
:
85
–
90
Evaluation Warning : The document was created with Spire.PDF for Python.