Download complete journal in PDF form - Academy Publish

ON BICRITERIA LARGE SCALE TRANSSHIPMENT PROBLEMS
Dr. Jasem M.S. Alrajhi, Dr. Hilal A. Abdelwali, Dr. Mohsen S. Al-Ardhi, Eng. Rafik El Shiaty
Z
n
n
d ij
x ij
2
i 1 j 1
With c ij = 0 for the quantity shipped from the source (S i ) to itself and
from destination (D j ) to itself.
Subject to:
n
i1
i
j
n
i1
i
j
x
x
ji
ij
x
x
x ij ≥ 0 for all i, j.
ij
ji
a
b
j
j
, j 1,2,...,
n.
, j 1,2,...,
n.
Bicriteria Multistage Transportation Problem with Transshipment
of the Second Kind (BMTSP 2):
This case represents bicriteria multistage transshipment problems in
which the transportation at any stage is independent of the
transportation at the other stages.
In order to obtain the mathematical formulation of the problem
representing this case let us assume that for k th stage, k= 1, 2, 3, …., N;
k
the availabilities are: ( a
jk
), jk
1,2,3 ,... nk
, n k is the number of
(sources + destinations) at the k th stage; the requirements are:
k
( b
jk
), jk
1,2,3,...
nk
; the transportation costs and deteriorations
are c k i j , d k
i j i = 1, 2, 3, ., n ; j = 1, 2, 3, ., n ;
k k k k
k k
k
quantity shipped from i k
k
to j k ; and
k
transshipped through point j k , x
j k j
0 .
k
Then the problem takes the form:
Min.
Z
Z
k
2
k
1
n
n
n
k k
ik
1 jk
1
n
k k
ik
1 jk
1
d
c
k
ik
jk
k
ik
jk
x
x
k
ik
jk
k
ik
jk
x
k
j k j k
x
k
i j denotes the
k k
is the net amount
k
With ci
j
0 for the quantity shipped from the source (S i ) to itself
k k
and from the destination (D j ) to itself.
Subject to:
k
ik
1
i j
k
n
k
k
ik
1
i j
k
n
k
x
x
k
j i
k k
k
i j
k k
x
x
k
j j
k k
k
j j
k k
a
b
k
j
k
k
j
k
, j
, j
k
k
1,2,...,
n.
1,2,...,
n.
x ij ≥ 0 for all i k , j k.
and the minimum transportation cost is given by:
MinZ
n
k1
MinZ
k
Bicriteria Multistage Transportation Problem with Transshipment
of the Third Kind (BMTSP 3):
This case represents bicriteria multistage transshipment problems with
some additional transportation restrictions on the intermediate stages
which does not affect the transshipment problem formulation at each
stage.
The mathematical formulation of the problem representing this case is
given as:
Min.
Z
...
n
N
1
n
N
iN
1 jN
1
Z
2
n1
n1
i1
1 j1
1
c i
x
N
N jN
n1
n1
i1
1 j1
1
...
n
N
n
k k
1 1
k k
ci
...
...
1 j
x
1 i1
j
c
1
ik
j
x
k ik
jk
N
iN
jN
n
ik
1 jk
1
n
k k
1 1
k k
di
...
...
1 j
x
1 i1
j
d
1
ik
j
x
k ik
jk
n
N
iN
1 jN
1
d i
x
N
N jN
N
iN
jN
n
ik
1 jk
1
k
with ci
k j and d k
k ik
j = 0 for the quantity shipped from the source ( k
S
k
i )
k
k
to itself and from the destination ( D
j ) to itself: k = 1, 2, …, N subject
k
to:
n1
1 1 1
x
j
,
1
1,2,...,
1i
x a j n
1 j1
j
1 j
1
1
i1
1
i j
1
n
1
1
i1
j
i1
1
.
.
.
n
i
i
1
k
k
k
j
1
n
k
k
ik
j
ik
1
.
.
k
x
1 1 1
i
,
1
1,2,...,
1 j
x b j n
1 j1
j
1 j
1
1
x
x
k k k
j j
x
j j
a
j
, j
k k k k k k
1,2,...,
k k k
i j
x
j j
bj
, j
k k k k k k
1,2,...,
n
n
k
k
AcademyPublish.org – Journal of Engineering and Technology Vol.2, No.2 22